Physics 250-06 “Advanced Electronic Structure”

Lecture 3. Improvements of DFT

Contents:

1. LDA+U.

2. LDA+DMFT.

3. Supplements: Self-interaction corrections, GW

Concept of delocalized and localized statesConcept of delocalized and localized states

Systems with d and f electrons show localized (atomic like) behavior.

Examples: cuprates, manganites, lanthanides, actinides, transition metal oxides, etc.

LDA is a static mean field theory and cannot describe many-bodyfeatures in the spectrum: example: atomic limit is with multipletsis missing in LDA.

When magnetic order exists, LSDA frequently helps!

(Anti)ferromagnets(Anti)ferromagnets

1( )

( )G k

k V

1

( )( )

G kk V

Splitting Vup-Vdn between up and down bands can be calculated in LSDA. It always comes out small (~1 eV). In many systems,it is of the order of 5-10 eV.

LDA+ULDA+U

ˆ ˆij i j i iij i

H t c c Un n

In LSDA splitting Vup-Vdn is controlled by Stoner parameter Iwhile on-site Coulomb interaction U can be much larger thanthat:

In simplest Hartree-Fock approximation:

~Ci

E Un n

dEV Un

dn

LDA+U functional with built-in Hubbard parameter U:

[ ( ), , ] [ ( )] [ ]LDA U LDA d d DC dd dE n r n n E n r Un n E n

Paramagnetic Mott InsulatorsParamagnetic Mott Insulators

1( )

( )LDALDA

G kk V

2

1 1( )

( ) ( )( )

4( )

1/ 2 1/ 2

( ) / 2 ( ) / 2

DMFT

d

G kUk

k

k U k U

How to recover the gap in the spectrum?Frequency dependence in self-energy is required:

LDA/LDA+U, other static mean field theories, cannot access paramagnetic insulating state because spin up and spin down solutions become degenerate

Effective (DFT-like) single particle spectrumalways consists of delta like peaks

Real excitational spectrumcan be quite different

Concept of Spectral FunctionsConcept of Spectral Functions

0[ ( ) ( )] ( , ) 1H k G k

Localized electrons: LDA+DMFTLocalized electrons: LDA+DMFT

2 ˆ ˆ[ ( ) ( ( ) )] ( ) ( )imp dcKS ff f kj kj kjV r V r r

Electronic structure is composed from LDA Hamiltonian for sp(d) electrons and dynamical self-energy for (d)f-electrons extracted from solving impurity problem

Poles of the Green function1

( , )kj

G k

have information about atomic multiplets and other many body effects.

N(N())

ddnn->d->dn+1n+1

Better description compared to LDA is obtained

ddnn->d->dn-1n-1

Spectral Density Functional TheorySpectral Density Functional Theory

Total EnergyTotal Energy and local excitational spectrum are accessed

Good approximation to exchange-correlation functionalis provided by local dynamical mean field theory.

Role of Kohn-Sham potential is played by a manifestly local self-energy operator (r,r’,). Generalized Kohn Sham equations for continuous distribution of spectral weight to be solved self-consistently.

(Kotliar et.al, RMP 2006)

A functional method where electronic spectral function is a variable would predict both energetics and spectra. A DMFT based electronic structure method - an approach where local spectral function (density of states) is at the center of interest. Can be entitled as

Spectral density functional theory

( , ', ) ( , ', ) ( , ')locG r r i G r r i r r

0 0( ) ( , , ) ( , , )i iDFT

i i

r G r r i e G r r i e

Family of Functionals

[ ( , ', )]locG r r i

[ ( )]DFT r

[ ( , ', )]BK G r r i 0[ ( , , ) ]iDFT

i

G r r i e

'r r

Local Green function FunctionalsLocal Green function Functionals

†( ) ( ')( , ', ) kj kj

kj kj

r rr r

G

[ ] [ ] [ ]SDF loc SDF loc SDF locG K G G

*( ) ( ')( , ', ) kj kj

DFTkj kj

r rG r r

[ ] [ ]SDF loc SDFK G K G

[ ] [ ]DFT DFT DFTK K G

Exactly as in DFT:

Generalization of Kohn Sham IdeaGeneralization of Kohn Sham Idea

To obtain kinetic functional:

introduce fictious particles which describe local Green function:

[ ] ( , ', ) ( , ', ) '

( ) ( ) [ ] [ ]

kj kj kj effkj i i

ext H xc loc

f r r i r r i drdr

r V r dr E G

M G

Spectral Density Functional looks similar to DFT

Effective mass operator is local by construction and playsauxiliary role exactly like Kohn-Sham potential in DFT

( , ', ) [ ( ) ( )] ( ')( , ', )

xceff ext H

loc

r r V r V r r rG r r

M

Energy dependent Kohn-Sham (Dyson) equations giverise to energy-dependent band structure

2 ( ) ( , ', ) ( ') ' ( )kj eff kj kj kjr r r r dr r M

have physical meaning in contrast to Kohn-Sham spectra. are designed to reproduce local spectral density

1

( )kjkj

fi E

Local Self-Energy of Spectral Density FunctionalLocal Self-Energy of Spectral Density Functional

kjE

LDA is not self-interaction free theory.

Simplest example: electron in hydrogen atom produces charge density cloudAnd would have excnage correlation potential according to LDA.

Perdew and Zunger (1984) proposed to subtract spirituous self-interactionenergy for each orbital from LDA total energy by introducing

Self-Interaction Corrections (SIC)

LDA-SIC theory produces orbital-dependent potential since one needs to define orbitals which self-interact.

SIC theory produces better total energies but wrong spectra in many cases.

Self-InteractionsSelf-Interactions

In GW (Hedin, 1965) spectrum is deduced from Dyson equation with approximate self-energy:

GW theory can be viewed as perturbation theory with respect to Coulomb interaction.

It produces correct energy gap in semiconductors which is an improvementon top of LDA

Being a weakly coupled pertrurbation theory it also has wrong atomic limit and does not produce atomic multiplets

GW Theory of HedinGW Theory of Hedin

2[ ( ) ( )] ( ) ( , ', ) ( ') ' ( )ext H kj GW kj kj kjV r V r r r r r dr r

Solve LDA equations and construct LDA Green functions and GW self-energy

Here, dynamically screened Coulomb interaction is calculated from the Knowledge of the dielectric function of the material:

Computing GW Self-EnergyComputing GW Self-Energy

2

21

[ ( ) ( ) ( )] ( ) ( )

( ) ( ')( , ', )

( , ', ) '' ( , '', ') ( '', ', ') '' '

( , ', ) ( , '', ) ''| '' ' |

ext H xc kj kj kj

kj kj

kj kj

GW

V r V r V r r r

r rG r r

r r dr G r r W r r dr d

eW r r r r dr

r r

1 0

01 CV

'0

' '

( , ', ) ( ) ( ) ( ') ( ')kj k qjkj k qj k qj kj

q kjj kj k qj

f fr r r r r r

E E