Part I: Elements of Functional Analysis · The functional in Example B is the total number of...

DefinitionsExample: Thomas-Fermi model of Atoms

Part I: Elements of Functional Analysis

Tomasz A. Wesolowski, Universite de Geneve

EPFL, Spring 2016

Tomasz A. Wesolowski, Universite de Geneve Part I: Elements of Functional Analysis


Functional

By a functional, we mean a correspondence which assigns a definite(real) number to each function (or a curve) belonging to some class.I.M. Gelfand and S.V. Fomin, ”Calculus of Variations” Prentice-Hall, Inc. 1963, page 1

Examples:

A[f ] = max f (x , y , z)

B[f ] =∫f (x , y , z)dxdydz

C [f ] =∫f 2(x , y , z)dxdydz

D[f ] = 12

∫ ∫ f (x,y ,z)f (x′,y ′,z′)[(x−x′)2+(y−y ′)2+(z−z′)2]1/2 dxdydzdx

′dy ′dz ′



Functional

In this lecture, mainly the functionals of the electron density denotedwith ρ will be considered i.e. f (x , y , z) = ρ(x , y , z).The functional in Example B is the total number of electrons

N =

∫ρ(x , y , z)dxdydz

The functional in Example D can be identified with the Coulomb integral(classical electron-electron repulsion).



Functional derivative

F [ρ+ δρ]− F [ρ] =∫δρ δF [ρ]

δρ dr + O(δρ)2

where δρ is an infinitesimally small function and O(δρ)2 denotesall the terms proportional to the higher powers of δρ (quadratic,cubic, etc.).The linear term in F [ρ+ δρ]− F [ρ] is called variation of thefunctional F and is denoted as δF .

δF =∫δρ δF [ρ]

δρ dr



Functional derivative: example

For F [ρ] =∫ρdr:

F [ρ+ δρ]− F [ρ] =

∫(ρ+ δρ) dr −

∫ρdr =

∫1 · δρdr

its functional derivative is:

δF [ρ]

δρ(r)= 1




For F [ρ] =∫v(r)ρ(r)dr:

F [ρ+ δρ]− F [ρ] =

∫v(r) (ρ(r) + δρ(r)) dr −

∫v(r)ρ(r)dr

=

∫v(r) · δρ(r)dr


δF [ρ]

δρ(r)= v(r)




For F [ρ] =∫ρk(r)dr:

F [ρ+ δρ]− F [ρ] =

∫(ρ(r) + δρ(r))k r −

∫ρk(r)dr

=

∫kρk−1(r) · δρ(r)dr + O(δ2ρ)


δF [ρ]

δρ(r)= kρk−1(r)



Functional derivative: exercise

Show that for the functional F [ρ] =∫ |∇ρ|2

ρ drits functional derivative is:

δF [ρ]

δρ(r)= −2

∇2ρ

ρ+|∇ρ|2

ρ2



Extremum of a functional

It is such a function ρ0 for which the functional F [ρ] is extremal(minimum or maximum).The necessary condition for the differentiable functional F [ρ] tohave an extremum for ρ(x , y , z) = ρ0(x , y , z) is that its variationvanish for ρ(x , y , z) = ρ0(x , y , z).

δF = 0

This condition can be expressed using the introduced definition ofthe functional derivative taking the form known as Euler equation:

δF [ρ]δρ(r) = 0

(Note the analogy with extremum of a function in elementarycalculus.)



Euler-Lagrange Equations

The function ρ might be subject of additional constraints of thegeneral form:

C [ρ] = 0

Example: The condition that the integral of the function ρ mustbe equal to a given number (the total number of electrons) can bewritten as:

C [ρ] =

∫ρ(r)dr − N = 0.0

The extremum of the functional F [ρ] in the presence of constraintsC [ρ] can be obtained from the Euler-Lagrange equation:

δF [ρ]δρ(r) − µ

δC [ρ]δρ(r) = 0

where µ is a constant to be found (Lagrange multiplier).Tomasz A. Wesolowski, Universite de Geneve Part I: Elements of Functional Analysis


Historical Introduction

In the 1920’s, Thomas and Fermi realized that statisticalconsideration can be used to approximate distribution of electronsin an atom. The model called now the Thomas-Fermi model is thefirst theory in which the description of an atom does not use thewavefunction Ψ(x1, x2, ..., xN) but it uses a much simpler quantity- the electron density ρ(r). The Thomas-Fermi model is of littlepractical use. It has serious known mathematical flaws. Theaccuracy of the obtained energies is not sufficient. TheThomas-Fermi model will be presented here because it can be seenas the simplest Density Functional Theory, introducing the sameconcepts and ideas as the more advanced contemporary theories.



Kinetic energy in non-interacting uniform gas of electrons (1)

We recall at first the exact solution for the energy levels of theparticle in a three-dimensional infinite well:

ε(nx , ny , nz) =h2

8ml2(n2x + n2

y + n2z

)=

h2

8ml2R2

where l is the length of the box and nx , ny , nz are integers (1, 2,3,...).The number of the energy levels below a given level ε equals to thenumber of such combinations of nx , ny , nz which satisfy:

n2x + n2

y + n2z ≤ ε

8ml2

h2




For sufficiently large l , the number of states satisfying the aboveinequality (φ(ε)) can be approximated by the volume of one octantof the sphere with radius R.

φ(ε) =1

8

(4πR3

3

)=π

6

(8ml2ε

h2

)3/2

As a consequence, the number of the energy states of the energiesbetween ε+ ∆ε and ε can be expressed as:

g(ε)∆ε = φ(ε+ ∆ε)− φ(ε)

=π

4

(8ml2ε

h2

)3/2

ε−1∆ε




To obtain the total energy, Thomas and Fermi introduced thestatistical gas theory. For electrons, the probability of the state ofa given energy (ε) to be occupied at a given temperature (T,β = 1

kT ) is known as the Fermi-Dirac distribution (f (ε)):

f (ε) =1

1 + exp[β(ε− µ)]

At zero temperature, the Fermi-Dirac distribution reduces to astep function:

f (ε) = 1 for ε < εF

f (ε) = 0 for ε > εF

where εF is called Fermi energy.




The total energy of the electrons can be obtained as the integralover energies from 0 to εF and multiplying by 2 as each state canbe occupied by 2 electrons with opposite spins:

E = 2

∫εf (ε)g(ε)dε = 4π

(2m

h2

)3/2

l3∫ εF

0ε3/2dε

=8π

5

(2m

h2

)3/2

l3εF5/2

We notice now that the total number of electrons in theconsidered three-dimensional infinite well can be also expressedusing f (ε) and g(ε):

N = 2

∫g(ε)f (ε)dε =

8π

3

(2m

h2

)3/2

l3εF3/2




Combining the two equation for N and for E by eliminating εFleads to the relation between the energy (which equals to thekinetic energy in the considered example) and the number ofelectrons in the well:

E = T =3

5NεF =

3h2

10m

(3

8π

)2/3

l3(N

l3

)5/3

Since l3 is the volume of the well (V), the above equation relatesthe density of the kinetic energy tk = T

V with the density of the

electron density ρ = NV .

tk =3h2

10m

(3

8π

)2/3

ρ5/3



Local Density Approximation for kinetic energy functional (1)

The whole space can be formally divided into small cells such thatin which cell the electron density can be considered to be constant.For each cell the relation tk = 3h2

10m ( 38π )2/3ρ5/3 holds. Therefore,

the total kinetic energy of a slowly-varying electron density can beobtained by integration over all cells:

T [ρ] ≈ TTF [ρ] = CF

∫ρ5/3(r)dr

where CF = 310 (3π2)2/3 = 2.871 in atomic units.



Local Density Approximation for kinetic energy functional (2)

The approximation in which the total energy is calculated as a sum(integral) over volume elements such that in each of them theelectron density is considered to be uniform is known as LocalDensity Approximation, LDA in modern density functionaltheory. Local Density Approximation is used not only to the kineticenergy functional but also to other functionals as well.



Thomas-Fermi model of atoms (1)

The introduction of the kinetic energy functional by Thomas andFermi, was a fundamentally new concept. The electronic kineticenergy is obtained in conventional quantum mechanics as anexpected value of the kinetic energy operatorT = (Ψ|

∑Ni [−1

2∇2i ]|Ψ) requiring the knowledge of the

wavefunction (Ψ) which as a complex function of 3N variables fora system with N electrons. In the Thomas-Fermi model, theelectronic kinetic energy is expressed as a functional of the electrondensity (ρ(r) which is a real function of 3 variables independent onthe number of electrons in the system. The underlying it LocalDensity Approximation can not be expected to be good forchemical molecules in which the electron density is characterizedby such features as cusps at nuclei and exponential behaviour farfrom them.




Applying the Thomas-Fermi functional to the energy expression ofan atom leads to the approximate functional of the total energy:

ETF [ρ(r)] = CF

∫ρ5/3dr − Z

∫ ρ(r)r dr + 1

2

∫ ∫ ρ(r)ρ(r′)|r−r′| drdr

′

The Thomas-Fermi approximate functional leads to no-nonsenseenergies of atoms, but it has a number of mathematical flaws suchas wrong asymptotic behaviour of the electrostatic potential. It isnot applicable for chemical problems because: this model predictsthat the chemical molecules are always unstable.




For the Thomas-Fermi energy functional, the Euler-Lagrangeequation reads:

δ(ETF [ρ]− µTF

(∫ρ(r)dr − N

))δρ(r)

= 0

leading to

µTF =5

3CTFρ

2/3(r)− φ(r) where φ(r) =Z

r−∫

ρ(r′)

|r − r′|dr

The function ρTF (r) solving the above equation has qualitativelywrong properties:

ρTF (r) ∝ 1r3/2 for r → 0 and ρTF (r) ∝ 1

r6 for r →∞



Kinetic energy as density functional: exercise

Follow the derivation of the Thomas-Fermi functional to obtain thedensity functional for the kinetic energy in:

a) one-dimensional uniform electron gas;b) two-dimensiomal uniform electron gas.


Part I: Elements of Functional Analysis · The functional in Example B is the total number of...

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