Quantum Mechanics: Density Functional Theory and Practical Application to Alloys

Quantum Mechanics: Density Functional Theory and Practical

Application to AlloysStewart Clark

Condensed Matter SectionDepartment of PhysicsUniversity of Durham

Introduction to Computer Simulation: Edinburgh, May 2010

Outline

• Aim: To simulate real materials and experimental measurements

• Method: Density functional theory and high performance computing

• Results: Brief summary of capabilities and performing calculations


What would we like to achieve?

• Computers get cheaper and more powerful every year.

• Experiments tend to get more expensive each year.

• IF computer simulation offers acceptable accuracy then

at some point it should become cheaper than experiment.

• This has already occurred in many branches of science

and engineering.

• Possible to achieve this for properties of alloys?


Property Prediction

•Property calculation of alloys provided link with experimental measurements:

- For analysis- For scientific/technological interest

•To enable interpretation of experimental results•To predict properties over and above that of experimental measurements

Aim of ab initio calculations


Atomic Numbers

Solve quantum mechanics for the material

Predict physical and chemical properties of systems


From first principles

Researchoutput

The equipment Application

“BaseTheory”(DFT)

Implementation(the algorithmsand program)

Setup model,run the code

Scientificproblem-solving

“AnalysisTheory”


Properties of materials

• Whole periodic table.• Periodic units containing thousands of atoms

(on large enough computers).• Structural optimisation (where are the

atoms?).• Finite temperature (atomic motion).• Lots of others…if experiments can measure it,

we try to calculate it – and then go further…• Toolbox for material properties


The starting point

€

ˆ H Ψ = EΨ

As you can see, quantum mechanics is “simply” an eigenvalue problem


Set up the problem

Let’s start defining various quantitiesAssume that the nuclei (Mass Mi) are at: R1, R2, …, RN

Assume that the electrons (mass me) are at:r1, r2, …, rm

Now let’s put some details in the SE

€

ˆ H Ψ R1,R2,...,RN ,r1,r2,...,rm( ) = EΨ R1,R2,...,RN ,r1,r2,...,rm( )


Summary of problem to solve

€

ˆ H N ,eΨN ,e RI{ }, ri{ }( ) = EN ,eΨN ,e RI{ }, ri{ }( )

Where

€

ˆ H N ,e = ˆ T N + ˆ T e + ˆ V N −N + ˆ V N −e + ˆ V e −e


The full problem

€

−h2

2∇I

2

M II∑ +

∇ i2

mei∑

⎛

⎝ ⎜

⎞

⎠ ⎟+

14πε 0

12

e2

ri − rj

−ZIe

2

ri − RIi,I∑

i≠ j∑ +

12

ZI ZJe2

RI − RJI ≠J∑

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

Ψ R1,R2,...,RN ,r1,r2,...,rm( ) = EΨ R1,R2,...,RN ,r1,r2,...,rm( )

•Why is this a hard problem? •Equation is not separable: genuine many-body problem•Interactions are all strong – perturbation won’t work•Must be Accurate --- Computation


Model Systems

In this kind of first-principles calculation– Are 3D-periodic– Are small: from one atom to a few

thousand atoms Supercells Periodic boundaries Bloch functions

Bulk alloy Slab for surfaces


First simplification

• The electron mass is much smaller than the nuclear mass

• Electrons remain in a stationary state of the Hamiltonian wrt nuclear motion

• Nuclear problem is separable (and, as we know, the nucleus is merely a point charge!)

€

ˆ H N ,eΨN ,e RI{ }, ri{ }( ) = ΘN ,e ri{ }( )ΦN ,e RI{ }( )


Electrons are difficult!

• The mathematical difficulty of solving the Schrodinger equation increases rapidly with N

• It is an exponentially difficult problem• The number of computations scales as eN

• With modern supercomputers we can solve this directly for a very small number of electrons (maybe 4 or 5 electrons)

• Materials contain of the order of 1026 electrons


Density functional theory

• Let’s write the Hamiltonian operator in the following way:

– T is the kinetic energy terms– V is the potential terms external to the electrons– U is the electron-electron term

• so we’ve just classified it into different ‘physical’ terms€

ˆ H = ˆ T + ˆ V + ˆ U


The electron density

• The electronic charge density is given by

• so integrate over n-1 of the dimensions gives the probability, n(r), of finding an electron at r

• This is (clearly!) a unique functional of the external potential, V

• That is, fix V, solve SE (somehow) for Q and then get n(r).

€

n(r) = ... Θ* r1,r2,...,rn( )∫∫∫ Θ r1,r2,...,rn( )dr2dr3 ...drn


DFT

• Let’s consider the reverse question: for a given n(r), does this come from a unique V?

• Can two different external potentials, V and V’, give rise to the same electronic density?


Method behind DFT• Assume two potentials V and V’ lead to the

same ground state density:

• We can do the same again interchanging the dashed and undashed quantities thus:€

E = Ψ | H | Ψ< Ψ'| H | Ψ' = Ψ'| H '+V −V ' | Ψ'= Ψ'| H ' | Ψ' + Ψ'|V −V ' | Ψ'= E '+ Ψ'|V −V ' | Ψ'

€

E ' < E + Ψ |V '−V | Ψ


Unique potential

• If we add these two final equations we are left with the contradiction

• so our initial assumption must be incorrect• That is, there cannot be two different external

potentials that lead to the same density• We have a one-to-one correspondence between

density, n(r), and external potential, V(r).€

E '+E < E + E '


Change of emphasis in QM

• But by the definition of the lowest energy state we must have

• And so the ‘variational principle’ tells us how to solve the problem

€

Ψ' | H | Ψ' > Ψ | H | ΨΨ'| F | Ψ' + Ψ'|V | Ψ' > Ψ | F | Ψ + Ψ |V | Ψ

n'(r)V (r)dr + F n'(r)[ ]∫ > n(r)V (r)dr + F n(r)[ ]∫E n'(r)[ ] > E n(r)[ ]


Don’t bother with the wavefunction!

• Express the problem as an energy

• And solve variationally with respect to the density

€

E n[ ] = Ts n[ ] + E H n[ ] +V n[ ] + Exc n[ ]

Degrees of freedomin the density, n, versus energy E[n]


QM using DFT

€

−h2

2me

∇2 +12

e2

4πε 0

1ri − rj

−e2

4πε 0

ZI

ri − RIi,I∑

i≠ j∑ +

12

e2

4πε 0

ZI ZJ

RI − RJI ≠J∑

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Θ r1,r2,...( ) = EΘ r1,r2,...( )

€

n exact r( ) = ... Θ* r1,r2,r3,...( )Θ r1,r2,r3,...( )dr2dr3 ...∫∫

N-body Schrödinger Equation

€

−h

2me

∇ i2 +

12

e2

4πε 0

n(r')r − r'

dr' −e2

4πε 0

ZI n(r')r' −RI

dr'+12

e2

4πε 0

ZI ZJ

RI − RJI ≠J∑∫

I∑∫ + μ xc n r( )[ ]

⎛

⎝ ⎜

⎞

⎠ ⎟φi r( ) = ε iφi r( )

€

nDFT r( ) = φi* r( )φi r( )

i∑

€

n exact r( ) = nDFT r( )

Density functional theory (Kohn-Sham equations)

Both equations


Kohn-Sham Equations

• Let’s collect all the terms into one to simplify

• Where– i labels each particle in the system– Vi

KS is the potential felt by particle i due to n(r)– n(r) is the charge density

€

−12

∇ i2 +Vi

KS n r( )[ ] ⎡ ⎣ ⎢

⎤ ⎦ ⎥ψ i r( ) = ε iψ i r( )


Kohn-Sham Equations

• The Kohn-Sham (KS) equations are formally exact

• The KS particle density is equal to the exact particle density

• We have reduced the 1 N-particle problem to N (coupled) 1-particle problems

• We can solve 1-particle problems!


Variational Method

Schrödinger’s Equation And the of use the Variational Principle

ˆ H ΨEΨ

E Ψ ˆ H ΨΨ Ψ

Minimising this

within DFT

Solve this

by


DFT: The XC approximation• Basically comes from our attempt to map 1 N-body

QM problem onto N 1-body QM problems• Attempt to extract single-electron properties from

interacting N-electron system• These are quasi-particles

“DFT cannot do…” : This statement is dangerous and usually ends incorrectly (in many publications!)

Should read:

“DFT using the ??? XC-functional can be used to calculate ???, but that particular functional introduces and error of ??? because of ???


Definition of XC

Exact XC interaction is unknown

∫∫ − '

|'|)',()(

21][ drdr

rrrrnrnnE xc

xc

This would be excellent if only we knew what nxc was!

This relation defines the XC energy.

It is simply the Coulomb interaction between an electron an r and the value of its XC hole nxc(r,r’) at r’.

Within DFT we can write the exact XC interaction as


Exchange-Correlation Approximations

∫ drrnrnrnE xcLDAxc )]([)()]([ homε

∫ ∇ drrnrnrnrnE GGAxc

GGAxc )](),([)()]([ ε

A simple, but effective approximation to the exchange-correlation interaction is

The generalised gradient approximation contains the next term in a derivative expansion of the charge density:


Hierarchy of XC appoximations

• LDA depends only on one variable (the density).• GGA’s require knowledge of 2 variables (the density and

its gradient).• In principle one can continue with this expansion.• If quickly convergent, it would characterise a class of

many-body systems with increasing accuracy by functions of 1,2,6,…variables.

• How fruitful is this? As yet, unknown, but it will always be semi-local.


Zoo of XC approximations

LDA

Semi-Empirical

RPBE

WC

WDA

CIPW91

PBE

EXXsX

CC

B3LYP

PBE0

Meta-GGA

HF

MP2

MP4

SDA

OEP


Structure Determination– Minimum energy corresponds to zero force– Much more efficient than just using energy

alone– Equilibrium bond lengths, angles, etc.– Minimum enthalpy corresponds to zero

force and stress– Can therefore minimise enthalpy w.r.t.

supercell shape due to internal stress and external pressure

– Pressure-driven phase transitions


Nuclear Positions?

• Up until now we assume we know nuclear positions, {Ri}

• What if we don’t?• Guess them or take hints from experiment• Get zero of force wrt {Ri}:

€

F = −∇R E(R)

€

E = φ H φ


Forces• If we take the derivative then:

€

∂ E∂R

=∂

∂Rφ H φ

=∂φ∂R

H φ + φ∂H∂R

φ + φ H∂φ∂R

= E∂φ∂R

| φ + φ |∂φ∂R

⎧ ⎨ ⎩

⎫ ⎬ ⎭+ φ

∂H∂R

φ

= E∂

∂Rφ | φ + φ

∂H∂R

φ

= φ∂H∂R

φ

€

H φ = E φ( )

Product rule

Product rule

€

∂∂R

Const( ) = 0


Forces II

• In DFT we have

• But only Vn-e and Vn-n depends on R and derivative are taken analytically

• We get forces for free!• Optimise under F to obtain {Ri}• Add in nuclear KE to obtain finite temperature

€

ˆ H (r,R) = −h2

2m∇r +Ve −e n r( )[ ] +Vn −e n r,R( )[ ] +Vn −n n R( )[ ] +Vxc n r( )[ ]


Example: Lattice Parameters

Volume

Ener

gy

Phase I

Phase IICommon tangent gives transition pressure:

P=-dE/dV

VII VI

•KS equations can be solved to give energy, E•What does that tell us?


Structures without experiment?

U(x)

x

start

stop

Relative energies of structures: examine phase stability


Summary so far

• Can get electronic density and energy• Can use forces (and stresses) to optimise

structure from an “intelligent” initial guess• Minima of energy gives structural phase

information


Alloys

• Alloys are complicated!

Phase separated Ordered Random


Ordered

• Ordered alloys are “easy” (usually!)• The have a repeated unit cell

Can perform calculation on this unit cell:•Electronic structure•Band Structure•Density of States•Etc…


Disordered

• There are two main approaches:• The Supercell approach– Make a large unit cell with species randomly

distributed as required– Characterises microscopic quantities

• The Virtual Crystal Approximation (VCA)– Make each atom behave as if it were an average of

various species AxB1-x

– Encapsulates only average quantities


Supercell Approach

• Need large unit cell• Computationally

expensive• A lot of atoms• Require check on

statistics (how many possible random configurations?)


Virtual Crystal Approximation

• What is an “average” atom?

• Put x of one atom and 1-x of the other atom at every site


VCA Example

• NbC1-xNx

• C and N are disordered

• How does electronic structure vary with x?


NbC1-xNx Electronic DoS


Electron by electron


Relation to experiment

• We solve the problem and get energy, E and density n(r) – experiments don’t measure these!

• An experiment:

Radiation orParticle (k)

Material (q)

“Different”Radiation orParticle(k-q)


Perturbation Theory

Based on compute how the total energy responds to a perturbation, usually of the DFT external potential v

Expand quantities (E, n, y, v)

...)2(2)1()0( EEEE

• Properties given by the derivatives

2

2)2()(

21particularin and

!1

EEE

nE n

nn


The Perturbations Perturb the external potential (from the nuclei and

any external field):• Nuclear positions phonons• Cell vectors elastic constants• Electric fields dielectric response• Magnetic fields NMR

But not only the potential, any perturbation to the Hamiltonian:• d/dk atomic charges• d/d(PSP) alchemical perturbation


Example: Phonons

• Perturb with respect to nuclear coordinates:

• This is equivalent to

€

E = E0 + E lk μulk μ

l,k,μ∑ +

12!

E lk μ l 'k 'νulk μ

ul 'k'ν+K

l,k,μl ',k ',ν

∑

€

f (x) = f x0( ) +dfdx x0

x +12!

d2 fdx 2

x0

x 2 +K


Atomic Motion

Eigenvectors of 2nd order energy give nuclear motion under phonon excitations


Summary• First principles electronic structure calculations:

Extremely powerful technique in condensed matter• Applicable to many different sciences

Condensed Matter Physics Chemistry Biology Material Science Surfaces Geology

• Simulate experimental measurements• Computationally possible, but still requires good

computer resources

Quantum Mechanics: Density Functional Theory and Practical Application to Alloys

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