Electron Density Functional Theory Page 1

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(I) Introductory background topics

a) The Schroumldinger equation

XXX

b) Bosons and Fermions

i Ground-state wave function of Bosons

Consider a one dimensional Schroumldinger equation for the ground state which

may be taken real We will ask is this wave function of the same sign every-

where What we mean is is it all non-negative or all non positive Of course

if it is all non-positive then we can multiply it by -1 and ask again is it all non-

negative Saying it is of a constant sign is the same as saying it has no nodes

ie it does not go through the x- axis anywhere

Exercise Prove the following theorem

Theorem The ground-state wave function of a 1D wave non-degenerate

Schroumldinger equation has no nodes

Proof Consider the functional

119864[120595] = 119879[120595] + 119881[120595] =ℏ

2119898int 120595 (119909) 119889119909

+ int 120595(119909) 119881(119909)119889119909

(Ib1)

The ground-state wave function is the minimize of this lrm(Ib1)functional over

all normalized real wave functions Note that 119879[120595] = 119879[|120595|] and 119881[120595] =

119881[|120595|]

Exercise (the kinetic energy functional appearing in Eq (Ib1)) Let 119879 [120595] =

minusℏ

int 120595(119909) 120595 (119909)119889119909

(the usual definition of kinetic energy) and 119879 [120595] =

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ℏ

int |120595 (119909)| 119889119909

Show that

1) 119879 [120595] = 119879 [120595] for any smooth 120595 that decays to zero as |119909| rarr infin

2) 119879 [120595 ] = minus119879 [120595 ] for 120595 (119909) = radic120572119890 | | Which of the two values is

physically unacceptable as kinetic energy

3) What do you think will happen when the two definitions of kinetic en-

ergies are applied to the family of functions 120595 (119909) = radic120572119890 radic Us-

ing numerical integration determine what is the limit lim rarr 119879 [120595 ] Is

it 119879 [120595 ] or 119879 [120595 ]

Based on your findings explain why 119879 is the more robust way to define

kinetic energy in the presence of discontinuities

We now show that if 120595(119909) has a node then one can build out of it function

having lower energy From this we will be able to deduce that the minimizing

wave function cannot have nodes For simplicity assume the node of 120595(119909) ap-

pears at the origin 119909 = 0

Figure lrmI-1 The absolute value of the wave function and the parabola in it

X1 X2

g(x)

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We will augment slightly the function 120595 by some 120575120595 and show that this

leads to reduction in energy Note that the energy change in an arbitrary per-

turbation is

120575119864[|120595|] = 120575119879[|120595|] + 2int |120595(119909)|(119881(119909) minus 119864[|120595|])|120575120595(119909)|119889119909

Exercise Verify this and show we explicitly assumed that the norm of 120595 is 1

Our augmentation smoothes the node as shown in Figure lrmI-1 determining a

large enough parameter 119886 such that the parabola

119901(119909) =1

2119886(119909 minus 119909

) + 119888

has the properties that it is tangent at some 119909 and 119909 to |120595(119909)| (where

119909 lt 0 lt 119909 ) The new wave function is nodeless

120594(119909) = |120595(119909)| 119909 lt 119909 119900119903 119909 gt 119909

119901(119909) 119909 lt 119909 lt 119909

Note that by decreasing 119886 the parabola becomes narrower and the following

limits are obtained

minus119909 rarr 119909 rarr |120595 (0)|119886 rarr 0

119888 rarr1

2 |120595 (0)| 119886 rarr 0

Note that because the function has a node at 119909 = 0 the potential energy

changes only to second order with 119886 119881[120594] minus 119881[120595] prop 119886 The kinetic energy on

the other hand has a linear dependence 119879[120594] minus 119879[120595] asymp minusℏ

|120595 (0)| 119886 But

note that it is negative Thus as 119886 rarr 0 the total change in energy is dominated

by the change in the kinetic energy and is negative This shows that 119864[120594] lt

119864[|120595|] = 119864[120595] Hence the ground state cannot have nodes otherwise one can

remove them lowering the energy further

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This proof can be extended to any number of dimensions It can be used to

prove that the ground state of a many-boson wave function has no nodes

Exercise Prove the corollary the ground-state of boson systems is non-

degenerate

ii Ground-state wave function of Fermions

The ground-state wave function of 119873 gt 1 Fermions is a much more compli-

cated object than that of 119873 gt 1 Bosons This is because of the Pauli principle

which states 120595(119909 119909 ) = minus120595(119909 119909 ) ie 120595 is bound to have nodes in general

Thus the Fermionic ground-state is actually a highly excited eigenstate of the

Hamiltonian It can still be described by a minimum principle namely the

normalized many-body antisymmetric wave-function which minimizes

119864[120595] = ⟨120595||120595⟩ This formulation of the ground state problem is very conven-

ient We do not have to look for many eigenstates of we need only search for

minimum Finding the minimal energy and the expectation values of various

physically important operators is the main challenge of electronic structure

theory Doing the same for low-lying excited states is the grand challenge of

this field

c) Why electronic structure is an important dif-

ficult problem

In this course we will study methods to treat electronic structure This prob-

lem is often considerably more complicated than nuclear dynamics of mole-

cules The most obvious reason is that usually there are many more electrons

than nuclei in a molecule However there are more reasons electrons in mol-

ecules have extreme quantum mechanical character while nuclei are more

classical This last statement requires probably some explanation We will

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discuss in this course ways of mapping the electrons in a molecule onto a sys-

tem of non-interacting particles In such a picture each electron has its own

orbital or wave-function Generally the orbitals of different electrons in mole-

cules strongly overlap Furthermore most of the valence orbitals are widely

spread over space encapsulating many molecular nuclei Nuclei on the other

hand are massive so their wave functions are strongly localized and hardly

overlap once again they behave much more like classical particles

Describing electronic structure is therefore a quantum mechanical problem

Now the electrons interact via the attractive Coulomb force not only with

the stationary nuclei but also with each other each electron repels each other

electron via the repulsive Coulomb force This makes the electronic structure

problem a many-body problem

Quantum many-body problems are very difficult to track down much more

difficult than classical mechanical ones The reason is that for 119873 electrons the

many-body wave function is then a function of 3119873 variables 120595(119955 hellip 119955 )

where ir (i=1hellip eN ) is 3 dimensional the position vector Highly accurate nu-

merical representation of such functions is next to impossible for 119873 gt 1

Thus electronic structure methods invariably include approximate methods

Typically the wave function encapsulates much more information than we

care to know about For example int|120595(119955 hellip 119955 )| 119889 119903 sdotsdotsdot 119889 119903 gives the proba-

bility for finding an electron at 119955 and an electron at 119955 and an electron at 119955

etc Indeed the total probability is 1

int|120595(119955 hellip 119955 )|

119889 119903 hellip119889 119903

= 1 (Ic1)

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d) The Born-Oppenheimer theory

iii The adiabatic theorem

Suppose a Hamiltonian is dependent parametrically on 119877 = (119877 119877 hellip ) We

write this as [119877] Denote the eigenstates of this Hamiltonian by

[119877]120595 [119877] = 119864 [119877]120595 [119877]

Now suppose we change 119877 with time so that we have a trajectory 119877(119905) The

Hamiltonian will become time dependent (119905) = [119877(119905)] Suppose the sys-

tem is placed in its ground state at time = 0 120601(119905 = 0) = 120595 [119877(0)] It will

evolve according to the TDSE

119894ℏ(119905) = (119905)120601(119905)

Now the adiabatic theorem says that if 119877(119905) changes very slowly then

120601(119905) = 119890 ( )120595 [119877(119905)]

Namely except for a purely TD phase the evolving state stays the instanta-

neous ground state of the Hamiltonian (119905)

To show this we use the instantaneous eigenstates of the Hamiltonian (119905) to

expand 120601(119905)

120601(119905) = sum119886 (119905)119890

ℏint ( ) 120595 [119877(119905)]

We want to plug this expansion into the SE So let us calculate

119894ℏ(119905) = sum[119894ℏ + 119864 119886 ]119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

And

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(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

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over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

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sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

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nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

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intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

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ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

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formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 2

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ℏ

int |120595 (119909)| 119889119909

Show that

1) 119879 [120595] = 119879 [120595] for any smooth 120595 that decays to zero as |119909| rarr infin

2) 119879 [120595 ] = minus119879 [120595 ] for 120595 (119909) = radic120572119890 | | Which of the two values is

physically unacceptable as kinetic energy

3) What do you think will happen when the two definitions of kinetic en-

ergies are applied to the family of functions 120595 (119909) = radic120572119890 radic Us-

ing numerical integration determine what is the limit lim rarr 119879 [120595 ] Is

it 119879 [120595 ] or 119879 [120595 ]

Based on your findings explain why 119879 is the more robust way to define

kinetic energy in the presence of discontinuities

We now show that if 120595(119909) has a node then one can build out of it function

having lower energy From this we will be able to deduce that the minimizing

wave function cannot have nodes For simplicity assume the node of 120595(119909) ap-

pears at the origin 119909 = 0

Figure lrmI-1 The absolute value of the wave function and the parabola in it

X1 X2

g(x)

Electron Density Functional Theory Page 3

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We will augment slightly the function 120595 by some 120575120595 and show that this

leads to reduction in energy Note that the energy change in an arbitrary per-

turbation is

120575119864[|120595|] = 120575119879[|120595|] + 2int |120595(119909)|(119881(119909) minus 119864[|120595|])|120575120595(119909)|119889119909

Exercise Verify this and show we explicitly assumed that the norm of 120595 is 1

Our augmentation smoothes the node as shown in Figure lrmI-1 determining a

large enough parameter 119886 such that the parabola

119901(119909) =1

2119886(119909 minus 119909

) + 119888

has the properties that it is tangent at some 119909 and 119909 to |120595(119909)| (where

119909 lt 0 lt 119909 ) The new wave function is nodeless

120594(119909) = |120595(119909)| 119909 lt 119909 119900119903 119909 gt 119909

119901(119909) 119909 lt 119909 lt 119909

Note that by decreasing 119886 the parabola becomes narrower and the following

limits are obtained

minus119909 rarr 119909 rarr |120595 (0)|119886 rarr 0

119888 rarr1

2 |120595 (0)| 119886 rarr 0

Note that because the function has a node at 119909 = 0 the potential energy

changes only to second order with 119886 119881[120594] minus 119881[120595] prop 119886 The kinetic energy on

the other hand has a linear dependence 119879[120594] minus 119879[120595] asymp minusℏ

|120595 (0)| 119886 But

note that it is negative Thus as 119886 rarr 0 the total change in energy is dominated

by the change in the kinetic energy and is negative This shows that 119864[120594] lt

119864[|120595|] = 119864[120595] Hence the ground state cannot have nodes otherwise one can

remove them lowering the energy further

Electron Density Functional Theory Page 4

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This proof can be extended to any number of dimensions It can be used to

prove that the ground state of a many-boson wave function has no nodes

Exercise Prove the corollary the ground-state of boson systems is non-

degenerate

ii Ground-state wave function of Fermions

The ground-state wave function of 119873 gt 1 Fermions is a much more compli-

cated object than that of 119873 gt 1 Bosons This is because of the Pauli principle

which states 120595(119909 119909 ) = minus120595(119909 119909 ) ie 120595 is bound to have nodes in general

Thus the Fermionic ground-state is actually a highly excited eigenstate of the

Hamiltonian It can still be described by a minimum principle namely the

normalized many-body antisymmetric wave-function which minimizes

119864[120595] = ⟨120595||120595⟩ This formulation of the ground state problem is very conven-

ient We do not have to look for many eigenstates of we need only search for

minimum Finding the minimal energy and the expectation values of various

physically important operators is the main challenge of electronic structure

theory Doing the same for low-lying excited states is the grand challenge of

this field

c) Why electronic structure is an important dif-

ficult problem

In this course we will study methods to treat electronic structure This prob-

lem is often considerably more complicated than nuclear dynamics of mole-

cules The most obvious reason is that usually there are many more electrons

than nuclei in a molecule However there are more reasons electrons in mol-

ecules have extreme quantum mechanical character while nuclei are more

classical This last statement requires probably some explanation We will

Electron Density Functional Theory Page 5

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discuss in this course ways of mapping the electrons in a molecule onto a sys-

tem of non-interacting particles In such a picture each electron has its own

orbital or wave-function Generally the orbitals of different electrons in mole-

cules strongly overlap Furthermore most of the valence orbitals are widely

spread over space encapsulating many molecular nuclei Nuclei on the other

hand are massive so their wave functions are strongly localized and hardly

overlap once again they behave much more like classical particles

Describing electronic structure is therefore a quantum mechanical problem

Now the electrons interact via the attractive Coulomb force not only with

the stationary nuclei but also with each other each electron repels each other

electron via the repulsive Coulomb force This makes the electronic structure

problem a many-body problem

Quantum many-body problems are very difficult to track down much more

difficult than classical mechanical ones The reason is that for 119873 electrons the

many-body wave function is then a function of 3119873 variables 120595(119955 hellip 119955 )

where ir (i=1hellip eN ) is 3 dimensional the position vector Highly accurate nu-

merical representation of such functions is next to impossible for 119873 gt 1

Thus electronic structure methods invariably include approximate methods

Typically the wave function encapsulates much more information than we

care to know about For example int|120595(119955 hellip 119955 )| 119889 119903 sdotsdotsdot 119889 119903 gives the proba-

bility for finding an electron at 119955 and an electron at 119955 and an electron at 119955

etc Indeed the total probability is 1

int|120595(119955 hellip 119955 )|

119889 119903 hellip119889 119903

= 1 (Ic1)

Electron Density Functional Theory Page 6

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d) The Born-Oppenheimer theory

iii The adiabatic theorem

Suppose a Hamiltonian is dependent parametrically on 119877 = (119877 119877 hellip ) We

write this as [119877] Denote the eigenstates of this Hamiltonian by

[119877]120595 [119877] = 119864 [119877]120595 [119877]

Now suppose we change 119877 with time so that we have a trajectory 119877(119905) The

Hamiltonian will become time dependent (119905) = [119877(119905)] Suppose the sys-

tem is placed in its ground state at time = 0 120601(119905 = 0) = 120595 [119877(0)] It will

evolve according to the TDSE

119894ℏ(119905) = (119905)120601(119905)

Now the adiabatic theorem says that if 119877(119905) changes very slowly then

120601(119905) = 119890 ( )120595 [119877(119905)]

Namely except for a purely TD phase the evolving state stays the instanta-

neous ground state of the Hamiltonian (119905)

To show this we use the instantaneous eigenstates of the Hamiltonian (119905) to

expand 120601(119905)

120601(119905) = sum119886 (119905)119890

ℏint ( ) 120595 [119877(119905)]

We want to plug this expansion into the SE So let us calculate

119894ℏ(119905) = sum[119894ℏ + 119864 119886 ]119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

And

Electron Density Functional Theory Page 7

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(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

Electron Density Functional Theory Page 8

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

copy Roi Baer

feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

copy Roi Baer

We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

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119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

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of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

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120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 3

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We will augment slightly the function 120595 by some 120575120595 and show that this

leads to reduction in energy Note that the energy change in an arbitrary per-

turbation is

120575119864[|120595|] = 120575119879[|120595|] + 2int |120595(119909)|(119881(119909) minus 119864[|120595|])|120575120595(119909)|119889119909

Exercise Verify this and show we explicitly assumed that the norm of 120595 is 1

Our augmentation smoothes the node as shown in Figure lrmI-1 determining a

large enough parameter 119886 such that the parabola

119901(119909) =1

2119886(119909 minus 119909

) + 119888

has the properties that it is tangent at some 119909 and 119909 to |120595(119909)| (where

119909 lt 0 lt 119909 ) The new wave function is nodeless

120594(119909) = |120595(119909)| 119909 lt 119909 119900119903 119909 gt 119909

119901(119909) 119909 lt 119909 lt 119909

Note that by decreasing 119886 the parabola becomes narrower and the following

limits are obtained

minus119909 rarr 119909 rarr |120595 (0)|119886 rarr 0

119888 rarr1

2 |120595 (0)| 119886 rarr 0

Note that because the function has a node at 119909 = 0 the potential energy

changes only to second order with 119886 119881[120594] minus 119881[120595] prop 119886 The kinetic energy on

the other hand has a linear dependence 119879[120594] minus 119879[120595] asymp minusℏ

|120595 (0)| 119886 But

note that it is negative Thus as 119886 rarr 0 the total change in energy is dominated

by the change in the kinetic energy and is negative This shows that 119864[120594] lt

119864[|120595|] = 119864[120595] Hence the ground state cannot have nodes otherwise one can

remove them lowering the energy further

Electron Density Functional Theory Page 4

copy Roi Baer

This proof can be extended to any number of dimensions It can be used to

prove that the ground state of a many-boson wave function has no nodes

Exercise Prove the corollary the ground-state of boson systems is non-

degenerate

ii Ground-state wave function of Fermions

The ground-state wave function of 119873 gt 1 Fermions is a much more compli-

cated object than that of 119873 gt 1 Bosons This is because of the Pauli principle

which states 120595(119909 119909 ) = minus120595(119909 119909 ) ie 120595 is bound to have nodes in general

Thus the Fermionic ground-state is actually a highly excited eigenstate of the

Hamiltonian It can still be described by a minimum principle namely the

normalized many-body antisymmetric wave-function which minimizes

119864[120595] = ⟨120595||120595⟩ This formulation of the ground state problem is very conven-

ient We do not have to look for many eigenstates of we need only search for

minimum Finding the minimal energy and the expectation values of various

physically important operators is the main challenge of electronic structure

theory Doing the same for low-lying excited states is the grand challenge of

this field

c) Why electronic structure is an important dif-

ficult problem

In this course we will study methods to treat electronic structure This prob-

lem is often considerably more complicated than nuclear dynamics of mole-

cules The most obvious reason is that usually there are many more electrons

than nuclei in a molecule However there are more reasons electrons in mol-

ecules have extreme quantum mechanical character while nuclei are more

classical This last statement requires probably some explanation We will

Electron Density Functional Theory Page 5

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discuss in this course ways of mapping the electrons in a molecule onto a sys-

tem of non-interacting particles In such a picture each electron has its own

orbital or wave-function Generally the orbitals of different electrons in mole-

cules strongly overlap Furthermore most of the valence orbitals are widely

spread over space encapsulating many molecular nuclei Nuclei on the other

hand are massive so their wave functions are strongly localized and hardly

overlap once again they behave much more like classical particles

Describing electronic structure is therefore a quantum mechanical problem

Now the electrons interact via the attractive Coulomb force not only with

the stationary nuclei but also with each other each electron repels each other

electron via the repulsive Coulomb force This makes the electronic structure

problem a many-body problem

Quantum many-body problems are very difficult to track down much more

difficult than classical mechanical ones The reason is that for 119873 electrons the

many-body wave function is then a function of 3119873 variables 120595(119955 hellip 119955 )

where ir (i=1hellip eN ) is 3 dimensional the position vector Highly accurate nu-

merical representation of such functions is next to impossible for 119873 gt 1

Thus electronic structure methods invariably include approximate methods

Typically the wave function encapsulates much more information than we

care to know about For example int|120595(119955 hellip 119955 )| 119889 119903 sdotsdotsdot 119889 119903 gives the proba-

bility for finding an electron at 119955 and an electron at 119955 and an electron at 119955

etc Indeed the total probability is 1

int|120595(119955 hellip 119955 )|

119889 119903 hellip119889 119903

= 1 (Ic1)

Electron Density Functional Theory Page 6

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d) The Born-Oppenheimer theory

iii The adiabatic theorem

Suppose a Hamiltonian is dependent parametrically on 119877 = (119877 119877 hellip ) We

write this as [119877] Denote the eigenstates of this Hamiltonian by

[119877]120595 [119877] = 119864 [119877]120595 [119877]

Now suppose we change 119877 with time so that we have a trajectory 119877(119905) The

Hamiltonian will become time dependent (119905) = [119877(119905)] Suppose the sys-

tem is placed in its ground state at time = 0 120601(119905 = 0) = 120595 [119877(0)] It will

evolve according to the TDSE

119894ℏ(119905) = (119905)120601(119905)

Now the adiabatic theorem says that if 119877(119905) changes very slowly then

120601(119905) = 119890 ( )120595 [119877(119905)]

Namely except for a purely TD phase the evolving state stays the instanta-

neous ground state of the Hamiltonian (119905)

To show this we use the instantaneous eigenstates of the Hamiltonian (119905) to

expand 120601(119905)

120601(119905) = sum119886 (119905)119890

ℏint ( ) 120595 [119877(119905)]

We want to plug this expansion into the SE So let us calculate

119894ℏ(119905) = sum[119894ℏ + 119864 119886 ]119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

And

Electron Density Functional Theory Page 7

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(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

Electron Density Functional Theory Page 8

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 4

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This proof can be extended to any number of dimensions It can be used to

prove that the ground state of a many-boson wave function has no nodes

Exercise Prove the corollary the ground-state of boson systems is non-

degenerate

ii Ground-state wave function of Fermions

The ground-state wave function of 119873 gt 1 Fermions is a much more compli-

cated object than that of 119873 gt 1 Bosons This is because of the Pauli principle

which states 120595(119909 119909 ) = minus120595(119909 119909 ) ie 120595 is bound to have nodes in general

Thus the Fermionic ground-state is actually a highly excited eigenstate of the

Hamiltonian It can still be described by a minimum principle namely the

normalized many-body antisymmetric wave-function which minimizes

119864[120595] = ⟨120595||120595⟩ This formulation of the ground state problem is very conven-

ient We do not have to look for many eigenstates of we need only search for

minimum Finding the minimal energy and the expectation values of various

physically important operators is the main challenge of electronic structure

theory Doing the same for low-lying excited states is the grand challenge of

this field

c) Why electronic structure is an important dif-

ficult problem

In this course we will study methods to treat electronic structure This prob-

lem is often considerably more complicated than nuclear dynamics of mole-

cules The most obvious reason is that usually there are many more electrons

than nuclei in a molecule However there are more reasons electrons in mol-

ecules have extreme quantum mechanical character while nuclei are more

classical This last statement requires probably some explanation We will

Electron Density Functional Theory Page 5

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discuss in this course ways of mapping the electrons in a molecule onto a sys-

tem of non-interacting particles In such a picture each electron has its own

orbital or wave-function Generally the orbitals of different electrons in mole-

cules strongly overlap Furthermore most of the valence orbitals are widely

spread over space encapsulating many molecular nuclei Nuclei on the other

hand are massive so their wave functions are strongly localized and hardly

overlap once again they behave much more like classical particles

Describing electronic structure is therefore a quantum mechanical problem

Now the electrons interact via the attractive Coulomb force not only with

the stationary nuclei but also with each other each electron repels each other

electron via the repulsive Coulomb force This makes the electronic structure

problem a many-body problem

Quantum many-body problems are very difficult to track down much more

difficult than classical mechanical ones The reason is that for 119873 electrons the

many-body wave function is then a function of 3119873 variables 120595(119955 hellip 119955 )

where ir (i=1hellip eN ) is 3 dimensional the position vector Highly accurate nu-

merical representation of such functions is next to impossible for 119873 gt 1

Thus electronic structure methods invariably include approximate methods

Typically the wave function encapsulates much more information than we

care to know about For example int|120595(119955 hellip 119955 )| 119889 119903 sdotsdotsdot 119889 119903 gives the proba-

bility for finding an electron at 119955 and an electron at 119955 and an electron at 119955

etc Indeed the total probability is 1

int|120595(119955 hellip 119955 )|

119889 119903 hellip119889 119903

= 1 (Ic1)

Electron Density Functional Theory Page 6

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d) The Born-Oppenheimer theory

iii The adiabatic theorem

Suppose a Hamiltonian is dependent parametrically on 119877 = (119877 119877 hellip ) We

write this as [119877] Denote the eigenstates of this Hamiltonian by

[119877]120595 [119877] = 119864 [119877]120595 [119877]

Now suppose we change 119877 with time so that we have a trajectory 119877(119905) The

Hamiltonian will become time dependent (119905) = [119877(119905)] Suppose the sys-

tem is placed in its ground state at time = 0 120601(119905 = 0) = 120595 [119877(0)] It will

evolve according to the TDSE

119894ℏ(119905) = (119905)120601(119905)

Now the adiabatic theorem says that if 119877(119905) changes very slowly then

120601(119905) = 119890 ( )120595 [119877(119905)]

Namely except for a purely TD phase the evolving state stays the instanta-

neous ground state of the Hamiltonian (119905)

To show this we use the instantaneous eigenstates of the Hamiltonian (119905) to

expand 120601(119905)

120601(119905) = sum119886 (119905)119890

ℏint ( ) 120595 [119877(119905)]

We want to plug this expansion into the SE So let us calculate

119894ℏ(119905) = sum[119894ℏ + 119864 119886 ]119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

And

Electron Density Functional Theory Page 7

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(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

Electron Density Functional Theory Page 8

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

copy Roi Baer

We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

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moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 5

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discuss in this course ways of mapping the electrons in a molecule onto a sys-

tem of non-interacting particles In such a picture each electron has its own

orbital or wave-function Generally the orbitals of different electrons in mole-

cules strongly overlap Furthermore most of the valence orbitals are widely

spread over space encapsulating many molecular nuclei Nuclei on the other

hand are massive so their wave functions are strongly localized and hardly

overlap once again they behave much more like classical particles

Describing electronic structure is therefore a quantum mechanical problem

Now the electrons interact via the attractive Coulomb force not only with

the stationary nuclei but also with each other each electron repels each other

electron via the repulsive Coulomb force This makes the electronic structure

problem a many-body problem

Quantum many-body problems are very difficult to track down much more

difficult than classical mechanical ones The reason is that for 119873 electrons the

many-body wave function is then a function of 3119873 variables 120595(119955 hellip 119955 )

where ir (i=1hellip eN ) is 3 dimensional the position vector Highly accurate nu-

merical representation of such functions is next to impossible for 119873 gt 1

Thus electronic structure methods invariably include approximate methods

Typically the wave function encapsulates much more information than we

care to know about For example int|120595(119955 hellip 119955 )| 119889 119903 sdotsdotsdot 119889 119903 gives the proba-

bility for finding an electron at 119955 and an electron at 119955 and an electron at 119955

etc Indeed the total probability is 1

int|120595(119955 hellip 119955 )|

119889 119903 hellip119889 119903

= 1 (Ic1)

Electron Density Functional Theory Page 6

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d) The Born-Oppenheimer theory

iii The adiabatic theorem

Suppose a Hamiltonian is dependent parametrically on 119877 = (119877 119877 hellip ) We

write this as [119877] Denote the eigenstates of this Hamiltonian by

[119877]120595 [119877] = 119864 [119877]120595 [119877]

Now suppose we change 119877 with time so that we have a trajectory 119877(119905) The

Hamiltonian will become time dependent (119905) = [119877(119905)] Suppose the sys-

tem is placed in its ground state at time = 0 120601(119905 = 0) = 120595 [119877(0)] It will

evolve according to the TDSE

119894ℏ(119905) = (119905)120601(119905)

Now the adiabatic theorem says that if 119877(119905) changes very slowly then

120601(119905) = 119890 ( )120595 [119877(119905)]

Namely except for a purely TD phase the evolving state stays the instanta-

neous ground state of the Hamiltonian (119905)

To show this we use the instantaneous eigenstates of the Hamiltonian (119905) to

expand 120601(119905)

120601(119905) = sum119886 (119905)119890

ℏint ( ) 120595 [119877(119905)]

We want to plug this expansion into the SE So let us calculate

119894ℏ(119905) = sum[119894ℏ + 119864 119886 ]119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

And

Electron Density Functional Theory Page 7

copy Roi Baer

(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

Electron Density Functional Theory Page 8

copy Roi Baer

(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

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119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

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of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 6

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d) The Born-Oppenheimer theory

iii The adiabatic theorem

Suppose a Hamiltonian is dependent parametrically on 119877 = (119877 119877 hellip ) We

write this as [119877] Denote the eigenstates of this Hamiltonian by

[119877]120595 [119877] = 119864 [119877]120595 [119877]

Now suppose we change 119877 with time so that we have a trajectory 119877(119905) The

Hamiltonian will become time dependent (119905) = [119877(119905)] Suppose the sys-

tem is placed in its ground state at time = 0 120601(119905 = 0) = 120595 [119877(0)] It will

evolve according to the TDSE

119894ℏ(119905) = (119905)120601(119905)

Now the adiabatic theorem says that if 119877(119905) changes very slowly then

120601(119905) = 119890 ( )120595 [119877(119905)]

Namely except for a purely TD phase the evolving state stays the instanta-

neous ground state of the Hamiltonian (119905)

To show this we use the instantaneous eigenstates of the Hamiltonian (119905) to

expand 120601(119905)

120601(119905) = sum119886 (119905)119890

ℏint ( ) 120595 [119877(119905)]

We want to plug this expansion into the SE So let us calculate

119894ℏ(119905) = sum[119894ℏ + 119864 119886 ]119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

And

Electron Density Functional Theory Page 7

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(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

Electron Density Functional Theory Page 8

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

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over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

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sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

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intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

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Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

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⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 7

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(119905)120601 = sum119886 119890

ℏint ( ) 119864 120595 [119877(119905)]

Equating the two expressions gives

0 = sum 119894ℏ 119890

ℏint ( ) 120595 [119877(119905)]

+ sum119886 119890

ℏint ( ) 119894ℏ [119877(119905)]

Now let us multiply by 120595 and integrate (remember that ⟨120595 |120595 ⟩ = 120575 )

0 = 119894ℏ 119890

ℏint ( ) + sum119886 119890

ℏint ( ) 119894ℏ⟨120595 | ⟩

Now we have

0 =119889

119889119905⟨120595 |120595 ⟩ = ⟨120595 | ⟩ + ⟨ |120595 ⟩

And since ⟨ |120595 ⟩ = ⟨120595 | ⟩ we find

120591 = ⟨120595 | ⟩ = minus⟨120595 | ⟩ = minus120591

The matrix 120591 = ⟨120595 | ⟩ is the matrix of ldquotime-dependent non-adiabatic

couplings (TDNACs)rdquo and it is an anti-Hermitean matrix Furthermore for

119899 = 119898 we see

119877119890⟨120595 | ⟩ = 0

Thus

= minus120591 119886 minus sum 119890 int ( ) 120591 119886

(Id1)

We see that the TDNACS 120591 create the non-adiabatic transitions ie the

transitions out of the ground state

Let us play a bit with 120591 Take the time derivative of the TISE

Electron Density Functional Theory Page 8

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

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lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 8

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(119905)120595 (119905) + (119905) (119905) = (119905)120595 (119905) + 119864 (119905) (119905)

Multiply by ⟨120595 | and get

⟨120595 || 120595 ⟩ + ⟨120595 || ⟩ = (119905)⟨120595 |120595 ⟩ + 119864 ⟨120595 | ⟩

Since ⟨120595 || ⟩ = 119864 ⟨120595 | ⟩ we have

⟨120595 || 120595 ⟩ = (119905)⟨120595 |120595 ⟩ + (119864 minus 119864 )⟨120595 | ⟩

If 119899 = 119898 then

(119905) = ⟨120595 || 120595 ⟩

This is called the Hellmann-Feynman theorem showing that the power is the

expectation value of the rate of change of the Hamiltonian If 119899 ne 119898

120591 = ⟨120595 | ⟩ =⟨120595 || 120595 ⟩

119864 minus 119864 (Id2)

This is called Epsteinrsquos theorem giving an expression for the TDNACs Re-

member that [119877(119905)] depends on 119905 through the positions of the nuclei Thus

[119877(119905)] =119889

119889119905[119877(119905)] =

part

partR [119877(119905)]

Thus

120591 = sum⟨120595 |

[119877(119905)]| 120595 ⟩

119864 [119877] minus 119864 [119877] (119905)

= sum120591 (119905)

Where the time-independent NACS are defined as

120591 [119877] = ⟨120595 |

120597

120597119877 120595 ⟩ =

⟨120595 [119877] |

[119877]| 120595 [119877]⟩

119864 [119877] minus 119864 [119877] (Id3)

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

copy Roi Baer

feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

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intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

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ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

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lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 9

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We see that the TDNACs depend on the velocity of the nuclei When nuclei

are small the couplings out of the ground state are small The 120591 are called

the ldquonon-adiabatic couplings (NACs)rdquo We find that as long as 119864 minus 119864 are not

zero on the trajectory one can always find small enough that makes the

TDNACs as small as we wish All we need for states 119899 and 119898 to stay decou-

pled is for the following conditions to be met

|120591 | = | (119905)120591 (119877 (119905))| ≪ 1

Thus we can make the NACs as small as we wish by taking the rate (119905) to

be small enough So all that is left in Eq lrm(Id1) is = minus120591 119886 Define the

non-dynamical phase as 120579 (119905) = 119894 int 120591 (119905 ) 119889119905

(note 120579 is real) and then

119886 (119905) = 119890 ( ) (119904119897119900119908 119901119903119900119888119890119904119904119890119904 = 119886119889119894119886119887119886119905119894119888 119901119903119900119888119890119904119904)

The total state is then

120601(119905) = 119886 (119905)119890

ℏint ( ) 120595 [119877(119905)] = 119890 ( ( ) ( ))120595 [119877(119905)]

where 120579 (119905) = minus

ℏint 119864 (120591)119889120591

is called the dynamical phase It is easy to prove

that if 119877(119905) traverses a closed loop the non-dynamical phase depends only on

that loop and not on the way it is traversed For example if we traverse the

same loop using different velocities the dynamic phase may change but the

geometric phase will not The closed loop geometric angle is called the Berry

phase The independence of path is a result of the fact that the non-dynamical

phase is a line integral and can be made with no reference to time

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 10

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120579 (119905) = 119894 int ⟨120595 (119877[120591])| (119877[120591])⟩119889120591

= 119894 int sum⟨120595 [119877]|120597

120597119877 120595 [119877]⟩

119889120591

= 119894 int sum120591 (119877)

119889119877

( )

( )

The dynamical phase for example is not a line integral and its value depends

not only on the path itself but also on the velocity taken along the way This

observation makes the non-dynamical phase a special quantity Berry has

shown that this quantity can give us information on the way the Hamiltonian

is dependent on its parameters For a real Hamiltonian for example 119890

around a closed path always equals either 1 or -1 If there is an even number

of degeneracies enclosed by the path it is 1 and if an odd number ndash it is -1

iv Motivation for the Born-Oppenheimer approximation classical nuclei

In a molecule we can think of the nuclei (having coordinates 119877) as heavy and

therefore classical particles which are slowly moving The electrons are light

and therefore quantum They have coordinates 119903 (119903 includes spin) and they

119877

119877

A path in parameter

ldquospacerdquo 119877 minus 119877

Electron Density Functional Theory Page 11

copy Roi Baer

feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 11

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feel the effect of slowly moving nuclei The electron dynamics is controlled by

the Hamiltonian

[119877] = + 119881(119903 119877)

The potential 119881(119903 119877) describes the Coulombic interactions between electrons

and nuclei (in atomic units)

119881(119903 119877) = minussum119885

| minus 119903 |

+1

2sum

1

119903

+1

2sum

119885 119885

119877

In general one wants to assume that the total energy of the molecule in this

classical approximation is

119864 = 119879 + ⟨120595 [119877]| [119877(119905)]|120595 [119877]⟩

Where 120595 [119877] is the ground state of the electrons at nuclear configuration 119877

The adiabatic theorem states that this is reasonable as long as nuclei move

slowly Thus the adiabatic theorem allows us to write

119864 = 119879 + 119881 (119877)

where 119881 (119877) is the ground state energy of the electrons at nuclear configura-

tion 119877 This energy is a usual classical mechanical energy and the Newton

equations of motion apply

119872 = 119865 = minus120597

120597119877 119881 (119877)

We see that the adiabatic theorem allows us to consider the nuclei as moving

in a potential well which is essentially the eigenvalues of the electrons This in

essence is the BO approximation when the nuclei are classical

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

copy Roi Baer

We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 12

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v The Born-Oppenheimer approximation in quantum nuclear case

The classical approach motivates a quantum treatment We are expecting that

nuclei will not excite electrons very efficiently That is the motivation for the

BO approximation

The Born and Oppenheimer development is similar to that of the adiabatic

theorem although there are no ldquoexternalrdquo fields Suppose we have a system of

fast particles with coordinates 119903 = (119903 119903 hellip ) and slow particles with coordi-

nates 119877 = (119877 119877 hellip ) The Hamiltonian can be written as

= + + 119881(119903 119877)

The Schroumldinger equation is

120595 (119903 119877) = 119864 120595 (119903 119877) (Id4)

Note that we can assume these wave functions are orthogonal

⟨120595 |120595 ⟩ = intint120595 (119903 119877)120595 (119903 119877)119889119903 119889119877 = 120575

Now to proceed let us first consider the fast r-Hamiltonian

[119877] = + 119881(119903 119877) (Id5)

In this ldquofast Hamiltonianrdquo [119877] the slow variables are simply parameters (it

contains no derivatives with respect to 119877) Thus it depends on 119877 parametrical-

ly The fast eigenvalue problem is

[119877]120601 (119903 119877) = 119882 (119877)120601 (119903 119877)

The eigenvalues are functions of the parameters 119877 They are called the adia-

batic (or Born-Oppenheimer) surfaces The notation with the semicolon be-

tween 119903 and 119877 is designed to emphasize that 120601 are wave functions in 119903 but

they depend only parametrically on 119877 This means for example that the over-

lap of the fast eigenstates (called adiabatic states) involves integration only

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

copy Roi Baer

We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 13

copy Roi Baer

over 119903 the dynamical coordinates while 119877 is held fixed since it is only a pa-

rameter

⟨120601 (119877)|120601 (119877)⟩ = int120601 (119903 119877)120601 (119903 119877)119889119903 = 120575

We cannot in fact say anything really meaningful about ⟨120601 (119877)|120601 (119877 )⟩ when

119877 ne 119877 (except when 119877 minus 119877 is infinitesimal but we will not go into this issue

here) Now we can expand the ldquorealrdquo wave-function as a linear combination

of the adiabatic functions

120595 (119903 119877) = sum120601 (119903 119877)Φ (119877)

(Id6)

We can do this because for any give 119877 120601 (119903 119877) span the space of wave func-

tions dependent on 119903 In fact the expansion coefficients Φ (119877) are given by

Φ (119877) = int120601 (119903 119877)120595 (119903 119877)119889119903

Now let us plug Eq lrm(Id6) into the SE lrm(Id4)

119864 120595 (119903 119877) = ( + )120595 (119903 119877)

= sum [120601 (119903 119877)Φ (119877)]

+ sum119882 (119877)120601 (119903 119877)Φ (119877)

(Id7)

Note that since = sum

where = minus119894ℏ

We have

[120601 (119903 119877)Φ (119877)]

= Φ (119877) [120601 (119903 119877)]

+ sum1

119872 ( 120601 (119903 119877)) ( Φ (119877))

+ 120601 (119903 119877) Φ (119877)

Multiplying Eq lrm(Id7) by 120601 (119903 119877) and integrating over 119903 gives

Electron Density Functional Theory Page 14

copy Roi Baer

sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 14

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sum(A + )Φ (119877)

+ ( + 119882 (119877))Φ (119877) = 119864 Φ (119877) (Id8)

Where

A = int120601 (119903 119877) [ 120601 (119903 119877)]119889119903 = ⟨120601 | 120601 ⟩

= sum

int120601 (119903 119877) [ 120601 (119903 119877)]119889119903

119872

= sum⟨120601 | 120601 ⟩

119872

= minussumℏ

119872 ⟨120601 |

120597

120597119877 120601 ⟩

120597

120597119877

= minussumℏ

119872 120591

120597

120597119877

The matrices 120591 = ⟨120601 |

120601 ⟩ are the non-adiabatic couplings in the ldquofastrdquo

system These are exactly the non-adiabatic coefficients in the adiabatic theory

(Eq lrm(Id3))

It is possible to show that

sum(A + )Φ (119877)

+ Φ (119877)

= sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

Thus Eq lrm(Id8) becomes

sumsumminusℏ

2119872 (

120597

120597119877 + 120591

)

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

This is a Schroumldinger ndashlike equation which determines the coefficients Φ (119877)

These are called the ldquoslow eigenfunctionsrdquo Once they are computed one has

an exact solution to the Schroumldinger equation However we do not really

want to solve this infinite set of coupled differential equations Thus we as-

sume that the quantities 120591 for 119895 ne 119896 can be neglected Note that here there is

no which can be taken as small as needed to make the effect of 120591 as mi-

Electron Density Functional Theory Page 15

copy Roi Baer

nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 15

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nute as we need Still we can hope that the fact that we chose 119877 to be slow de-

grees of freedom allow us to make just this approximation This was the idea

of Born and Oppenheimer who neglected 120591

summinusℏ

2119872

120597

120597119877 Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877)

The resulting equation is a Schroumldinger equation for the slow degrees of free-

dom which move under the force of electrons derived from a potential 119882 (119877)

When applied to molecules the slow degrees of freedom are usually the nuclei

and the fast - the electrons There is a problem with neglecting 120591 because of

their non-dynamical effect Taking them into account results in treating them

as a magnetic field

sumℏ

2119872 (119894

120597

120597119877 minus 120591 )

Φ (119877)

+ 119882 (119877)Φ (119877) = 119864 Φ (119877) (Id9)

The BO approximation breaks the molecular SE into two consecutive tasks

First the electronic SE Eq lrm(Id5) must be solved for any relevant clamped po-

sition of nuclei 119877 Then the nuclear equation lrm(Id9)

Further reading on this subject M Baer Beyond Born-Oppenheimer elec-

tronic non-adiabatic coupling terms and conical intersections (Wiley Hobo-

ken NJ 2006)

e) Electron correlation

vi The electronic wave function of two non-interacting electrons

We have stressed that the electronic structure problem is difficult because of

the complexity of the many-body wave function when electrons interact In

order to appreciate this complexity of the electronic wave function let us first

study a simple system of two non-interacting electrons in a 1D atomic well

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

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lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

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change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

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120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 16

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We consider an atomic well given by the potential 119907 (119909) and we place in it

an electron The Hamiltonian is

ℎ = minusℏ

2119898 nabla + 119907 (119909)

(Ie1)

Now consider a 2-electron problem Assume we have two electrons Fermi-

ons which are non-interacting in the well The Hamiltonian is

= ℎ(1) + ℎ(2) (Ie2)

The notation ℎ(119894) means the Hamiltonian of Eq lrm(Ie1) with the coordinate of

the i-th electron

What are the eigenstates in this case First since each electron can have a

spin we must decide on the spin of the state For now let us assume the state

is spin-polarized ie that the total spine is 1 both electrons are in spin-up or-

bitals We try the following form as a wave function

Ψ(119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]120572(1)120572(2) (Ie3)

Notice that the spatial part is anti-symmetric while the spin part is symmetric

This renders the entire wave-function anti-symmetric in accordance with the

Pauli principle The notation 120572(1)120572(2) means both electrons have spin projec-

tion up (120572) We do not yet know if and under what conditions this wave func-

tion can actually describe eigenstates of the two electrons the

an lrm(Ie2) We assume that 120595 (119909) and 120595 (119909) are orthonormal This makes the

state normalized since

Electron Density Functional Theory Page 17

copy Roi Baer

intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 17

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intΨ(119909 119909 ) 119889119909 119889119909 =

1

2int(120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 ))

119889119909 119889119909

=1

2int [(120595 (119909 )120595 (119909 ))

+ (120595 (119909 )120595 (119909 ))

minus 2120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 )] 119889119909 119889119909

(Ie4)

The first and second terms are both equal to

int(120595 (119909 )120595 (119909 )) 119889119909 119889119909 =int120595 (119909 )

119889119909 int120595 (119909 ) 119889119909 =1 times 1

= 1 (Ie5)

The third term in lrm(Ie4) is

int120595 (119909 )120595 (119909 )120595 (119909 )120595 (119909 ) 119889119909 119889119909

=int120595 (119909 )120595 (119909 ) 119889119909 int120595 (119909 )120595 (119909 ) 119889119909 = 0 times 0 = 0 (Ie6)

Indeed the entire wave function is orthonormal (thanks to the factor 1radic2

in lrm(Ie3))

Now let us see under what condition the wave function in Eq (Ie3) is an ei-

genstate of the Hamiltonian in lrm(Ie2)

Ψ(119909 119909 ) = (ℎ(1) + ℎ(2))Ψ(119909 119909 )

=1

radic2120595 (119909 )ℎ(1)120595 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

minus1

radic2120595 (119909 )ℎ(1)120601 (119909 ) + 120595 (119909 )ℎ(2)120595 (119909 )

(Ie7)

This should be equated to

119864Ψ(119909 119909 ) =1

radic2119864120595 (119909 )120595 (119909 ) minus 119864120595 (119909 )120595 (119909 ) (Ie8)

If we choose the orbitals 120595 (119909) and 120595 (119909) to be eigenstates of h (which are

orthogonal so that is compatible with our previous assumption)

Electron Density Functional Theory Page 18

copy Roi Baer

ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

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Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

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⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 18

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ℎ120595 (119909) = 120598 120595 (119909) 119894 = 01 hellip (Ie9)

Thus

Ψ(119909 119909 ) =1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

minus1

radic2120598 120595 (119909 )120595 (119909 ) + 120598 120595 (119909 )120595 (119909 )

= (120598 + 120598 )Ψ(119909 119909 )

(Ie10)

And we see that indeed Ψ(x x ) is an eigenstate with energy E = ϵ + ϵ

vii Correlation in action a wave function of 2 interacting particles in an

Harmonic trap

We now build a simple electronic structure model that will allow us to study

in some detail the most basic concepts For this we suppose that the electrons

are in a harmonic atom that is the potential well

119907 (119909) =1

2120583120596 119909 (Ie11)

The two lowest eigenstates of a Harmonic oscillator are

120595 (119909) = 119873 119890

ℏ

120595 (119909) = 119873 radic120583120596

ℏ119909119890

ℏ

(Ie12)

The normalization constants are

119873 = (120583120596

120587ℏ)

119873 = (4120583120596

120587ℏ)

(Ie13)

And the eigenvalues are

120598 = (119899 +1

2)ℏ120596 119899 = 01 hellip (Ie14)

The groundstate energy of the 2-electron system in the triplet state describes

one electron in 120595 (119909) and another in 120595 (119909)

Electron Density Functional Theory Page 19

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119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

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lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 19

copy Roi Baer

119864 = 120598 + 120598 = 2ℏ120596 (Ie15)

As discussed above singlet and triplet two-electron ground state wave func-

tions composed of two orbitals must be space-symmetric or antisymmetric

respectively We consider below 3 wave functions The first Ψ is the ground

state singlet where both electrons are in 120595 The second and third are a singlet

and a triplet made from one electron in 120595 and the other in 120595

Ψ (119909 119909 ) = 120595 (119909 )120595 (119909 ) = 119873 119890

ℏ(

)

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) + 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 + 119909 )

Ψ (119909 119909 ) =1

radic2[120595 (119909 )120595 (119909 ) minus 120595 (119909 )120595 (119909 )]

= 119873 119890

ℏ(

)(119909 minus 119909 )

(Ie16)

The normalization factors are

119873 = 119873 = radic

120583120596

120587ℏ 119873 =

1

radic2radic120583120596

ℏ119873 119873 =

120583120596

120587ℏ

(Ie17)

Eq lrm(Ie16) describes the distribution of positions of both electrons in their

corresponding states We now want to ask how much are the electrons in this

state aware of each other Do they correlate their motion in some way One

way to measure correlation is to consider the quantity lang119909 119909 rang minus lang119909 ranglang119909 rang If

electrons are completely unaware of each other this quantity called the posi-

tion-autocorrelation function is zero because then the average of the product

of their position must decompose to the product of the average Any deviance

from zero indicates some degree of correlation For the triplet wave function

in Eq lrm(Ie16) we find

Electron Density Functional Theory Page 20

copy Roi Baer

lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

copy Roi Baer

The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 20

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lang119909 rang = ⟨Ψ |119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |120595 ⟩)

=1

2(⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩) = 0

(Ie18)

Of-course the same result would be obtained if we calculate ⟨119909 ⟩ because the

electrons are equivalent Furthermore

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ =1

2⟨120595 120595 minus 120595 120595 |119909 119909 |120595 120595 minus 120595 120595 ⟩

=1

2(⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ + ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩

minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ minus ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩)

= minus|⟨120595 |119909|120595 ⟩| = minus

1

2

ℏ

120583120596

(Ie19)

We thus find that

lang119909 119909 rang minus lang119909 ranglang119909 rang = minus1

2

ℏ

120583120596

(Ie20)

The position autocorrelation functional is non-zero even though the electrons

are non-interacting The correlation is due to the Pauli principle It is negative

because the Pauli principle pushes the electrons apart and since most of the

time they both want to be near the origin this gives preference to them being

placed on opposite sides (when one electron has positive x coordinate the

other has negative and vice versa) Letrsquos see what happens in the singlet wave

function Ψ Here too lang119909 rang = lang119909 rang = 0 Then

lang119909 119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ = ⟨120595 120595 |119909 119909 |120595 120595 ⟩

= ⟨120595 |119909 |120595 ⟩⟨120595 |119909 |120595 ⟩ = 0 (Ie21)

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

copy Roi Baer

We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 21

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The singlet ground state shows no correlation because the Pauli principle is

applied here to the spins not the spatial part of the wave function Note how-

ever this does not mean that all singlet wave functions of 2 electrons have no

correlation Indeed this is the situation in Ψ where the development is

very similar to Eq lrm(Ie19) except that the minus sign is now a plus sign so

lang119909 119909 rang minus lang119909 ranglang119909 rang = ⟨Ψ |119909 119909 |Ψ ⟩ minus 0

=1

2⟨120595 120595 + 120595 120595 |119909 119909 |120595 120595 + 120595 120595 ⟩ = |⟨120595 |119909|120595 ⟩|

=1

2

ℏ

120583120596

(Ie22)

Here we find positive autocorrelation indicative of the fact that spin-opposite

non-interacting pairs of electrons want to stick together (when one has posi-

tive x the other wants to have as well and when one has negative the other

one wants negative as well) like non-interacting bosons

Since there is no interaction between the electrons the correlation in these

wave functions arises only from the Pauli principle ie because we impose

the fact that electrons are Fermions This is called Fermi correlation Our les-

son is this

1) Wavefunctions that are mere products of singe-particle orbitals have no correlation 2) If the products are symmetrized like in the case of the excited singlet the correlation

lang119909 119909 rang is positive indicating that the particles ldquolike to be togetherrdquo ie both on the right or both on the left of the origin

3) If the products are anti-symmetrized like in the case of the triplet the correlation lang119909 119909 rang is negative indicating that the particles ldquolike to be separatedrdquo

Up to now we assumed no e-e interaction So now letrsquos include it and add to

the Hamiltonian an interaction term

= ℎ(1) + ℎ(2) + 119881(119909 119909 ) (Ie23)

Electron Density Functional Theory Page 22

copy Roi Baer

Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

copy Roi Baer

Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

copy Roi Baer

When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

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119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

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of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 22

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Let us take one case of coupling which is simple enough to yield to analytical

analysis

119881(119909 119909 ) = 120583120574 119909 119909 (Ie24)

With 120574 lt 120596 In this case the Hamiltonian is

=119901

2120583+

119901

2120583+

1

2120583120596 119909

+1

2120583120596 119909

+ 120583120574 119909 119909 (Ie25)

The interaction seems strange at first sight because it does not seem to depend

on the distance between the particles as we are used to from electrostatics

yet it does describe a repulsion We can see this by writing the Hamiltonian

as

=119901

2120583+

119901

2120583+

1

2120583(120596 + 120574 )119909

+1

2120583(120596 + 120574 )119909

minus1

2120583120574 (119909 minus 119909 )

(Ie26)

We now see that the Hamiltonian describes two particles in a Harmonic po-

tential 119881 (119909) =

120583(120596 + 120574 )119909 interacting by a Harmonic potential

minus

120583120574 (119909 minus 119909 )

The potential drops as |119909 minus 119909 | grows ie it is repulsive

Exercise lrmI-1

Find the eigenvalues and eigenfunctions of this Hamiltonian (You can define

new coordinates 119883 and 119909 by

radic = 119909 and

radic = 119909 )

Exercise lrmI-2

Write the ground-state energy and wave function for the triplet state of the

system in the previous exercise Determine the effect of interaction on the en-

ergy by calculating 119903 =

and on the correlation function 119888 = lang119909 119909 rang minus

lang119909 ranglang119909 rang

Solution The result is shown in Figure lrmI-2 (left panel)

Electron Density Functional Theory Page 23

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Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

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⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 23

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Figure lrmI-2 (Left panel) The ratio = (Right panel) the correlation ratio ( ) (

)

vs interaction strengths where = (

)

f) The electron density is a much simpler object

than the wave function

The complexity of the electronic wave function is overwhelming It includes

all the information we can have on the electrons in a molecule in a certain

state However all these intricacies and details are often uninteresting for us

in many cases we simply have no direct use for them Take for instance the

electronic energy - our primary target in the Born-Oppenheimer picture It

depends only on the relative distance of pairs of particles This is because the

electron-electron Coulomb repulsion is a pair-wise interaction

One interesting quantity besides energy that can be extracted from the elec-

tronic wave function is the electron density 119899(119955) This 3D function tells us the

expectation value of the density of electrons That is 119899(119955)119889 119903 is the expectation

value of the number of electrons at a small volume 119889 119903 around point r Thus

we can write

119899(119955) = ⟨120595|(119955)|120595⟩ (If1)

We use the notation

00 01 02 03 04 05

04

05

06

07

08

09

10

r

00 01 02 03 04 050

5

10

15

cc

2

Electron Density Functional Theory Page 24

copy Roi Baer

⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

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mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

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If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

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change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 24

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⟨120595|120601⟩ equiv int120595 (119955 119955 hellip 119955 )120601(119955 119955 hellip 119955

)119889 119903 119889 119903 hellip119889 119903

(If2)

Here (119955) is the operator corresponding to the electron number density Since

electrons are point particles and the position operator for the ith electron is 119955

this operator is defined by

(119955) = sum120575( minus 119955)

(If3)

We used the definition of a 120575-function according to which

int120575(119955 minus 119955)119891(119955 )119889 119903 = 119891(119955)

(If4)

The function 120575( minus 119955) is the density of electron 119894 at 119955

Exercise lrmI-3

Calculate the expectation value of the electron number density and show that

119899(119955) = 119873 int120595(119955 119955 hellip 119955 )120595(119955 119955 hellip 119955

)119889 119903 hellip119889 119903

(If5)

Looking at Eq lrm(If5) we see that 119899(119955) involves integrating out a huge amount

of wave-function details It is as if only the data concerning the density distri-

bution of a single electron remains This is multiplied by 119873 in Eq lrm(If5) so

119899(119955) accounts for the combined density of all electrons Indeed integrating

over the entire space one obtains from lrm(If5)

int119899(119955)119889 119903 = 119873 (If6)

Expressing the fact that the total number of electrons is 119873

Exercise lrmI-4

Calculate and plot the 1D electron density of the triplet ground state from Ex-

ercise lrmI-2 as a function of the interaction parameter

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

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moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

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formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 25

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When we look for the ground-state energy of a complicated system with

Hamiltonian the variational principle is often extremely important It says

quite simply that the ground-state energy is a minimum of a functional of

wave functions The functional is

119864[120601] =⟨120601||120601⟩

⟨120601|120601⟩ (If7)

And the variational principle states that

119864 equiv 119864[120601 ] le 119864[120601] (If8)

For all 120601 What this principle allows us is to determine which of the two wave

functions 120601 and 120601 gives an energy closer to 119864 without knowing 119864 Sim-

ple the one with lower energy is closer

We can build a parameterized family of functions 120601 and determine the op-

timized parameter as the one that gives minimum to 119864(120574) equiv 119864[120601 ]

Exercise lrmI-5

Consider the quantum ground-state problem for a particle in a well 119907(119909) =

119896119909 We consider the family of functions

120601 (119909) =119890

radicradic2120587120590 (If9)

Here is a positive parameter What is the best function of this type for rep-

resenting the ground state

g) Dilation relations

Consider the normalized 120595(119909) From it we can define a family of normalized

wavefunctions

120595 (119909) = 120574 120595(120574119909) (Ig1)

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

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moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 26

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We can check that each member is indeed normalized

⟨120595 |120595 ⟩ = int|120595 (119909)| 119889119909 = 120574int|120595(120574119909)| 119889119909 = int|120595(119910)| 119889119910

(Ig2)

This operation of stretching the argument of a function is called dilation Di-

lation affects functionals Consider the kinetic energy functional

119879[120595] equiv ⟨120595||120595⟩ = int120595(119909) (minus1

2

119889

119889119909 )120595(119909)119889119909

(Ig3)

Exercise lrmI-6

Prove the dilation relation of 119879[120595 ] and 119879[120595]

119879[120595 ] = 120574 119879[120595] (Ig4)

The potential between all particles is called ldquohomogeneous of order 119899rdquo if

119881(120574119909 hellip 120574119909 ) = 120574 119881(119909 hellip 119909 ) (Ig5)

Examples are 119899 = 2 is for Harmonic potential wells and harmonic interactions

and 119899 = minus1 is for the Coulomb potential wells and Coulombic interactions

Exercise lrmI-7

Prove the dilation relation for homogeneous potentials of order 119899

119881[120595 ] = 120574 119881[120595] (Ig6)

We combine the results from Eqs lrm(Ig4) and lrm(Ig6) and obtain an interesting

property of the total energy for systems with homogeneous interactions

119864[120595 ] = 119879[120595 ] + 119881[120595 ] = 120574 119879[120595] + 120574 119881[120595] (Ig7)

For a molecule the interaction between the particles (electrons and nuclei) is

the Coulomb potential 119881(119955 119929) equiv 119881( 119955 hellip 119955 119929 hellip 119929

) which is homoge-

neous of order 119899 = minus1 one finds the energy of a molecule obeys

Electron Density Functional Theory Page 27

copy Roi Baer

119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

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moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 27

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119894ℏ(119905) = (119905)120601(119905)119864[120595 ] = 119879[120595 ] + 119881[120595 ]

= 120574 119879[120595] + 120574119881[120595] (Coulomb) (Ig8)

viii The concept of the virial in classical mechanics

First let us define the virial For a system with coordinate 119902 collectively de-

noted as 119954 the virial in classical mechanics is the time average of 119954 sdot 119917 where

119917 is the force vector

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ =1

120591int 119954(119905 ) sdot 119917(119905 )119889119905

It can be shown that for bound systems[5]

119907119894119903119894119886119897 = ⟨119954 sdot 119917⟩ = minus2⟨119879⟩

For conservative systems the force is a gradient of the potential 119917 = minusnabla119907(119954)

The viral relates to dilation of the coordinates through

119889

119889120574119907 (119954) = 119954 sdot nabla119907(120574119954) = 119954 sdot nabla119907 (119954) (Ig9)

For homogeneous potentials we have 120574

119907 (119954) = 119899120574 119907(119954) = 119899119907 (119954) thus

119899119907 (119954) = 120574119954 sdot nabla119907 (119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig10)

In particular for 120574 = 1

119899119907(119954) = 119954 sdot nabla119907(119954) (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899) (Ig11)

We will especially be interested in Coulomb systems where 119899 = minus1 then

119894ℏ(119905) = (119905)120601(119905) (Ig12)

119907(119954) + 119954 sdot nabla119907(119954) = 0 (119907 ℎ119900119898119900119892119890119899119890119900119906119904 119900119903119889119890119903 119899 = minus1) (191)

Exercise lrmI-8

Eq lrm(Ig11) is known as Eulerrsquos equation after its discoverer In Thermody-

namics it is extremely useful Thermodynamics stipulates that if you know

Electron Density Functional Theory Page 28

copy Roi Baer

the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 28

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the energy of a system 119880(119878 119881 ) as a function of its macroscopic parameters

119878 119881 then you have complete thermodynamics knowledge The first and se-

cond laws of thermodynamics state that

119894ℏ(119905) = (119905)120601(119905) (Ig13)

119889119880 = 119879119889119878 minus 119901119889119881 + 120583 sdot 119889 (192)

where 119879 119901 and 120583 are respectively the temperature the pressure and the chem-

ical potentials A second stipulation is that 119880 is a homogeneous function of

order 1

From this show that for all thermodynamical systems

1 119880 = 119879119878 minus 119875119881 + 120583 sdot

2 (

) =

For homogeneous potentials the viral theorem becomes

119894ℏ(119905) = (119905)120601(119905) (Ig14)

119907119894119903119894119886119897 = minus119899⟨119907⟩ = minus2⟨119879⟩ (193)

For Coulomb systems 119899 = minus1

119894ℏ(119905) = (119905)120601(119905) (Ig15)

⟨119907⟩ = minus2⟨119879⟩ (194)

From which

119894ℏ(119905) = (119905)120601(119905) (Ig16)

119864 = ⟨119907⟩ + ⟨119879⟩ = minus⟨119879⟩ =⟨119907⟩

2 (195)

We now show that this relation holds also in quantum mechanics

Electron Density Functional Theory Page 29

copy Roi Baer

ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 29

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ix The Virial Theorem in quantum mechanics

Now if 120595 is the ground-state energy then 119864[120595 ] obtains the minimum and

therefore

119894ℏ(119905) = (119905)120601(119905) (Ig17)

119889

119889120574119864[120595 ] = 0 (196)

Plugging this into Error Reference source not found one obtains

119894ℏ(119905) = (119905)120601(119905) (Ig18)

0 = 2120574119879[120595] minus 119899120574 119881[120595] (197)

Since we know that the optimal value of 120574 is 1 then (dropping the [120595] symbol

and remembering that all following quantities are ground state expectation

value

119894ℏ(119905) = (119905)120601(119905) (Ig19)

2119879 = 119899119881

119864 = 119879 + 119881 = (1 +2

119899)119879 = (

119899

2+ 1)119881

(198)

These relations show that the virial theorem (Eq lrm(Ig16) holds in quantum me-

chanics provided 120595 is the full molecular eigenstate For molecules 119899 = minus1

and one has

119894ℏ(119905) = (119905)120601(119905) (Ig20)

2119879 = minus119881 (Coulomb)

119864 = 119879 + 119881 = minus119879 =1

2119881

(199)

A subtle issue all energies and potentials in the above expressions are abso-

lute We usually give energy and potentials only to within an additive con-

Electron Density Functional Theory Page 30

copy Roi Baer

stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

copy Roi Baer

Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

copy Roi Baer

We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 30

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stant However the fact that the potential is homogenous it cannot tolerate

addition of a constant (119909 is homogeneous of order 2 but 119909 + 119886 is not)

x Slaterrsquos theory of the chemical bond using the viral theorem

What happens when 120595 is the electronic eigenstate in the Born Oppenheimer

approximation If we look only at the electronic wave function we do not ex-

pect Eq lrm(Ig19) to be valid Indeed using = + 119881(119955 119929) where = (minus

nabla )

and 119881(119955 119929) =

|119955 119929| (with obvious summation on nuclei and electrons)

[119929]120595[119929](119955) = 119864[119929]120595[119929](119955) (Ig21)

Upon dilation 120595 [119929](119955) = 120574 120595[119929](120574119955) and note that 119929 is not dilated Then

we have for the KE and potential

119879 [119929] equiv ⟨120595 ||120595 ⟩ = int120595 [119929](119955) (minus1

2nabla )120595 [119929](119955)119889 119903

= 120574 int120595[119929](120574119955) (minus1

2nabla )120595[119929](120574119955)119889 119903

= 120574 int120595[119929](119956) (minus1

2nabla )120595[119929](119956)119889 119904 = 120574 119879[119929]

(Ig22)

And we define

119882 [119929 119929 ] equiv ⟨120595 [119929]|[119929 ]|120595 [119929]⟩ = int120595 (119955 119929) 119881(119955 119929 )119889 119903

= 120574 int120595(120574119955 119929) 119881(119955 119929 )119889 119903

= int120595(119956 119929) 119881(120574 119956 119929 )119889 119904

= 120574int120595(119956 119929) 119881(119956 120574119929 )119889 119904 = 120574119882[119929 120574119929 ]

(Ig23)

We have defined 119882 a two 119929 quantity Of course the physical interaction po-

tential is 119881[119929] = 119882[119929119929] The reason we defined 119882 this way is that we can

now write

Electron Density Functional Theory Page 31

copy Roi Baer

119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 31

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119864 [119929] = 120574 119879 [119929] + 120574119882[119929 120574119929] (Ig24)

Note that we inserted the subscript 119890 to the KE since now it is only the KE of

the electrons

One still has the variational principle ie 119889119864 [119929]119889120574| = 0 (Eq (196))

0 = 2120574119879 [119929] + 119882[119929 120574119929] + 120574(119825 sdot nabla119929 119882)[119929119929 ]119929 119929 (Ig25)

Here we used the fact that

119891(120574119929) = [119929 sdot nabla119891](120574119929) Putting in the solution

120574 = 1 we find

0 = 2119879 [119929] + 119881[119929] + 119825 sdot nabla119929 119882[119929119929 ]119929 119929 (Ig26)

From the Hellmann-Feynman theorem (see section XXX) nabla119929 (119882[119929119929 ])119929 119929 =

nabla 119864(119929) Adding to the electronic energy 119864 the nuclear potential energy

119864 =

sum

|119929 119929 | gives the BO energy 119864 = 119864 + 119864 and since the nuclear

energy is homogeneous of order -1 we can use Eq (191) and

0 = 119864 + sum119825 sdot nabla

119864 (Ig27)

Plugging all this into eq Error Reference source not found we find Slaterrsquos

formula[6]

119879 = minus119864 minus 119825 sdot nabla 119864 (Ig28)

From this using 119864 (119929) = 119879 + 119881 (where 119881 = 119881 + 119864 is the total PE of the

molecule) we find also

119881 = 2119864 + 119825 sdot nabla 119864 (1910)

We point out an interesting property of Eq

Error Reference source not found is that we can obtain electronic properties

such as kinetic energy 119879 or potential energy 119881 directly from the potential sur-

face 119864 This will be important for analyzing properties of DFT quantities

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

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mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 32

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Note that at stationary points on the Born Oppenheimer PES where nabla 119864 =0

the usual virial relation eq (198) holds (with neglect of nuclear kinetic ener-

gies)

Slater derived Eq Error Reference source not found in a different manner

not from variational or dilation considerations He used it for analyzing the

general nature of chemical bonding of a diatomic molecule We follow in his

footspteps here Consider a diatomic molecule Suppose the BO curve is give

as a potential of the following form in terms of the inter-nuclear distance 119877

119864 (119877) =119860

119877 minus

119861

119877 + 119864 (Ig29)

Where 119860 and 119861 are positive constants and typically 119898 ≫ 119899For example in the

case the PE curve is given by the Lennard-Jones potential then 119898 = 12 and

119899 = 6 This potential describes long-range attraction and short-range repul-

sion It has a minimum at

119877 = (119861119899

119860119898)

(Ig30)

One then has

119879 =119861(1 minus 119899)

119877 minus

119860(1 minus 119898)

119877 minus 119864 (Ig31)

When 119877 rarr infin we have 119879 asymp minus ( )

minus 119864 Let us consider what happens to

the KE as the distance 119877 is reduced Clearly it initially descends however at

some distance 119877 it reaches a minimum and below this distance it begins to

increase Since

119889

119889119877119879 = minus

119861119899(1 minus 119899)

119877 +

119860119898(1 minus 119898)

119877 (Ig32)

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

copy Roi Baer

mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

copy Roi Baer

nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 33

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We find the minimum kinetic energy is obtained at a distance somewhat larg-

er than 119877

119877 = (119860119898(119898 minus 1)

119861119899(119899 minus 1))

= (119898 minus 1

119899 minus 1)

119877 gt 119877 (Ig33)

For example for 119898 = 12 119899 = 6 we find 119877 = 114119877 The sum of electronic

and nuclear potential energy is

119881 =119860(2 minus 119898)

119877 minus

119861(2 minus 119899)

119877 + 2119864 (Ig34)

At large 119877 119881 + 119864 actually increases as the atoms approach The potential en-

ergy does not rise indefinitely At some inter-nuclear distance larger than the

distance at which the BO potential obtains its minimum it reaches a maxi-

mum and then starts decreasing We have

119860119898(2 minus 119898)

119877 =

119861119899(2 minus 119899)

119877

119877 = (119860119898(119898 minus 2)

119861119899(119899 minus 2))

= (119898 minus 2

119899 minus 2)

119877 gt 119877

(Ig35)

For the Lennard Jones potential we have 119877 = 116119877 For 119877 lt 119877 we find

that the kinetic energy grows sharply as 119877 decreases while the potential en-

ergy decreases This sharp behavior can be interpreted as a condensation of

the electron cloud around each nuclei as such a condensation causes increase

in kinetic energy 119879 and decrease in electronic potential energy 119881 The total BO

energy continues to drop as long as 119877 gt 119877 and then starts rising

As an application of this theory consider the 119883hellip119884 system where 119883 is a neu-

tral atom and 119884 a distant atomic ion (can also be a molecular ion) The ener-

gy at large distance 119877 is that of Eq Error Reference source not found with

119860 = 0 and 119861 = ( )

where 120572(119883) is the polarizability of 119883 and 119899 = 4 This form

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

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mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

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change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

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120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 34

copy Roi Baer

of the potential surface results from a simple exercise in electrostatics Quan-

tum mechanics is needed only for calculating the value of the polarizability of

119883 120572(119883) Slaterrsquos analysis states that

119879 = minus3120572(119883)

2119877 minus 119864

119881 =120572(119883)

119877 + 2119864

(Ig36)

So the force due to the total potential energy is repulsive while that due to the

KE is attractive The KE attraction outweighs the PE repulsion by a factor of

three halves Slaterrsquos analysis shows that the interaction here is fundamentally

different from that of electrostatics where the force attributed entirely to the

Coulombic attraction

Another application is to the simplest chemical bond that of the 119867 molecule

The text-book analysis of this molecule assumes that the molecular orbital is a

ldquobondingrdquo linear combination of the two atomic orbitals (LCAO) both having

the 1s shape 120595 (119903) = 119890

but each centered on its corresponding nucleus

Quantitatively this picture is very inaccurate While it does show binding its

predicted a bond length is almost 14 Å (while the true bond length is 106Å)

and its estimate of atomization energy is 17 eV while the experimental value

is close to 27eV There are no electron correlation effects here so the only

source of inaccuracy is the limited efficacy of the small basis

Electron Density Functional Theory Page 35

copy Roi Baer

Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

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mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

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If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

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119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

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change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 35

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Figure lrmI-3 The optimal dilation parameter as a function of the inter-nuclear distance 119929

in where the wave function is the sum of 2 1s-type orbitals (localized on each of the

two nuclei)

One can add a dilation factor to the wave function 120595 (119903) = 119890

and perform

a variational determination of 120577 Then one can obtain for each value of the

inter-nuclear distance the optimal 120577 denoted 120577 (119877) At 119877 rarr infin the dilation pa-

rameter should be equal to 1 as then the 1s orbital is exact At 119877 rarr 0 the mole-

cule becomes the He ion and 120577 should approach the exact value of 2 See Fig-

ure lrmI-3 But the optimal value 120577 is not monotonic At large 119877 it becomes small-

er than 1 indicating that the electron cloud expands the kinetic energy drops

and the potential energy rises Thus long-range attraction is driven by the ki-

netic energy At 119877 = 52119886 the value of 120577 is again 1 and then grows monoton-

ically as 119877 is reduced Now the orbital is contracting kinetic energy is growing

and potential energy is decreasing In this range chemical binding is driven

by the potential energies At the distance minimizing the BO energy the value

of 120577 is 120577 asymp 123 ie the orbitals are then considerably contracted (relative to

the free H atom) With such contracted orbitals the calculated bond length of

the LCAO calculation is 106Å The bond energy computed by subtracting the

calculated H2+ energy from that of the hydrogen atom gives the bond energy

of 235eV The first to report this type of calculation is in ref [7] a clear exposi-

tion of the calculation is given in refs [8] These improved theoretical esti-

Electron Density Functional Theory Page 36

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mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 36

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mates with a two-function basis achieved with just a single dilation parame-

ter show that orbital shapes are crucial for a good description of chemical

bonding

This result contradicts our intuition regarding the behavior of electrons in a

covalent bond It seems that a major source of the chemical bond energy in H

is not due to the electrostatic benefit resulting from the placement of electrons

in the middle region separating the two nuclei Rather it is due to the contrac-

tion of the electron wave function near each of the nuclei induced by change

in Coulomb potential near that nucleus due to the presence of the other nu-

clei Around one nucleus of H the nuclear Coulombic potential becomes

deeper and distorted towards the second nucleus As a result a contraction of

the electron density around the nucleus is obtained and the charge placed be-

tween the atoms is not in the ldquomidwayrdquo region of the bond it is between them

alright but very close to each nucleus Each nuclei seems to share the electron

not by placing it midway between them but rather by having the electrons

much closer to each nucleus

Combining this with the above result we find a very general conclusion

Rule 1 of chemical attraction As atoms approach from afar (for 119877 gt 119877 ) the

kinetic energy decreases and the potential energy increases and thus it is the

lowering of kinetic energy which is responsible for the distant attraction between

molecules

Rule 2 of chemical bonding For distances near but larger than the bonding

distance the attraction is more subtle The electronic kinetic energy rises

sharply and the potential energy drops even more sharply Thus a major

source of energy is the contraction of orbitals around their respective nuclei

inspired by the deepening of potential wells due to the presence of close-by

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 37

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nuclei The electronic PE is thus the major glue when atoms are close to each

other It has to offset the nuclear repulsion and the repulsion due to kinetic

energy of electrons

xi Other Dilation Facts

Consider the Schroumldinger equation

minusℏ

2119898120595 (119909) + 119907(119909)120595(119909) = 119864120595(119909) (Ig37)

We now dilate we assume 120595 (119909) = 120595(120582119909) is an eigenstate and find the corre-

sponding Hamiltonian Note that 120595 (119909) = 120582 120595 (120582119909) thus

minusℏ

2119898120582 120595

(119909) + 119907(120582119909)120595 (119909) = 119864120595 (119909) (Ig38)

We see that the required Hamiltonian involves a dilated potential and a scalled

mass by a factor 120582 The energy is left intact Alternatively we can multiply

through by the square of the dilation factor 120582 and obtain

minusℏ

2119898120595

(119909) + 120582 119907(120582119909)120595 (119909) = 120582 119864120595 (119909) (Ig39)

Now the potential is dilated and scaled 119907(119909) rarr 120582 119907(120582119909) when then the wave

function is dilated by 120595(119909) rarr 120595(120582119909) and the total energy is also scaled by

119864 rarr 120582 119864 This is general for basically any Schroumldinger equations and holds for

any number of spatial dimensions

Now consider homogeneous potentials in 3D 119907(120582119955) = 120582 119907(119955) In this case the

potential transform is indeed a multiplicative scaling 119907(119955) rarr 120582 119907(119955) In par-

ticular in Coulomb systems when 119899 = minus1 we find that the potential scaling

is simply by a factor 120582 We can turn around the discussion and state

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 38

copy Roi Baer

If the Coulomb coupling constant is scaled by a factor 120582 then the total

energy is scaled by a factor 120582 and the wave function is dilated by a fac-

tor 120582 In particular the density is dilated and scaled as 119899(119955) rarr 120582 119899(120582119955)

According to the virial theorem the potential energy and the total energy are

related by

= minus119879 = 119864 thus the expectation value of the potential and kinetic

energies is scaled by a factor 120582 as well

Exercise lrmI-9

An application of the above discussion allows an interesting exact conclusion

concerning the ground-state energy per particle of the homogeneous electron

gas of density 119899 (see section XXX for details) Denote this quantity

120598 (119899 119890 ) and the Coulomb coupling constant 119890 4120587120598 Show that

3119899120597120598

120597119899+ 119890

120597120598

120597119890 = 2120598 (Ig40)

Hint Start from 120598 (120582 119899 λ119890 ) = λ 120598 (119899 119890 ) Then take the derivative with

respect to 120582 Then set 120582 = 1

h) Introduction to Functionals

xii Definition of a functional and some examples

A formula that accepts a function and maps it to a number is called a function-

al We denote the number a functional 119865 maps to a function 119899(119955) by this by

the symbol 119865[119899] Take for example the simplest functional 119865[119899] = 0 This

functional maps any function 119899(119955) to the number zero A more interesting

functional is

119865119955 [119899] = 119899(119955 ) (Ih1)

This functional maps to each function 119899(119955) its value at a certain point 119955

Next consider the functional

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 39

copy Roi Baer

119868 [119899] =1

119881int119899(119955)119889 119903

(Ih2)

mapping each function its average value in a given volume 119881

Another familiar example is the kinetic energy functional for a particle of

mass 120583 in some wave function 120595(119955)

119879[120595] = minusℏ

2120583int120595 (119955)nabla 120595(119955)119889 119903

(Ih3)

Or the energy for a given potential 119881(119955)

119864[120595] = 119879[120595] + 119881[120595] (Ih4)

Where

119881[120595] = int119881(119955)|120595(119955)| 119889 119903 (Ih5)

xiii Linear functionals

A functional is called linear if for any two functions 119899(119955) and 119898(119955) one has

119865[119899(119955) + 119898(119955)] = 119865[119899(119955)] + 119865[119898(119955)] and if 119865[120572119899(119955)] = 120572119865[119899(119955)] for any num-

ber 120572 The functionals above 119865119955 [119899] and 119868 [119899] are examples of such functionals

please check Any linear functional can be written in the form

119865[119899] = int119891 (119955)119899(119955)119889 119903

(Ih6)

The function 119891 (119955) is called the linear response kernel of 119865 For example the

linear response kernel of 119865119955 is 119891 (119955) = 120575(119955 minus 119955 ) And that of 119868 is 119891 (119955) =

xiv Functional Derivatives

Let us now introduce the concept of a functional derivative Consider the

functional 119865119955 [119899] of Eqlrm(Ih1) A functional derivative denoted

() measures

how 119865[119899] changes when we make a small localized change of 119899(119955) at

Electron Density Functional Theory Page 40

copy Roi Baer

Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 40

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Let us give a more detailed definition Suppose we change the argument func-

tion in 119865[119899] by making an arbitrarily small change in (119955) rarr 119899(119955) + 120598Φ(119955) We

use an arbitrary function Φ(119955) multiplied by an infinitesimal number 120598 (that

is a number 120598 we intend to make vanishingly small at a later statge) In prin-

ciple for any finite 120598 the change

() in 119865 has a very complicated dependence

on 120598 But since 120598 is assumed small we can Taylor expand with respect to 120598Φ

and then ldquothrow outrdquo all terms beyond the term linear in 120598Φ This makes

sense since we assume that 120598 is a small as we want Thus the linear part of 120575119865

is a functional of 120598Φ and it is a linear functional We denote the linear re-

sponse kernel of this functional by the symbol

() called ldquothe functional de-

rivativerdquo Therefore we can write

120575119865 equiv 119865[119899 + 120598Φ] minus 119865[119899] = int120575119865

120575119899()120598Φ()119889 + 119874(120598 )

(Ih7)

Note that this the linear response kernel

() is independent of Φ(119955) and 120598

One technique to obtain the kernel is taking Φ(119955) = 120575(119955) = 120575(119955 minus ) To

make this notion precise we define

120575119865

120575119899()equiv lim

rarr

119865[119899 + 120598120575] minus 119865[119899]

120598

(Ih8)

Where 120575 = 120575(119955 minus ) is a delta function expressing the fact that we make a

localized change in 119899(119955) at r

Examples

Apply definition lrm(Ih6) to 119865119955 [119899] and 119868 [119899] Find the functional derivatives

and check that Eq lrm(Ih5) holds

Two shortcuts when ldquofunctional integratingrdquo one can often use regular dif-

ferentiation rules All one needs to remember is the following shortcuts

Electron Density Functional Theory Page 41

copy Roi Baer

120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

copy Roi Baer

For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 41

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120575119899(119909)

120575119899(119909 )rarr 120575(119909 minus 119909 )

120575119899prime(119909)

120575119899(119909 )rarr 120575 ( )

(Ih9)

And use them within ldquochain rulesrdquo ( ( ))

( )rarr 119891 (119899(119909))120575(119909 minus 119909 ) (where

119891 (119899) equiv

119891(119899)) An example of the use of this rule is the previous exercise

120575119868 120575119899()

equiv 119881 int120575119899(119955)

120575119899()119889 119903 = 119881 int120575(119955 minus ) 119889 119903 = 119881

(Ih10)

Another example is the following functional related to the kinetic energy

119869[119891] = int 119891 (119909) 119889119909

(Ih11)

The functional derivative is

120575119869[119891]

120575119891()= int

120575(119891 (119909) )

120575119891()119889119909

= int 2119891 (119909)120575119891 (119909)

120575119891()119889119909

= int 2119891 (119909)120575prime(119909 minus )119889119909

= minus2119891 () (Ih12)

Here we used the identity int 119892(119909)120575 (119909 minus )119889119909 = minus119892 () and the shortcut

( )

( )= 120575prime(119909 minus )

xv Invertible function functionals and their functional derivatives

Consider a ldquocontinuous set of functionalsrdquo which is ldquoindexedrdquo by a continu-

ous variable 119955 in some domain 119863 More precisely 119907[119899](119955) assigns a functional

to each 119955 in 119863 Let us suppose further that 119899 is a function of 119955 defined on the

same domain 119863 An example is the relation between the electrostatic potential

and the charge density

119907 [119899](119955) = int119899(119955 )

|119955 minus 119955 |119889 119903

(Ih13)

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 42

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For any charge distribution 119899(119955 ) the integral forms a the electrostatic poten-

tial 119907 (119955) The potential at a give point 119955 depends in principle on the charge

density at all points in space This is a very non-local dependence

What about the functional derivative If one makes a small change in the

function 119899 at the point 119955 120575119899(119955 ) how will that affect the functional at 119955 We

write this as a definition of the functional derivative

120575119907[119899](119955) = int120575119907[119899](119955)

120575119899(119955 )120575119899(119955 )119889 119903

(Ih14)

In the example above it is immediately noticeable that

120575119907 [119899](119955)

120575119899(119955 )=

1

|119955 minus 119955 |

(Ih15)

An interesting concept can arise when one knows in advance that the relation

119907[119899] is unique ie

119907[119899](119955) = 119907[119899 ](119955) for all 119955 hArr 119899(119955) = 119899 (119955) for all 119955

In this case one can also consider 119899 to be a functional of 119907 Indeed

120575119899[119907](119955) = int120575119899[119907](119955)

120575119907(119955 )120575119907(119955 )119889 119903

Once can combine this equation withlrm(Ih15) and see that

120575119899(119955) = int120575119899[119907](119955)

120575119907(119955 )int

120575119907[119899](119955 )

120575119899(119955 )120575119899(119955 )119889 119903 119889 119903

(Ih16)

This shows that

int120575119899(119955)

120575119907(119955 )

120575119907(119955 )

120575119899(119955 )119889 119903 = 120575(119955 minus 119955 )

(Ih17)

Thus there is an inverse relation between the functional derivatives This is

similar to what happens in usual derivatives with inverse functions When a

function 119892(119909) is invertible we can speak of 119909(119892) The change in 119892 due to a

Electron Density Functional Theory Page 43

copy Roi Baer

change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

copy Roi Baer

The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 43

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change in 119909 is 119889119892(119909) =

119889119909 Similarly 119889119909 =

119889119892 Therefore (

) =

(( ( )

)

) ( )

xvi Convex functionals

A function 119891(119909) is said to be convex if for any pair of abscisas 119909 and 119909 the

curve (119909 119891(119909)) described by 119891(119909) in the interval 119909 le 119909 le 119909 is always below

the straight line connecting (119909 119891(119909 )) and (119909 119891(119909 )) A more formal defini-

tion is that for any 0 lt 120582 lt 1 we have

119891(120582119909 + (1 minus 120582)119909 ) le 120582119891(119909 ) + (1 minus 120582)119891(119909 ) (Ih18)

A useful interpretation of the above definition for convexity is in terms of av-

erages We can view 120582119909 + (1 minus 120582)119909 as a weighted average of the two points

with positive weights (such an average is usually called a convex average) In

this sense we have convexity obey

119891(⟨119909⟩) le ⟨119891(119909)⟩ (Ih19)

This relation is much more general than just 2 points We can easily show that

for convex functions it holds for any averaging procedure ⟨119909⟩ = sum 119888 119909 and

⟨119891(119909)⟩ = sum 119888 119891(119909 ) with 119888 ge 0 and sum 119888 = 1 Eq lrm(Ih19) is sometimes called

Jensenrsquos inequality

One of the important implications of a convex function is that it cannot have a

local minimum either there is no minimum (for example 119891(119909) = 119890 ) or there

is just one global minimum If a function is known to be convex and to have a

minimum then any ldquodescentrdquo algorithm is surely to find the (global) mini-

mum This is useful

We can Taylor expand 119891(120582119909 + (1 minus 120582)119909 ) = 119891(120582(119909 minus 119909 ) + 119909 ) with respect to

120582 assuming now 120582 ≪ 1 Then

Electron Density Functional Theory Page 44

copy Roi Baer

120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

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For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

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2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 44

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120582119891(119909 ) + (1 minus 120582)119891(119909 ) ge 119891(120582(119909 minus 119909 ) + 119909 )

= 119891(119909 ) + nabla119891(119909 ) sdot 120582(119909 minus 119909 ) + 119874(120582 ) (Ih20)

After rearrangement and division by 120582 we set 120582 to zero and obtain

119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih21)

This relation if obeyed for all pairs of points is equivalent to the convexity of

the function By exchanging 119909 and 119909 we obtain also

nabla119891(119909 ) sdot (119909 minus 119909 ) ge 119891(119909 ) minus 119891(119909 ) ge nabla119891(119909 ) sdot (119909 minus 119909 ) (Ih22)

From Error Reference source not found we further have for 120582 rarr 0

119891(119909 ) + 120582nabla119891(119909 ) sdot 119906 le 119891(119909 + 120582119906)

= 119891(119909 ) + 120582nabla119891(119909 ) sdot 120575119909 + 120582 1

2120575119909 sdot nablanabla119891(119909 ) sdot 120575119909

+ 119874(120582 |119906| )

(Ih23)

From this after cancelling and dividing by 120582 we find

119906 sdot nablanabla119891(119909 ) sdot 119906 ge 0 for

arbitrary 119906 which means the Hessian nablanabla119891(119909 ) is positive definite another

characteristic of convex functions

A similar definition of convexity can be applied to functionals

119865[120582119899 + (1 minus 120582)119899 ] le 120582119865[119899 ] + (1 minus 120582)119865[119899 ] (Ih24)

And they too have the equivalent differential property which will be useful

below

119865[119899 ] minus 119865[119899 ] ge int(δ119865

120575119899(119955))

(119899 (119955) minus 119899 (119955))119889 119903

(Ih25)

An example for a convex functional useful for density functional theory is the

so-called Hartree energy functional

119864 [119899] =1

2∬

119899(119955)119899(119955 )

|119955 minus 119955 |119889 119903119889 119903

(Ih26)

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

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move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

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moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 45

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The second functional derivative of this function is |119955 minus 119955 | which is a mani-

festly positive matrix (this is easily seen by Fourier transforming the function

119907(119955) =

with result (119954) =

which is positive) Another example is the von-

Weizsacker kinetic energy functional

119864 [119899] = minus1

2intradic119899(119955)nabla radic119899(119955)119889 119903

(Ih27)

The physical meaning of this functional is that it gives 119873 times the kinetic en-

ergy of a single-particle with ground state wave function 120595(119955) = radic (119955)

where

119873 = int119899(119955)119889 119903 Again by showing the Hessian is positive definite we can

straightforwardly show this functional is convex First rewrite it as

119864 [119899] =1

2int(nablaradic119899(119955))

119889 119903 =

1

8int

(nabla119899)

119899119889 119903 (Ih28)

We drop the explicit 119955 in the last expression for brevity Then make an arbi-

trary but small perturbation 120598Φ(119955)

119864 [119899 + 120598Φ] =1

8int

(nabla119899 + 120598nablaΦ)

119899 + 120598Φ119889 119903 (Ih29)

And expand the denominator to second order in 120598 (119899 + 120598Φ) rarr

(1 minus 120598

+

120598 (

) + ⋯) Expanding the numerator and performing the multiplication

order by order one can then read off the second order contribution and after

some manipulation bring it to the form

120598 120575 119864 [119899 + 120598Φ] =120598

8int

Φ

119899(nabla119899

119899minus

nablaΦ

Φ)

119889 119903 (Ih30)

Clearly this second order term is absolutely non-negative for all perturba-

tions and at all physical densities This shows that the underlying functional

derivative Hessian is positive definite and thus the functional is convex

Electron Density Functional Theory Page 46

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A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 46

copy Roi Baer

A similar condition exists for the second functional derivative The Jensen in-

equality holds here as well

119865[⟨119899⟩] le ⟨119865[119899]⟩ (Ih31)

This relation in combination with the fact that 119891(119909) = 119890 is convex is useful

for developing mean field approaches in Statistical mechanics It was used by

Gibbs and later in a quantum context by Peierls[9]

i) Minimizing functionals is an important part

of developing a DFT

xvii Minimization of functions

We start with the problem of minimization of a simple 1D function Given a

function 119891(119909) we want to find a point 119909 such that the function is minimal It

is clear that the slope of 119891(119909) at 119909 = 119909 must be zero If the slope is positive

then we can go left (decrease 119909 ) and reduce 119891(119909) Same logic (but to right) if

it was negative Thus a necessary condition for a minimum is

119891 (119909 ) = 0 (Ii1)

Let us expand in a Taylors series the function around the point 119909 Clearly we

have

119891(119909) = 119891(119909 ) + 119891 (119909 )(119909 minus 119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii2)

However taking Eq Error Reference source not found into account

119891(119909) = 119891(119909 ) +1

2119891 (119909 )(119909 minus 119909 ) + ⋯ (Ii3)

When 119909 is extremely close to 119909 we may neglect high order terms and then we

see that Eq Error Reference source not found is the equation of a parabola

In order that 119891(119909) be a minimum we must have an ascent of 119891(119909) when we

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 47

copy Roi Baer

move away from 119909 no matter which direction we take (left or right) For

small displacements 119909 rarr 119909 we see from Eq

Error Reference source not found that the change in 119891(119909) is

119891 (119909)(119909 minus

119909 ) Since (119909 minus 119909 ) is always positive no matter the direction we move we

must demand that 119891 (119909 ) gt 0 as well Thus our necessary conditions for a

minimum are

119891 (119909 ) = 0 119886119899119889 119891 (119909 ) gt 0 (Ii4)

Now let us consider the case of functions of two variables 119891(119955) where

119955 = (119909119910) equiv (119909 119910) Notice that we use the transpose symbol 119879 to turn a col-

umn vector into a row vector Let us Taylor expand 119891(119955) around a point 119955

119891(119955) = 119891(119955 ) + nabla119891(119955 ) (119955 minus 119955 ) +1

2(119955 minus 119955 ) nablanabla119891(119955 )(119955 minus 119955 ) + ⋯

= 119891(119955 ) + 119944 (119955 minus 119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯

(Ii5)

Note that we use the notations

119944 = 119944(119955 ) = nabla119891(119955 ) =

(

120575119891

120575119909120575119891

120575119910)

119867 = 119867(119955 ) =

(

120575 119891

120575119909

120575 119891

120575119910120575119909

120575 119891

120575119909120575119910

120575 119891

120575119910 )

119955

(Ii6)

The vector 119944(119955) is called the gradient and the matrix 119867(119955) the Hessian at 119955

Note that the Hessian is a symmetric matrix since the order of differentiation

is immaterial When 119955 is a minimum moving infinitesimally away from it in

any direction will not change the function Why We can show this leads to a

contradiction Let 119941 be an arbitrary direction If the function decreases when

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 48

copy Roi Baer

moving from 119955 in that direction no matter how small the step size then this

contradicts that 119955 is a minimum Now if 119941 is an ascent direction then minus119941

will be a descent direction - again a contradiction to 119955 being a minimum

Thus we conclude that 119891 cannot change (to first order) or in other words the

gradient 119944 of 119891(119955) at 119955 must be zero Then Eq

Error Reference source not found becomes

119891(119955) = 119891(119955 ) +1

2(119955 minus 119955 ) 119867 (119955 minus 119955 ) + ⋯ (Ii7)

Continuing in order for this function to have a minimum at 119955 the second

term on the right hand side must always be positive Thus Hessian matrix

must be such that for any vector ne 119907 119919119907 gt 0 A matrix having this property

is called positive definite (PD) When a PD matrix is symmetric then its ei-

genvalues must be strictly positive We can summarize the necessary condi-

tions for a minimum at 119955

119944 = nabla119891(119955 ) = 0 119886119899119889 119919 119894119904 119875119900119904 119863119890119891 (Ii8)

As an example let us take the function 119891(119909) = 119909 + 119910 This function is always

non-negative so its minimum is at 119909 = 119910 = 0 where it obtains the value 0

The gradient is indeed zero at (119909 119910) = (00)

(120597119891

part119910)( )

= (2119910)( ) = 0 (Ii9)

And the Hessian is 2119868 where 119868 is the 2times2 identity matrix thus it is trivially

PD

xviii Constrained minimization Lagrange multipliers

Next we discuss constrained minimization Suppose we want to minimize

119891(119909 119910) under a constraint that 119910 = 119892(119909) This is ldquoeasyrdquo since it can be trans-

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

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Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 49

copy Roi Baer

formed into an unconstrained minimization All we have to do is minimize

119865(119909) = 119891(119909 119892(119909)) This will give

119865 (119909) = (120597119891

120597119909)( ( ))

+ (120597119891

120597119910)( ( ))

119892 (119909)

119865 (119909) = (120597 119891

120597119909 )( ( ))

+ 2(120597 119891

120597119909120597119910)( ( ))

119892 (119909) + (120597119891

120597119910)( ( ))

119892primeprime(119909)

+ (120597 119891

120597119910 )( ( ))

119892 (119909)

(Ii10)

And we can then search for 119909 such that 119865 (119909 ) = 0 and 119865 (119909 ) gt 0

But sometimes it is not possible to write the constraint directly as 119910 = 119892(119909) A

more general form is ℎ(119909 119910) = 0 This is also more symmetric Consider the

situation of minimizing 119891(119909 119910) under the constraint ℎ(119909 119910) = 0 Suppose the

point depicted in the following figure is this minimum

This means that moving slightly along the isopleth of ℎ = 0 from 119955 will not

change 119891 if it decreases 119891 this is not a minimum and if it increases 119891 we will

move in the opposite direction and 119891 will decrease

h=1

h=0 h=-1

nabla119891

nabla119891

nablaℎ

nablaℎ

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

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1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 50

copy Roi Baer

Thus at 119955 the gradient of 119891 nabla119891(119955 ) must be normal to the line ℎ = 0 Fur-

thermore by definition the gradient of ℎ is normal to its isopleth Thus the

necessary condition for minimum is that both vectors point to the same direc-

tion and so are proportional We denote the proportionality constant by 120582

nabla119891(119955 ) = 120582 nablaℎ(119955 )

ℎ(119955 ) = 0 (Ii11)

Lagrange noticed that this relation and wrote a Lagrangian for the constrained

problem

119871(119955 120582) = 119891(119955) minus 120582ℎ(119955) (Ii12)

The new variable 120582 is called a Lagrange multiplier The necessary conditions

for a minimum in Eq Error Reference source not found translate into the

condition of finding a stationary point of 119871

nabla119871(119955 120582 ) = 0 (Ii13)

Where nablaequiv (120597 120597 120597 120597 ) The derivative with respect to 120582 gives us back the

constraint As an example consider minimizing 119891(119909 119910) = 119909 + 119910 under the

constraint ℎ(119909 119910) = 119909 + 119910 minus 1 (ie find the point on the unit circle for which

119909 + 119910 is minimal) We have

119871(119955 120582) = (119909 + 119910) minus 120582(119909 + 119910 minus 1) (Ii14)

The necessary condition is

nabla119871 = (1 minus 2120582 119909 1 minus 2120582 119910 minus(119909 + 119910 minus 1)) = (000)

rArr 119909 = 119910 =1

2120582 2 (

1

2120582 )

= 1 (Ii15)

The solution is thus

Electron Density Functional Theory Page 51

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120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 51

copy Roi Baer

120582 = plusmn1

radic2 119909 = 119910 = plusmn

1

radic2

(Ii16)

Since these are necessary conditions we need to consider them further (they

might correspond to a maximum) The minimum corresponds only to the

negative solution

The method of Lagrange multipliers is often more convenient to work with

than direct replacement The problem is thus that of finding a minimum of

119891(119955 hellip 119955 ) under the 119872 lt 119873 constraints 119945(119955 hellip 119955 ) = or ℎ (119955 hellip 119955 ) = 119888

119894 = 1hellip 119872 is

find 119955 such that 119945(119955 ) = for which ∶ 119891(119955 ) le 119891(119955)

for any 119955 such that 119945(119955) = (Ii17)

To facilitate such a search we formulate the Lagrangian function

119871(119955 120640 ) = 119891(119955) minus sum120582 (ℎ (119955) minus 119888 )

(Ii18)

Our plan is to find the position of 119955 which minimizes 119871 for any choice of 120640 and

then change until 119945(119955) = At 120640 we have 119871 assuming a minimum at a point

119955 A necessary condition for the constrained minimum to be achieved at the

point 119955 and with the Lagrange multipliers 120640 is

nabla 119871( 120524 ) = 0

nabla120524119871(119955 120640 ) = 119945(119955 ) minus = 0

(Ii19)

Note that 120640 is not necessarily a minimizer of 119871 In fact the opposite is tue 120640

is a maximizer of 119871 and the search for constrained minimizations is a search

for saddle points (see ref [10] for a method to solve such a problem on the

large scale)

Electron Density Functional Theory Page 52

copy Roi Baer

It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 52

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It is interesting now to ask how 119891(119955) changes if we change the value of the

constraint 119888 Indeed when the constraints are changed the optimized point

and Lagrange multiplier can change so the Lagrangian is changed

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 ) (Ii20)

Now because of Eq lrm(Ii19) we find

120575119871(119955 120640 ) = 119871(119955 + 120575119955 120640 + 120575120640 + 120575 ) minus 119871(119955 120640 )

= 119871(119955 120640 + 120575 ) minus 119871(119955 120640 ) (Ii21)

Thus

119889

119889119888 119871(119955 120582 ) =

120575

120575119888 119871(119955 120582 ) = 120640

(Ii22)

Since 119871(119955 120640 ) = 119891(119955 ) we find

119889

119889119888 119891(119955 ) = 120640

119900119903 nabla 119891(119955 ) = 120640

(Ii23)

This equation reveals the ldquomeaningrdquo of the Lagrange multipliers 120582 at the op-

timal point they are equal to the rate at which the optimal value of the mini-

mized function 119891 changes when 119888 the value of the 119894 constraint is changed

This is an important result which we use below whenever we want to give

physical significance to Lagrange multipliers

xix Minimization of functionals

The same considerations for functions apply for functionals Given a func-

tionalI f a necessary condition for its minimum is

120575119868

120575119891(119955)= 0

(Ii24)

For example consider a 1D classical particle of mass 119899 in a potential well 119907(119909)

The action 119878 is a functional of the trajectory of this particle

Electron Density Functional Theory Page 53

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119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 53

copy Roi Baer

119878[119909] = int [1

2119898(119905) + 119907(119909(119905))] 119889119905

(Ii25)

For any trajectory 119909(119905) between times 119905 and 119905 119871[119909] returns a number La-

grange showed that finding the trajectory that makes 119871[119909] stationary (alt-

hough not necessarily minimal) under the condition that 119909(119905 ) = 119909 and

119909(119905 ) = 119909 are given and thus not varied is equivalent to solving Newtons

equations under these same constraints The functional differentiation of the

kinetic energy is performed with

120575

120575119909(119905)int (119905 ) 119889119905

= 2int (119905 )120575(119905 )

120575119909(119905)119889119905

= 2int (119905 )(119905 minus 119905 )119889119905

= minus2int (119905 )120575(119905 minus 119905 )119889119905

= minus2(119905) (Ii26)

The condition for stationarity of the action under changes in the trajectory

which leave the edges intact give

0 =120575119878

120575119909(119905)= minus119898(119905) minus 119907 (119909(119905))

(Ii27)

From which we obtain Newtons equation 119898(119905) = minus119907 (119909(119905)) This equation

must be solved subject to the given constraints at the endpoints ie 119909(119905 ) = 119909

and 119909(119905 ) = 119909

Minimizing functionals with constraints can again be done using Lagrange

multipliers Then one defines

119871[119891 120582] = 119868[119891] + 120582ℎ[119891] (Ii28)

And the necessary conditions are

120575119871

120575119891(119955)|

= 0 120575119871

120575120582|

= ℎ[119891 ] = 0 (Ii29)

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 54

copy Roi Baer

For example let us solve the following problem Find the shape in a plane of a

closed contour encircling maximal area under the constraint of a given cir-

cumference 119897 We need to find the contour given by 119903(120579) with 119903(120579 + 2120587) =

119903(120579) which gives zero variation to

119871[119903 120582] = 119878[119903] + 120582(119871[119903] minus 119897 )

Where 119878 is the area and 119871 is the circumference We limit ourselves to curves

that can be uniquely defined A point on the curve is defined by the angle 120579

and the distance from the origin 119903(120579) The vector to a point on the curve and

its derivative is

119955(120579) = 119903(120579)(cos 120579 sin 120579)

119955 (120579) = 119903 (120579)(cos 120579 sin 120579) + 119903(120579)(minussin 120579 cos 120579)

The area of a small arc from 120579 to 120579 + 119889120579 is 119889119878 =

|119955(120579) times 119955(120579 + 119889120579)| =

|119955(120579) times 119955prime(120579)|119889120579 Notice that |119955 times 119955 | = 119903

and so 119889119878 =

119903(120579) 119889120579

The square circumference is 119889119871 = (119955(120579 + 119889120579) minus 119955(120579)) = (119955 (120579))

119889120579 Thus

119889119871 = |119955 (120579)|119889120579 = radic119903(120579) + 119903 (120579) 119889120579 Thus the area is a functional is

119878[119903] = int1

2119903(120579) 119889120579

119871[119903] = int radic119903(120579) + 119903 (120579) 119889120579

The functional derivative of S is easy

120575119878[119903]

120575119903(120579)= 119903(120579)

The functional derivative of 119871 is computed by adding a ldquosmallrdquo function 120598(120579)

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 55

copy Roi Baer

2 2 2

02

2 2

0

22 2

2 20

2 2

2 21

l r r d l

r r r r d l

r rr r d l

r r

Then we can use 211 1

2O so

2

2 20

r rl d

r r

Integrating by parts the second term (remembering that the end terms drop

because 120579 = 2120587 is the same point as 120579 = 0

120575119897 = int119903120598

radic119903 + 119903 119889120579

minus int (119903

radic119903 + 119903 )

120598119889120579

We thus get

120575119897

120575119903(120579)=

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

We define the Lagrangian

119871[119903] = 119878[119903] minus 120582(119897[119903] minus 119897 )

And get the maximum from

0 =120575119871

120575119903(120579)= 119903 minus 120582 (

119903

radic119903 + 119903 minus (

119903

radic119903 + 119903 )

)

Clearly a solution is 119903(120579) = 120582 a constant since then all derivatives are zero

The curve is then circle of radius 120582 and given the circumference we have

120582 = 119897 2120587

EXERCISES FOR CHAPTER 1

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital

Electron Density Functional Theory Page 56

copy Roi Baer

1) Consider a system of two particles in a harmonic well Each particle repels the other

The potential energy is 119881(119909 119909 ) =

119896119909

+

119896119909

minus

119902(119909 minus 119909 )

We assume that

the system is bound (so 0 lt 2119902 lt 119896) (a) Write down the exact Hamiltonian for this system assuming the mass are 119898 and 119898 (b) Make a transformation separating

the Hamiltonian into two uncoupled particles the center of mass 119883 =

and the reduced mass 119909 = 119909 minus 119909 Write the Hamiltonian of each part and deter-mine the mass of each particle What are the masses when 119898 ≫ 119898 (c) Write down the energy levels of the system and the eigenfunctions (d) Now make a Born-Oppenheimer approximation Assuming 119898 ≫ 119898 determine the Born Oppenhei-mer potential surface (obtained when 119909 is considered a parameter) Solution The Hamiltonian is

119867 = minusℏ

2119898 120597 minus

ℏ

2119898 120597 +

1

2119896119909

+1

2119896119909

minus1

2119902(119909 minus 119909 )

In the new coordinates

119883 =(119898 119909 + 119898 119909 )

119872 119909 = 119909 minus 119909 119909 = 119883 minus

119898

119872119909 119909 = 119883 +

119898

119872119909

If 119891(119909 119909 ) is any function than we can define the ldquosamerdquo scalar function as

119865(119909 119883) = 119891 (119883 minus

119909 119883 +

119909)

2) Calculate the correlation in the case of a singlet where both electrons are in the ground state orbital and when one is in the ground state orbital and the other in the first excited orbital