Solid State TheorySolid State TheoryPhysics 545Physics 545

Density Functional TheoryDensity Functional Theory

Eigenstates of electronsEigenstates of electronsEigenstates of electronsEigenstates of electronsFor optical absortion etc one needs theFor optical absortion etc one needs theFor optical absortion, etc., one needs the For optical absortion, etc., one needs the spectrum of excited statesspectrum of excited statesF h d i d h i hF h d i d h i hFor thermodynamics and chemistry the For thermodynamics and chemistry the lowest states are most importantlowest states are most importantIn many problems the temperature is low In many problems the temperature is low compared to characteristic electroniccompared to characteristic electroniccompared to characteristic electronic compared to characteristic electronic energies and we need only theenergies and we need only the ground stateground state–– Phase transitionsPhase transitions–– Phonons, etc.Phonons, etc.

Fundamental laws for the propertiesFundamental laws for the propertiesFundamental laws for the properties Fundamental laws for the properties of matter at low energiesof matter at low energies

Atomic scale (chemical bonds etc.)

Yes BUTElectrons and nuclei

(simple Coulomb interactions)

=> Quantum Mechanics ˆ ({ }) ({ })i iH r E rΨ = Ψ

Schrödinger’s EquationSchrödinger’s EquationSchrödinger s EquationSchrödinger s Equation

Complete descriptionComplete descriptionImpossible to solveImpossible to solve t

txitxtxVx

txm ∂

Ψ∂=Ψ+

∂Ψ∂

−),(),(),(),(

2 2

22

Impossible to solve Impossible to solve except for very simple except for very simple systems such as neutralsystems such as neutral

txm ∂∂2

systems such as neutral systems such as neutral HydrogenHydrogen

Numerical methods and the fastest computers

t h t lare not enough to solve Schrödinger’s Equation for non trivial systemsfor non-trivial systems

Time-independent and nonrelativistic Schrödinger equation:system consists of M nuclei and N electrons

2 2ˆ 12

12

1 1N MA

N N N

i

MM

A

MA B

m rZ ZR

Zr

+ += ∇ −− ∇ − ∑∑ ∑ ∑∑∑∑ ∑H11 111 122 i A ii A j iA A Bi Ai ABjAm r Rr= >= == = >=

ˆ ˆ

• elec • nucT V

• elec-nuc attraction

• elec-elec repulsion

nuc nuc repulsion• nuc-nuc repulsion

ManyMany--particle problemparticle problemyy p pp pSchroedinger’s equation is exactly Schroedinger’s equation is exactly solvable forsolvable for

Two particles (analytically)Two particles (analytically)-- Two particles (analytically)Two particles (analytically)-- Very few particles (numerically)Very few particles (numerically)

The number of electrons and nuclei The number of electrons and nuclei in a pebble is 10in a pebble is 10 23in a pebble is ~10 in a pebble is ~10 23

ˆ ( ) ( )H r r r E r r rΨ = Ψ

> APPROXIMATIONS> APPROXIMATIONS

1 2 1 2( , ,..., ) ( , ,..., )N NH r r r E r r rΨ = Ψ

=> APPROXIMATIONS=> APPROXIMATIONS

BornBorn--OppenheimerOppenheimer

1>>nm 1>>em

⇒Nuclei are much slower than electrons

(1) (2)

slower than electrons

(1) (2)

electronic/nuclear decoupling

2 21 1 1ˆ Z Z ZH μ μ ν∇ ∇ + +∑ ∑ ∑ ∑ ∑

, , ,2 2 ii i j i iij i

HM r r R

μ μμ

μ μ μ ν μμ μ μν> >

= − ∇ − ∇ + − +∑ ∑ ∑ ∑ ∑

})({121ˆ

}{2

}{ iextR

iji iji

i

elR rV

rH

μμ++∇−= ∑∑

>

})({})({})({ˆ elelelel rRErH ΨΨ

2 , iji iji r>electronselectronselectronselectrons

})({})({})({ }{,}{,}{ iRnniRnR rRErHμμμ μ Ψ=Ψ

1 21ˆ ({ })2

elnH E R

M μ μμ μ

= − ∇ +∑l il il il i μ μnucleinucleinucleinuclei

0 ({ })elF E R∂=Classical =>Classical =>Classical =>Classical => 0 ({ })F E R

Rν μν∂

Classical =>Classical =>Classical =>Classical =>

The Ground StateThe Ground StateThe Ground StateThe Ground State

General idea: Can use minimization General idea: Can use minimization methods to get the lowest energymethods to get the lowest energymethods to get the lowest energy methods to get the lowest energy statestateWhy is this difficult ?Why is this difficult ?It is aIt is a ManyMany Body ProblemBody ProblemIt is a It is a ManyMany--Body Problem Body Problem ΨΨii ( r( r11, r, r22, r, r33, r, r44, r, r55, . . . ), . . . )ΨΨii ( r( r11, r, r22, r, r33, r, r44, r, r55, . . . ), . . . )How to minimize in such a large How to minimize in such a large space?space?

ManyMany--electron problemelectron problemOld and extremely hard problem!

Different approachesDifferent approaches • Quantum Chemistry (Hartree-Fock, CI…)• Quantum Monte Carlo• Perturbation theory (propagators)Perturbation theory (propagators)• Density Functional Theory (DFT)

V ffi i d lVery efficient and generalBUT implementations are approximatep pp

and hard to improve (no systematic improvement)

(… actually running out of ideas …)

Materials simulationMaterials simulationAppliedApplied Quantum Mechanics (other methods)Quantum Mechanics (other methods)AppliedApplied Quantum Mechanics (other methods)Quantum Mechanics (other methods)

Materials simulationMaterials simulationEmpiricalPotentials

Ab initio Quantum Monte Carlo

Cascade

+ Surfaces

H-F Hartree DFT

LDA GGA+ dynamics

LDA GGA

Pseudopotentials PAWPseudopotentials

FFTReal space Wavelets

FirstFirst--principles calculationsprinciples calculationsFirstFirst principles calculationsprinciples calculationsFundamental laws of physicsFundamental laws of physics

S f “ d” i iS f “ d” i iSet of “accepted” approximations Set of “accepted” approximations to solve the corresponding equations on a computerto solve the corresponding equations on a computerto solve the corresponding equations on a computerto solve the corresponding equations on a computer

No empirical inputNo empirical inputp pp p

PREDICTIVE POWERPREDICTIVE POWER

Density-Functional Theory

)(})({ rr ρ→Ψ1 particle particle particle particle )(})({ rri ρ→Ψ1. densitydensitydensitydensity2. As if non-interacting electrons in an

effective (self-consistent) potential

Density Functional TheoryDensity Functional TheoryDensity Functional TheoryDensity Functional Theory1998 N b l P i i Ch i t1998 N b l P i i Ch i t1998 Nobel Prize in Chemistry 1998 Nobel Prize in Chemistry to Walter Kohnto Walter KohnA prescription for replacingA prescription for replacingA prescription for replacing A prescription for replacing Schrödinger’s Equation with Schrödinger’s Equation with similar, decoupled equationssimilar, decoupled equations––similar, decoupled equationssimilar, decoupled equationswhich we can solve with fast which we can solve with fast computers and clever algorithmscomputers and clever algorithmsKohnKohn--Sham EquationsSham EquationsAppliedApplied Quantum Mechanics Quantum Mechanics (other methods)(other methods)An important long range for the An important long range for the field: protein foldingfield: protein folding

Density Functional TheoryDensity Functional TheoryDensity Functional TheoryDensity Functional Theory1998 N b l P i i Ch i t1998 N b l P i i Ch i t1998 Nobel Prize in Chemistry 1998 Nobel Prize in Chemistry to Walter Kohnto Walter KohnA prescription for replacingA prescription for replacingA prescription for replacing A prescription for replacing Schrödinger’s Equation with Schrödinger’s Equation with similar, decoupled equationssimilar, decoupled equations––

hi h l ith f thi h l ith f tt

txitxtxVx

txm ∂

Ψ∂=Ψ+

∂Ψ∂

−),(),(),(),(

2 2

22

which we can solve with fast which we can solve with fast computers and clever computers and clever algorithmsalgorithmsggKohnKohn--Sham EquationsSham EquationsAppliedApplied Quantum Mechanics Quantum Mechanics (other methods)(other methods)An important long range for the An important long range for the field: protein foldingfield: protein foldingfield: protein foldingfield: protein folding

Density Functional TheoryDensity Functional TheoryDensity Functional TheoryDensity Functional Theory1998 N b l P i i Ch i1998 N b l P i i Ch i ΨΨH1998 Nobel Prize in Chemistry 1998 Nobel Prize in Chemistry to Walter Kohnto Walter KohnA prescription for replacingA prescription for replacing and

)()()( where,

+++=Ψ=Ψ

xceene

nknknk

rVrVrVKHH ε

A prescription for replacing A prescription for replacing Schrödinger’s Equation with Schrödinger’s Equation with similar, decoupled equationssimilar, decoupled equations––

,2

and

22

∇−=m

Ksimilar, decoupled equationssimilar, decoupled equationswhich we can solve with fast which we can solve with fast computers and clever algorithmscomputers and clever algorithms

)(

,||

)( 2∑′−

−=a a

ane Rr

ZerV

KohnKohn--Sham EquationsSham EquationsAppliedApplied Quantum Mechanics Quantum Mechanics

)(

,||

)()( 32 ∫ ′−′

−=

xc

ee

EV

rdrr

rerV

δ

ρ

(other methods)(other methods)An important long range for the An important long range for the ,|)(|)(

,)(

2∑ Ψ=

=

nknk

xcxc

rfr

rV

ρδρ

field: protein foldingfield: protein folding nk

Density Functional TheoryDensity Functional TheoryDensity Functional TheoryDensity Functional Theory1998 N b l P i i Ch i1998 N b l P i i Ch i Materials simulation1998 Nobel Prize in Chemistry 1998 Nobel Prize in Chemistry to Walter Kohnto Walter KohnA prescription for replacingA prescription for replacing

Materials simulationEmpiricalPotentials

Ab initio

A prescription for replacing A prescription for replacing Schrödinger’s Equation with Schrödinger’s Equation with similar, decoupled equationssimilar, decoupled equations––

Cascade

+ Surfaces

H-F Hartree DFT

LDA GGAsimilar, decoupled equationssimilar, decoupled equationswhich we can solve with fast which we can solve with fast computers and clever algorithmscomputers and clever algorithms

+ dynamicsLDA GGA

Pseudopotentials PAW

KohnKohn--Sham EquationsSham EquationsAppliedApplied Quantum Mechanics Quantum Mechanics (other methods)(other methods)An important long range for the An important long range for the

FFTReal space Wavelets

field: protein foldingfield: protein folding

HohenbergHohenberg--Kohn Theorems Kohn Theorems ggand and

KohnKohn--Sham ApproachSham Approachpppp

Wave function based methodsWave function based methodsdi t d t i ti f thdi t d t i ti f th ti l f titi l f ti bbdirect determination of thedirect determination of the many particle wave function many particle wave function ψψ by by

solving the Schrödinger equationsolving the Schrödinger equation

Density Functional Theory (DFT)Density Functional Theory (DFT)1964 Hohenberg & Kohn: Birth of DFT1964 Hohenberg & Kohn: Birth of DFTintroducing theintroducing the electron density electron density ρρ as a basic variableas a basic variablerecall:recall: 2ρ(r ) = N (x x x ) ds dx dxΨ∫ ∫central idea: express total energy as functional of central idea: express total energy as functional of ρρ:: E = E[E = E[ρρ]]

1 1 2 N 1 2 Nρ(r ) = N ... (x ,x ,..., x ) ds dx ...dxΨ∫ ∫

What’s so great about DFT?What’s so great about DFT?ggNormallyNormally, we need to describe solids by a wavefunction , we need to describe solids by a wavefunction ofof all electronsall electronsof of all electronsall electronsThen, we need to find solutions for more than Then, we need to find solutions for more than 10102323

electronselectrons and combine themand combine themThis is still an This is still an impossible taskimpossible task

In Density Functional Theory (DFT) we only need to find In Density Functional Theory (DFT) we only need to find the the charge distributioncharge distribution throughout our systemthroughout our systemTh d ibTh d ib i l li l l i ii iThen, we can describe Then, we can describe single electronssingle electrons moving in a moving in a crystal mean field of all ions and other electronscrystal mean field of all ions and other electronsIn this way we can calculate solids of up to a fewIn this way we can calculate solids of up to a fewIn this way, we can calculate solids of up to a few In this way, we can calculate solids of up to a few thousand atomsthousand atomsAnd that is all we need, usuallyAnd that is all we need, usually, y, y

OverviewOverview

HohenbergHohenberg--Kohn Theorems (1964)Kohn Theorems (1964)HohenbergHohenberg Kohn Theorems (1964)Kohn Theorems (1964)

–– First Theorem: Proof of Existence First Theorem: Proof of Existence –– Second Theorem: Variational Principle Second Theorem: Variational Principle –– ConstrainedConstrained--Search Approach by Levy (1979)Search Approach by Levy (1979)

KohnKohn--Sham Approach (1965)Sham Approach (1965)

–– Orbitals and NonOrbitals and Non--Interacting Reference SystemInteracting Reference System–– KohnKohn--Sham EquationsSham Equations–– Effective Potential and ExchangeEffective Potential and Exchange--Correlation PotentialCorrelation Potential

Hohenberg Hohenberg -- KohnKohngg)(})({ rri ρ→Ψ

∑++=N

iextee rVVTH )(ˆFor our manyFor our many--electron problemelectron problemFor our manyFor our many--electron problemelectron problem=i 1

3 ( ) ( ) [ ( )]d r V r r F rρ ρ≡ +∫ E≥1111 [ ( )]E rρ ( ) ( ) [ ( )]extd r V r r F rρ ρ≡ +∫ GSE≥1.1.1.1. [ ( )]E rρ

(universal functional)(depends on nuclear positions)

[ ( )]E r Eρ =2222

(universal functional)(depends on nuclear positions)

[ ( )]GS GSE r Eρ =2.2.2.2. PROBLEM:PROBLEM:Functional unknown!Functional unknown!PROBLEM:PROBLEM:Functional unknown!Functional unknown!

Kohn Kohn -- ShamShamIndependent particles in an effective potentialIndependent particles in an effective potentialIndependent particles in an effective potentialIndependent particles in an effective potential

3 1[ ] [ ] ( )[ ( ) ( )] [ ]E T d r r V r r Eρ ρ ρ ρ= + + Φ +∫They rewrote the functional as:They rewrote the functional as:They rewrote the functional as:They rewrote the functional as:

0 2[ ] [ ] ( )[ ( ) ( )] [ ]ext xcE T d r r V r r Eρ ρ ρ ρ= + + Φ +∫Kinetic energy for systemKinetic energy for systemKinetic energy for systemKinetic energy for system Hartree potentialHartree potentialHartree potentialHartree potentialKinetic energy for system Kinetic energy for system with no ewith no e--e interactionse interactionsKinetic energy for system Kinetic energy for system with no ewith no e--e interactionse interactions

Hartree potentialHartree potentialHartree potentialHartree potential

The rest:The rest:The rest:The rest:The rest: The rest: exchange exchange correlationcorrelation

The rest: The rest: exchange exchange correlationcorrelation

Equivalent to independent Equivalent to independent particles under the potentialparticles under the potentialEquivalent to independent Equivalent to independent particles under the potentialparticles under the potentialparticles under the potentialparticles under the potentialparticles under the potentialparticles under the potential

[ ]( ) ( ) ( ) xcEV r V r r δ ρ= + Φ +( ) ( ) ( )

( )extV r V r rrδρ

= + Φ +

First Hohenberg-Kohn Theorem

• First Hohenberg-Kohn Theorem: Proof of Existenceg

For a given external potential Vext the electron density of the ground state of a system uniquely determines the ground state wave function Ψ0

0ρ (r)state of a system uniquely determines the ground state wave function Ψ0

and hence all properties A of the ground state:i.e. there is for any given system only one density of charge ρ

0 00ˆA | A |= Ψ Ψ

0 0 0 0 0 0 0 00ˆ[ρ ] A A[ρ ] [ρ ] | A | [ρ ]Ψ = Ψ ⇒ = = Ψ Ψ

• Note: no explicit analytical forms of and except expectation values of local one-particle operators, e g the external (nuclear) potential V:

0 0[ρ ] Ψ 0A[ρ ]

e.g. the external (nuclear) potential V:

0 00V V[ρ ] = ρ (r)V(r)dr= ∫

First Hohenberg-Kohn Theorem

• Proof: reductio ad absurdo

– Assuming two external potentials, Vext and , which differ by more than a constant but both give rise to the same ground state electron density:

'extV

'0 0ρ ρ=

– Both Hamiltonians, and , belong to twodifferent ground state wave functions, Ψ and Ψ', and two correspondingdifferent ground state energies

'ee ext

ˆ ˆ ˆ ˆH' = T+V +Vee extˆ ˆ ˆ ˆH = T+V +V

'E E≠different ground state energies .

– Variational principle in general: U i Ψ' t i l f ti f

0 0E E≠

H

0 0 0 trial trial trialˆ ˆ| | = E E | | H HΨ Ψ ≤ = Ψ Ψ

0 ˆ ˆ ˆ ˆE ' | H | ' ' | H' | ' ' | H-H' | '< Ψ Ψ = Ψ Ψ + Ψ Ψ

Using Ψ' as a trial function for H

( )ˆ ˆ ˆ ˆH H' + H - H'=

' '0 0 ext extE E ' | V -V | '< + Ψ Ψ

( )' '0 0 0 ext extE < E + ρ (r) V -V dr∫

' '00V = ρ (r)V'(r)dr∫

'0 0ρ (r) ρ (r)=( )0 0 0 ext extE E ρ (r) V V dr∫ (I) 0 0ρ (r) ρ (r)

First Hohenberg-Kohn Theorem

( )' '0 0 0 ext extE < E + ρ (r) V -V dr∫ (I)

– Now using Ψ as a trial function for and redoing the same steps:

( ) (I)

H'' ˆ ˆ ˆ ˆE | H' | | H | | H' H |< Ψ Ψ Ψ Ψ + Ψ Ψ ( )ˆ ˆ ˆ ˆH' H + H' H

( )' '∫

0E | H' | | H | | H'-H |< Ψ Ψ = Ψ Ψ + Ψ Ψ

' '0 0 ext extE E | V -V |< + Ψ Ψ 00V = ρ (r)V(r)dr∫

( )H' H + H' - H=

( )0 0 0 ext extE < E ρ (r) V -V dr + ∫

(II)( )' '0 0 0 ext extE < E ρ (r) V -V dr− ∫

' '0 0 0 0E + E E + E 0 < 0< ⇔ (I) + (II)

– Adding equations (I) and (II):

(I) (II)

– Therefore there cannot be two different Vext that yield the same ground state electron density, or the ground state density uniquely specifies Vext.state electron density, or the ground state density uniquely specifies Vext.

Hohenberg-Kohn Functional I

• Separating the energy functional E0[ρ] into its contributions: ee Neˆ ˆ ˆ ˆH = T+V +V

][E][E][T|H|][E 0Ne0ee00000 ρ+ρ+ρ=ΨΨ=ρ

T[ρ ] kinetic energy of the electrons

Ne 0 0 NeE [ρ ] = ρ (r)V dr∫

Eee[ρ0] electron-electron repulsion energyT[ρ0] kinetic energy of the electrons

potential energy due to the nuclei-electron attraction

• E [ρ] can further be separated: E [ρ] J[ρ] E [ρ]= +

Ne 0 0 Ne[ρ ] ρ ( )∫ p gy

• Eee[ρ] can further be separated: ee nclE [ρ] J[ρ] E [ρ]= +

1 2 1 2

12

1 ρ(r )ρ(r )J[ρ] dr dr classical Coulomb part2 r

= ∫ ∫

Encl[ρ] non-classical contributions to the electron-electron interaction (self-interaction correction, exchange and Coulomb correlation)

122 r

Hohenberg-Kohn Functional II

S ti f E[ ] i t t d di th t l t i E [ ] d• Separation of E[ρ] into parts depending on the actual system, i.e. ENe[ρ], and parts which are universal (independent of N, RA and ZA), i.e. T[ρ] and Eee[ρ].

• Collecting the system independent parts into a new quantity, we introduce the Hohenberg-Kohn functional HK nclF [ρ] T[ρ] J[ρ] E [ρ]= + +

• If FHK[ρ] was known exactly the Schrödinger equation would be solved not approximately, but exactly.

• Since FHK[ρ] is a universal functional independent of the system, it applies equally well to the hydrogen atom as to gigantic molecules such as DNA !

• The major challenge of DFT is to find explicit expressions for the unknown functionals T[ρ] and Encl[ρ] !

Second Hohenberg-Kohn Theorem

• Second Hohenberg-Kohn Theorem: Variational PrincipleThe functional FHK[ρ] that delivers the ground state energy E0 of the system,The functional FHK[ρ] that delivers the ground state energy E0 of the system, delivers the lowest energy if and only if the input density is the true ground state density ρ0. Since every system has only one ground state, which for H is described by ψ the charge related to this ground state must be

ρ

for H is described by ψ, the charge related to this ground state must be the one, where the energy of a system reaches a minimum

• Variational principle of DFT0 0E E[ρ] for ρ ρ < ≠0 0

0 0

[ρ] ρ ρE = E[ρ] for ρ = ρ

ext

ρ(r) 0, ρ(r) dr = NV -representability

≥ ∫

• Proof: Any trial density defines its own Hamiltonian and hence its own wave function . This wave function can be taken as the trial wave function for the H ilt i A i i th i ti l i i l

ρΨ

HHamiltonian . Again using the variational principle:

ee ext 0 0 0 0ˆ ˆ| H | = T[ρ] V [ρ] ρ(r)V dr = E[ρ] E [ρ ] = | H |Ψ Ψ + + ≥ Ψ Ψ∫

H

Summary of Hohenberg-Kohn Theoremsy g

• All properties of a system defined by an external potential are determined by th d t t d itthe ground state density ρ0.

• The ground state energy associated with a density is available through theρThe ground state energy associated with a density is available through the potential including the Hohenberg-Kohn functional FHK:

0 HK Ne HK nclE [ρ] = F [ρ] + ρ(r)V dr with F [ρ] = T[ρ] J[ρ] E [ρ]+ +∫

ρ

• This functional attains its minimum value with respect to all allowed densities if and only if the input density is the true ground state density, i.e. .0ρ(r) ρ (r)≡if and only if the input density is the true ground state density, i.e. .

• DFT is usually termed a ground state theory ! Though ρ0 contains information

0ρ( ) ρ ( )

about excited states no practical way is known so far to extract this information.

Constrained-Search Approach by Levy

• Recall the variational principle: 0N N

ˆE E[ ] = | H | with | 1min minΨ→ Ψ→

= Ψ Ψ Ψ Ψ Ψ =

From all antisymmetric N-electron wave functions the one that yields the lowest expectation value of (i.e. the energy E) is the ground state wave function Ψ0.H

• Connection to DFT by performing a two-step constrained-search:

ˆE E[ ] | H | ith ( )d Ni i i⎛ ⎞

Ψ Ψ ∫⎜ ⎟

(1) Search over all antisymmetric wave functions that yield a particular density ρ and

0ρ N ρ N ρ

E E[ρ] = | H | with ρ ρ(r)dr = Nmin min min→ → Ψ→

= Ψ Ψ = ∫⎜ ⎟⎝ ⎠

(1) Search over all antisymmetric wave functions that yield a particular density ρ andidentify the wave function that yields the lowest energy.

(2) Extend the search over all densities and finally identify that density as the ground(2) Extend the search over all densities and finally identify that density as the groundstate density for which the wave function characterized in the first step delivers the lowest energy.

Universal Functional

• Separation of the energy functional E[ρ] into T + Vee and VNe that is simply determined by ρ and therefore independent of the wave function generating ρ:determined by ρ and therefore independent of the wave function generating ρ:

ρ N ρ ρ N0 ee Ne

ψ Nˆ ˆ ˆE |H| = | T + V | ρ(r)V drmin min min min

→ Ψ→ → →

⎛ ⎞⎛ ⎞= Ψ Ψ Ψ Ψ + ∫⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ρ ρ ρ ψ N→⎝ ⎠ ⎝ ⎠

( )0 Neρ N

E F[ρ] ρ(r)V drmin→

= + ∫

• Introducing the Universal functional F[ρ]:

• So given a density, delivers the corresponding energy and

eeρ

ˆ ˆF[ρ] | T + V |minΨ→

= Ψ Ψ

NeF[ρ] ρ(r)V dr+ ∫g y, p g gyupon minimization, the ground state density and energy are obtained.

N

Ne[ρ] ρ( )∫

ˆ ˆF [ ] | T V |Ψ Ψ• Note: In contrast to FHK[ρ], F[ρ] is defined for all densities that originate from an antisymmetric wave function Ψ; for the ground state density .0 HK 0F[ρ ] F [ρ ]=

HK eeF [ρ] | T + V |= Ψ Ψ

y ; g y 0 HK 0[ρ ] [ρ ]

Ground State Wave Function in DFT

• Do we know the Ground State wave Function in DFT ?

– The Levy constrained-search strategy provides only a formal route to extract the ground state wave function Ψ0 from the ground state denisty ρ0( hi h i id d b i i i i th f ti l F[ ])(which is provided by minimizing the functional F[ρ]).

– There are many N-electron wave functions that yield the same density via The correct ground state wave2ρ (r) = N (x x x ) ds dx dxΨ∫ ∫ . The correct ground state wave

function Ψ0 is the one which yields the lowest energy.

I l li ti h t ll th f ti Th

0 1 2 N 1 2 Nρ (r) = N ... (x ,x ,..., x ) ds dx ...dxΨ∫ ∫

– In real applications we have no access to all these wave functions. Thus there is no way to identify the correct wave function associated with a particular density.

– Hence for all practical purposes, there is no wave function in DFT !

Discussion of the Hohenberg-Kohn Theorems

• Employing the electron density as the central variable containing all information to describe an atomic or molecular system is physically soundinformation to describe an atomic or molecular system, is physically sound.

• Both theorems tell us that a unique mapping between the ground state density ρ and the ground state energy E exists in principle but do notdensity ρ0 and the ground state energy E0 exists in principle, but do not tell us how to construct the corresponding functional F[ρ].

• The constrained-search scheme is only of theoretical value because we areThe constrained search scheme is only of theoretical value because we are not able to search through all wave functions and therefore it is impossible to set up the universal functional F[ρ].

• The variational principle applies only to the exact functional F[ρ]; but in any real application of DFT we are forced to use an approximation for F[ρ].

DFT i i i l i HF C fi i I i (CI)– DFT is not variational in contrast to HF or Configuration Interaction (CI)

– is not an indicator for the quality of the trial function

energies obtained from approximate DFT can be lower than the exact ones

ˆE | H |= Ψ Ψ

– energies obtained from approximate DFT can be lower than the exact ones

Kohn-Sham Approach

• How to use Density Functional Theory in practice ? y y p

– ground state energy ( )0 Neρ N

E F[ρ] ρ(r)V drmin→

= + ∫

– how to construct the universal functional F[ρ]

ee eeˆ ˆF[ρ] | T + V | T[ρ] E [ρ]min= Ψ Ψ = +ee ee

ρF[ρ] | T V | T[ρ] E [ρ]min

Ψ→Ψ Ψ +

P bl N h i l bi di i Th F i th d !5/3

TFT [ρ] ρ (r)dr∫∼

– Problem: No chemical binding in Thomas-Fermi methods !

– Reason: too simple functional form for

– Task: determine T[ρ] with better control of accuracy

Independent particlesIndependent particles21ˆ ( )

2h V r= − ∇ + ( )

2ˆ ( ) ( )h r rψ ε ψ=( ) ( )n n nh r rψ ε ψ=

2( ) | ( ) |occ

∑ε2( ) | ( ) |n

nr rρ ψ= ∑

Non-Interacting Reference System

• Kohn-Sham model system (1965)

– For each system of interacting electrons in an external potential Vext with ground state density ρ0 there exists a reference system of non-interactingg y ρ0 y gelectrons with a suitably chosen effective potential VS exhibiting the same ground state density ρ0

N KSS i

iˆH = f with ∑ KS 2

S1ˆ f = - V (r)2

∇ +– Reference system:

{ }1

– Eigenfunctions: Slater determinants ΘS built of the Kohn-Sham orbitals ϕi

{ }S 1 1 2 2 N N1 det (x ), (x ),..., (x )N!

ϕ ϕ ϕΘ =

Kohn-Sham Equations

• Kohn-Sham equations KSi i if ε i = 1, ..., Nϕ ϕ=

KS 2s

1f V (r)2

= − ∇ +KSf one-electron Kohn-Sham operator

i Kohn-Sham orbitalsϕi Kohn Sham orbitalsϕ

iε orbital energies

• VS is chosen such that the density ρS of the non-interacting system equals the ground state density ρ0 of the interacting system

N 2S i 0

i sρ (r) (r,s) ρ (r)ϕ= =∑∑

Exchange-Correlation Energy

• Separation of F[ρ]: eeF[ρ] E [ρ] + T[ρ]=

1 21 2

12

1 ρ(r )ρ(r )J[ρ] dr dr 2 r

= ∫ ∫Eee[ρ] = J[ρ] + Encl[ρ] classical Coulomb interaction

N 2S i i

i

1T | |2

ϕ ϕ= − ∇∑T[ρ] = TS[ρ] + TC[ρ] kinetic energy of KS model system

E [ ] (T[ ] T [ ]) + (E [ ] J[ ]) T [ ] + E [ ]

• Exchange-correlation energy EXC

XC S ee C nclE [ρ] (T[ρ] - T [ρ]) + (E [ρ] - J[ρ]) = T [ρ] + E [ρ]≡

• Note: Different concepts for exchange and correlation energy in DFT and HF !

Energy Functional

• Determination of the Kohn-Sham orbitals ϕi

1 2S 1 2 XC Ne

1 ρ(r )ρ(r )= T [ρ] + dr dr E [ρ] + ρ(r)V dr2 r

+∫ ∫ ∫

S XC NeE[ρ] = T [ρ] + J[ρ] + E [ρ] + E [ρ]

122 rN N N N M22 22 A

i i i 1 j 2 1 2 XC i 1 1i i j i A12 1A

1 1 1 Z= - | | + (r ) (r ) dr dr + E [ρ] - (r ) dr 2 2 r r

ϕ ϕ ϕ ϕ ϕ∇∑ ∑ ∑ ∑ ∑∫ ∫ ∫ ∫

• Applying the variational principle, i.e. what condition must the KS orbitals fulfil in order to minimize E[ρ] under the constraint | = δϕ ϕfulfil in order to minimize E[ρ] under the constraint i j ij| = δ ϕ ϕ

M2 2 2 A2 XC 1 i eff 1 i i i

A12 1A

1 ρ(r ) Z 1dr V (r ) V (r ) ε2 r r 2

ϕ ϕ ϕ⎛ ⎞⎡ ⎤ ⎛ ⎞− ∇ + + − = − ∇ + =∑∫⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎝ ⎠⎣ ⎦⎝ ⎠12 1A ⎝ ⎠⎣ ⎦⎝ ⎠

KS 2S

1ˆ f = - V (r)2

∇ +KSi i if εϕ ϕ=• Comparing with Kohn-Sham equation:

2

Effective Potential

• Effective potential Veff

M 2 A

S eff 2 XC 1 C XC NeA12 1A

ρ(r ) ZV (r) V (r) = dr V (r ) = V + V + V r r

≡ + − ∑∫

VC classical Coulomb potential VXC exchange-correlation potentialV t ti l d t l iVNe potential due to nuclei

V l d d d th d it d th th KS bit l• Veff already depends on the density and thus on the KS orbitals

→ self-consistent field (SCF) problem that has to be solved iteratively

• Note: KS equations are formally simpler than HF equations because Veff is localat variance with the non-local nature of exchange term of Fock operatorg

SelfSelf--consistencyconsistencySelfSelf consistencyconsistency

PROBLEM: The potential (input) depends PROBLEM: The potential (input) depends on the density (output)on the density (output)

PROBLEM: The potential (input) depends PROBLEM: The potential (input) depends on the density (output)on the density (output)

ρ V ρ ρinρ V ρ outρ

| |1n nρ ρ ε− >| |1n nρ ρ −

Exchange-Correlation Potential

• Exchange-correlation potential VXC

– potential due to the exchange-correlation energy EXC

– functional derivative of EXC with respect to ρ XCXC

δEV ≡XC p ρ XC δρ

• Kohn-Sham approach is in principle exact and takes electron correlation• Kohn-Sham approach is in principle exact and takes electron correlation fully into account since it does not contain any approximations so far !

• Goal of DFT: find better and better approximations to EXC and VXC

– Local Density Approximation (LDA)

– Generalized Gradient Approximation (GGA)

EE & V& V [ ]xcEV δ ρEExc xc & V& Vxcxc[ ]( )

xcxcV

rρ

δρ≡

Local Density Approximation (LDA)Local Density Approximation (LDA)Local Density Approximation (LDA)Local Density Approximation (LDA)

))((][ rVV xcxc ρρ ≈ (function parameterised for the homogeneous (function parameterised for the homogeneous electron liquid as obtained from QMC)electron liquid as obtained from QMC)

(function parameterised for the homogeneous (function parameterised for the homogeneous electron liquid as obtained from QMC)electron liquid as obtained from QMC)

Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)Generalised Gradient Approximation (GGA)

))(),((][ rrVV xcxc ρρρ ∇≈(new terms parameterised for heterogeneous (new terms parameterised for heterogeneous

electron systems (atoms) as obtained from QC)electron systems (atoms) as obtained from QC)(new terms parameterised for heterogeneous (new terms parameterised for heterogeneous

electron systems (atoms) as obtained from QC)electron systems (atoms) as obtained from QC)

Kinetic energyKinetic energy

Assumption: the kinetic energy is the same as the kinetic energyfor every individual electron in its ground state.

Then the kinetic energy Ts[r] is a sum over integrals:gy [ ] g

Error: the electrons in a solid are not necessarily in their groundstate since motion of electrons in this case is usually coupledstate since motion of electrons in this case is usually coupled(motion one electron will have an effect on all other electrons)

Electron Electron –– ion interactionion interaction

Assumption: the electrons interact via electrostatic fields withAssumption: the electrons interact via electrostatic fields withthe ion cores of the solid. The interaction can then be calculatedas a sum over all ion cores and the Coulomb interaction potential:

Error: none, if the ions don’t move. If the ions move, which they do under thermal excitation, the error is still very small since theyIons move much slower than the electrons.

ElectronElectron--electron interactionelectron interactionThere are two parts:

One, coming from the electrostatic repulsion of electrons, orthe Coulomb interaction. This is the easy bit:

The second one comes from the correlated motion of electrons (if one electron moves, then it changes the electromagnetic fields in its vicinity) and the difference ifelectromagnetic fields in its vicinity), and the difference, if electrons of different spins are interacting.

This part is called the exchange correlation interaction and itThis part is called the exchange-correlation interaction, and itcan only be approximated. This is the hard bit.

Effective Schrodinger equationEffective Schrodinger equationg qg qPutting all these contributions to the energy together in one effectivepotential we can formulate an effective Schrodinger equation for electrons in a solid, which can be solved:

Practical solutions in current research:1. Guess an initial charge density (usually from the electron states1. Guess an initial charge density (usually from the electron states

in isolated atoms) for N electrons2. Calculate all the contributions to the effective potential3 Solve the Schrodinger equation and find N electron states3. Solve the Schrodinger equation and find N electron states4. Fill the eigenstates with electrons starting from the bottom5. Calculate the new charge density6 C l l t ll th t ib ti t th ff ti t ti l6. Calculate all the contributions to the effective potential …

… until the charge density does not change any more.

Summary of the Kohn-Sham Approachy pp

• Non-interacting reference system: VS → ρS = ρ0 and represented by ΘS(ϕi)S0Ψ

• ϕi are solutions of KS equations and define KSi i if εϕ ϕ=

N 2S i i

i

1T | |2

ϕ ϕ= − ∇∑

1• How to generate VS in KS operator such that ρS = ρ0 ?KS 2S

1f = - +V (r)2

∇

• Separation: E[ρ] = TS[ρ] + ENe[ρ] + J[ρ] + EXC[ρ] with EXC[ρ] = TC[ρ] + Encl[ρ]Sepa at o : [ρ] S[ρ] Ne[ρ] J[ρ] XC[ρ] w t XC[ρ] C[ρ] ncl[ρ]

• Variational principle: VS = VNe + VC + VXC is local and VS [ρ] → SCF problem

• VS → → KS equations → KS orbitals → N 2

S i 0i s

ρ (r) (r,s) ρ (r)ϕ= =∑∑KSf

• Inserting ρ0 into E[ρ] yields exact ground state energy E0

• Applications: Approximations for V and E (LDA or GGA)• Applications: Approximations for VXC and EXC (LDA or GGA)

RecapRecapRecapRecapBornBorn--Oppenheimer: Oppenheimer: electronelectron--nuclear decouplingnuclear decouplingpppp p gp g

ManyMany--electron electron --> > DFT DFT (LDA, GGA)(LDA, GGA)

OneOne particle problem in effective selfparticle problem in effective self consistentconsistentOneOne--particle problem in effective selfparticle problem in effective self--consistent consistent potential (iterate)potential (iterate)B i t S l i i t tB i t S l i i t tBasis set => Solving in two steps:Basis set => Solving in two steps:

1. Calculation of matrix elements of H and S1. Calculation of matrix elements of H and S2. Diagonalisation2. Diagonalisation

Extended crystals: Extended crystals: PBC + k samplingPBC + k samplingyy p gp g

Outlook

• Detailed Discussion of the Kohn-Sham Approach

– local Kohn-Sham potential VS

– exchange-correlation energy EXC is different to Hartree-Fockexchange correlation energy EXC is different to Hartree Fock

– meaning of the Kohn-Sham orbitals

– Is the Kohn-Sham approach a single determinant method ?

– unrestricted Kohn-Sham formalism

– degeneracy, excited states and multiplet problem

Some materials’ propertiesSome materials’ propertiesp pp pExp.Exp. LAPWLAPW Other PWOther PW PWPW DZPDZP

a a ((Å)Å) 3.573.57 3.543.54 3.543.54 3.533.53 3.543.54(( ))CC BB (GPa)(GPa) 442442 470470 436436 459459 453453

EEcc (eV)(eV) 7.377.37 10.1310.13 8.968.96 8.898.89 8.818.81((Å)Å) 5 435 43 5 415 41 5 385 38 5 385 38 5 405 40a a ((Å)Å) 5.435.43 5.415.41 5.385.38 5.385.38 5.405.40

SiSi BB (GPa)(GPa) 9999 9696 9494 9696 9797EEcc (eV)(eV) 4.634.63 5.285.28 5.345.34 5.405.40 5.315.31a a ((Å)Å) 4.234.23 4.054.05 3.983.98 3.953.95 3.983.98

NaNa BB (GPa)(GPa) 6.96.9 9.29.2 8.78.7 8.78.7 9.29.2EE (eV)(eV) 1 111 11 1 441 44 1 281 28 1 221 22 1 221 22EEcc (eV)(eV) 1.111.11 1.441.44 1.281.28 1.221.22 1.221.22a a ((Å)Å) 3.603.60 3.523.52 3.563.56 -- 3.573.57

CuCu BB (GPa)(GPa) 138138 192192 172172 -- 165165EEcc (eV)(eV) 3.503.50 4.294.29 4.244.24 -- 4.374.37a a ((Å)Å) 4.084.08 4.054.05 4.074.07 4.054.05 4.074.07

AuAu BB (GPa)(GPa) 173173 198198 190190 195195 188188AuAu BB (GPa)(GPa) 173173 198198 190190 195195 188188EEcc (eV)(eV) 3.813.81 -- -- 4.364.36 4.134.13

Absence of DC conductivity in -DNAλP. J. de Pablo et al. Phys. Rev. Lett. 85, 4992 (2000)

Effect of sequence disorder and vibrations on the electronic structureq

=> Band-like conduction is extremely unlikely: DNA is not a wire

Pressing nanotubes for a switchgPushed them together, relaxed &

l l t d d ti t th t t SWITCHcalculated conduction at the contact: SWITCH

M. Fuhrer et al. Science 288, 494 (2000)Y.-G. Yoon et al. Phys. Rev. Lett. 86, 688 (2001)

Pyrophyllite, illite & smectiteStructural effects of octahedral cation substitutions

C. I. Sainz-Diaz et al. (American Mineralogist, 2002)

WET SURFACESOrganic molecules intercalated between layers

WET SURFACESg y

M. Craig et al. (Phys. Chem. Miner. 2002)