Exchange-Correlation functionals for the ground state: from LDA up to the

Optimized Effective Potential method

FHI, Berlin, 13th July 2011Hands-on Tutorial Workshop 2011 on Ab Initio Molecular Simulations:

Toward a First-Principles Understanding of Materials Properties and Functions

Pag. 1

Dr. Fabio Della SalaNational Nanotechnology Laboratories, Lecce, Italy

www.theory-nnl.it

Kohn-Sham (KS) Density Functional Theory

)()( rr ρ⇔extv

Coulomb Potential:

(Nuclei Fixed)HK theorem

KS equations

Pag. 2

)()()(.

2

rrr exactocc

a

KS

asn ρφρ →=∑

Exchange Correlation Potential:

(ns=2, closed-shell)Electronic Density:

SC

F

Density of a single Slater-determinant(non-interacting electrons) )(rextv

][ρKSΦ )(rρ

KSΦ

Kohn-Sham Total Energy

Coulomb Energy:

Exchange-Correlation

Energy:44 344 2144 344 21

T

ni

U

Jeexc TTEVE ])[][()][][(][ ρρρρρ −+−=

Pag. 3

Kinetic Energy:

Energy:

KS kinetic energy is an

orbital-dependent functional

?][ =ρxcE“Exact”

??][ =ρniT

MB

ee

MB

ee

HKVTTVF

MBΨ+Ψ=+=

→Ψ

ˆˆmin][][][ρ

ρρρ

cxc TU

( )∑ ∇=occ

a

asn 2

2)( rr φτ

)(2

1][ 2

.

rrτφφρ ∫∑ =∇−→ dnT a

occ

a

asni

Outline

Section I� Exact definitions / Functional Form� Uniform Electron Gas (LDA)� Gradient Expansion (PBE-like functionals)

Section II� Orbital Dependent Functional � Exact Exchange

Section III

Pag. 4

Section III �Hybrid Functionals / Generalized KS

Summary

Purpose: Introduction to XC functional performance and development

Not covered: VDW, Magnetism, RPA (see other talks)

Exchange Functional

as in Hartree-Fockbut with KS orbitals

Exact ExchangeExact Exchange][][][ ρρρ cxxc EEE +=

1e1e-- Density MatrixDensity Matrix

∫∫

∫∫∑∑

′′

−=

′′−

′′−=

=−ΦΦ=

rrrr

rrrr

rrrr

ddn

ddn

EVE

KS

s

KS

b

KS

b

KS

a

KS

aocc

a

occ

b

s

J

KS

ee

KSEXX

x

2

. .

),(

)()()()(

2

][][][][

γ

φφφφ

ρρρρ

Pag. 5

Ex is a KS-Density Matrix Functional(implicit density functional )

∫ ∫∫ ′−

′== rr

rr

rrrrrr dd

hdE x

xx

),()(

2

1)()(][

ρερρ

rrr

rrr ′

′−

′= ∫

3),(

2

1)( d

hxxε 0

)(

),(),(

2

≤′

−=′r

rrrr

ρ

γ KS

s

EXX

x nh

Exact exchange-hole:Exchange-energy density:

][ργγ KSKS =∫∫ ′

′−−= rr

rrdd

ns),(

2

Correlation Functional

MB wavefunction KS single Slater-Det.

As in Many-Body theory (where there are different densities)

: Potential contributioncU

ˆˆˆˆˆˆ HF

eeext

HFMB

eeext

MBMB

c VVTVVTE Φ++Φ−Ψ++Ψ=

0 ][ˆˆ][ˆˆmin][ ≤Φ+Φ−Ψ+Ψ=→Ψ

ρρρρ

KS

ee

KSMB

ee

MBDFT

c VTVTEMB

Pag. 6

Tc as large as Uc

∫ ∫∫ ′−

′== rr

rr

rrrrrr dd

hdE c

cc

),()(

2

1)()(][

ρερρ

: Kinetic contribution

: Pair-Density

cT

( )[ ] rrrrr

rrrr

rrrr

rr dn

ddE

KSs

KS

c

),(),(2

),(),(

2

1 ][

∫∫

∫∫

′−′∇⋅∇−

′′′−

′Γ−′Γ=

′ γγ

ρ

),( rr ′Γ

Jacob’s Ladder (Perdew)

∫= ))((...,)(][ rrr τερρ dE

))((...,)(][ 3

∫= rrr EXX

xxcxc dE εερρ

)},{(...,)(][ 3

∫= ALL

iixcxc dE εφερρ rr

3. Meta-GGA (TPSS)

4. Hyper-GGA, EXX,Hybrid , local hybrid

5. All orbitals/eigenvalues(RPA,GLPT, double hybrid)

Com

puta

tional

cost

Acc

ura

cy

Pag. 7

∫= ))(()(][ rrr ρερρ xcxc dE

∫= ))((...,)(][ rrr τερρ xcxc dE

0. Hartree 0][ =ρxcE

1. Local Density Approx.(LDA)

2. Gener. Gradient Approx. (GGA) ∫ ∇= ))((...,)(][ rrr ρερρ xcxc dE

3. Meta-GGA (TPSS)

Com

puta

tional

cost

Acc

ura

cy

Uniform Electron Gas (UEG, HEG, Jellium)

ˆˆ1

) exp(1

)()(),(

322

.

12

.

2

*

121

k

iV

k

occoccKS

F

=⋅=

=⋅==

∫∫∫

∑∑ φφγ

krk

rkrrrrkk

kk

V

i )exp()(

rkrk

⋅=φ

)(tQ

∞→V

Density Matrix:

Uniform background (positive) density

Pag. 8

)( 6

ˆ)ˆexp( )2(

1122

2

ˆ

12

0

2

3rkQ

kdikkdk F

F

=⋅= ∫∫∫ ππ

krk

k

sn/ρ3/1

26

=

s

Fn

kρ

π( )3

4

3

srπρ =sr

Seitz Radius

(in real metals, rs=2..5)

Electron densityis uniform

Fermi wavevector

LDA Exchange

( )2

12

2

21 )()(

)(

),(),( rkQ

nnh F

s

KS

s

LDA

x

r

r

rrrr

ρ

ρ

γ−=

′−=

3/1

3/1

2

12

2

12

)(4

3

2

3

)(

9)(

2

1

)()(

2

1),(

2

1)];([

rr

r

rr

rrr

rrr

ρπ

πρ

ρρε

43421

−=

−=

=′−=′′−

′= ∫∫

sFs

F

s

xLDA

x

nkn

dr

rkQ

nd

h

>>

Pag. 9

Dirac-Exchange 1930

7386.0

42)(243421

=

xA

sFs nkn

rr dAE x

LDA

x ∫−= 3/4)(][ ρρ

Quite good performance (underestimation)

for atoms where the density is not uniform !

>>>>>>>>>>>))(()];([ rr ρερε LDA

x

LDA

x →simple function

(not a functional) of the density

Exchange Hole

),( 4 2

1),(

2

1),(

2

1)];([ 33

uhududu

hd

hx

xxx ru

urrr

rr

rrr πρε ∫∫∫ =

+=′

′−

′=

π4

)ˆ,( ˆ),(

2 urrur

uhduh

x

x

+=∫

Ex depends only on

spherically-averaged

exchange hole :

Neon

atom

Pag. 10

π4x

u [a.u.]

s

KS

sxn

nh)(

)(

),(),(

2

r

r

rrrrr

ρ

ρ

γ−=−==′

1)(

)(),(

)(),(

yidempotenc32

3 −= →′′−=′′ ∫∫s

sKSsx

n

nd

ndh

r

rrrr

rrrr

ρ

ργ

ρ

Exact Conditions for the exchange hole:

Satisfied by LDA! Satisfied by LDA!

atom

UEG correlation

High Density ( ) ...)(ln)()(ln)(),( 1100 +−+−= sssc rbrabra ζζζζζρε

Low Density( )

...)()(

),( 2

2/3

10 +−

++−

−=s

s

ss

xc

r

cf

r

f

r

cf ζζζρε

∞→sr

0→sr

<<<

Fits between the two limitsPerdew, Wang 1992

Perdew, Zunger 1981

Vosko,Wilk,Nusair 1980

Sun, Perdew, Seidl 2010

Spin polarization

Pag. 11

<<<<<<<

Overestimation for atoms:

Error cancellation with exchange !

Perdew, Zunger 1981 Sun, Perdew, Seidl 2010

Rung 2: Gradient Expansion

( ) rr dsE x

LDA

xx ...1][)( ][ 2 ++= ∫ µρερρ

ρρ

ρ

ρπ

ρ

)(2)3(2 3/43/12

Fks

∇=

∇=

Reduced gradient(adimensional)

81

10=xµ

Antoniewicz, Kleiman 1985

N-atomExact second order (GE2)In solids

2.28.0 max ≤≤ s

Pag. 12

( ) rr dtE c

LDA

cc ...)(][ )(][ 2 ++= ∫ ρβρερρ

6/7

6/1

2

34

)(2ρ

π

ρ

ρρ

ρ

∇=

∇=

skt

Reduced gradient for correlation

Ma,Brueckner 1968

066725.0)0( =→srβ

Gradient Expansion (GE2, GE4..) works badly ….

Generalized Gradient Expansion

rrr 3)())(()( ][ dsFE x

LDA

x

GGA

x ∫= ρερρ

Exchange Enhancement FactorPW86 Perdew Wang 1986 8148 cit.B86 Becke 1986 917 cit.

B88 Becke 1988 19414 cit.

PW91 Perdew Wang 1991 2817 cit.

PBE Perdew, Burke, Ernzerhof 1996 20630 cit.revPBE Zhang Yang 1998 521 cit.

Pag. 13

revPBE Zhang Yang 1998 521 cit.

RPBE Hammer, Hansen Norskov 1999 1678 cit.

AM05 Armiento Mattsson 2005 105 cit.

WC Wu Cohen 2006 244 cit.

PBEsol Perdew el at 2008 208 cit.

SOGGA Zhao Truhlar 2008 42 cit.

RGE Ruzsinszky Csonka Scuseria 2009

PBEint Fabiano Constantin Della Sala 2010

APBE Constantin Fabiano Laricchia Della Sala 2011

Many are optimized for certain propertiesFew are not empirical

(Citations from google-schoolar)

Fx Limits

PW91

RPBE

B88

PBE

revPBE

804.1)( <sFx

rd

hr

xx

)(

2

1),(

2

1)(

0

3 rr

rr

rrr

ρε →′

′−

′=

→∫13/1

2 )2( −= xAc

(fitted on atoms)

3/12

2 )6(2 π=c

0042.0=βBecke88

Exact asymptotic property

X

Pag. 14

][273.2][][ ρρρ LDA

xxcx EEE ≥≥

Lieb-Oxford boundSOGGA

PBEsol

Exact GE2 LDA 9.8

PBE 0.8

B88 0.2s

MARE(%)

for atoms

X

Fx

GGA performance

LDA (MAE=58mA)

Lattice constants for 60 solids

B0

underestimate

Pag. 15

PBE=55mA

TPSS=47mAPBEsol=29mA

SOGGA=29mA

AM05=35mA

WC=31mA

overestimate

PBE: quest for parameters

κ

µ

κκ

2

1

1)(s

sFPBE

x

+

−+= κ+=⇒∞→ 1xFs

210 sFs x µ+=⇒→

( ) rr dtHELDA

cc ),,(],[ )(],[ 0 ζρζρερζρ += ∫PBE correlation

Pag. 16

To satisfy the LDA response

1234.081

10≈=xµ

21961.0)3/( 2 ≈= πβµ MB

PBE

x

atoms improoves

revPBE)in as 1.2( large ≈κ

solid improoves

SOGGA)in as 0.5 ( small ≈κ

804.0=PBEκ

(PBEsol)

APBE: Constantin,Fabiano,Laricchia, Della Sala, Phys. Rev. Lett. 106, 186406 (2011)

26.0=xµSemiclassical neutral atom:

asymptotic-Z expansion

(APBE)

(LO bound)

Mole

cule

sAtomiz. Energy Bond-lenghts

too large

Too largeToo large

too small Too largeToo large

too small

Pag. 17

Solid

s

Too largeToo large

Two dimensional PBE scan: Fabiano, Constatin, Della Sala submitted

smalltoo large

too small

(Perdew) GGA cannot be exact for solid and molecules: PBE (APBE) best on the average

We need to go beyond GGA : meta-GGA, OEP, hybrids ...

Section II

Orbital Dependent Functional Exact Exchange

Pag. 18

Exact Exchange

Exchange potential for Hydrogen atom

exp(r)

n

C=

−−

−−== −−

4434421434212

13

1

13/1

4

3

4

3

3

4

)()(

PBE

x

PBE

xxx

GGA

xx

ds

dFs

ds

dsu

ds

dFstFnA

Ev

rr

δρ

δPBE

PBE1

PBE2� PBE diverges at nucleus (LDA,TPSS ok)

ren α

π

α 23

−=

Hydrogenic

density

Pag. 19

-C/r

s

� PBE diverges at nucleus (LDA,TPSS ok)� PBE asymptotically exponentially decreasing (TPSS,LDA too; B88 )

)()( rr uvx −Devia

tion

2r

a−→

1-e Self-interaction

1-electron (ns=1) or 2-electron (ns=2) systems

)(2

1)(

2

1)( 3 rr

rr

rr u

nd

n ss

x −=′′−

′−= ∫

ρε

s

c

s

xn

Edd

nE −=

′−

′−= ∫∫ rr

rr

rr 33)()(

2

1][

ρρρ

s

KS

s

EXX

xn

nh)(

)(

),(),(

2

r

r

rrrr

′−=

′−=′

ρ

ρ

γ

udv

)()()()()( x rr

rrrr −=′

′+= ∫δε

ρε

Pag. 20

For 1-e systems exchange cancelscoulomb exactly

GGA/Hybrid functionals:

large self-interaction errror

2) Orbital dependent Functionals

1) Perdew,Zunger (SIC) 9517 cit.

s

xn

udv

)(

)(

)()()()( x

x

r

r

rrrrr −=

′′+= ∫ δρ

δερε

)(

1)(

)22exp()(

exactKS,

H

4

−=

− →

− →

⇒− → ∞→

∞→

∞→

Ir

v

rv

rIrexact

c

rexact

x

rexact

ε

αρ r

r

r

Almbladh, von Barth 1985 (atoms) 1-e Self-interaction-free

Perdew, Levy 1997 Della Sala, Gorling 2002 (molecules)

Kohn-Sham Eigenvalues

Rydbergseries of virtual

orbitals

Difficulties in representing

virtual unbound orbitalin local basis sets

Common LDA/GGA gives largely incorrect largely incorrect absolute

eigenvalues (3eV), but acceptable energy differences (ok

for solid-state). LDA/GGA LUMO more diffuse

Exact-KSLDA/GGA

Pag. 21

HOMO energy in HF and Exact-KS issimilar and close to the Exp. I.P.

Exp. I.P.

Exact KS eigenvalues:

Response properties (TD-DFT, NMR,...)

-1/r corrected Functional

Leeuwen, Baerends 1994 (LB94) 714 cit.

rx

xvv

rLDA

xcxc

1

arcsinh(x) 31)()(

23/1 − →

+−= ∞→

ββρrr

3/4 x;05.0

ρ

ρβ

∇==

Becke, Roussel 1989 (BR) 193 cit.

dnvrbaba 1)()()()(

)(occ.

3 − →′′′

−= ∞→∑ ∫ rrrrr

rφφφφ

Pag. 22

rrr dvE slatx ∫= )()(2

1ρ

( )r

xxxxx

uhuduvr

xslat

1)exp(

2

1)exp(1

)3/exp(

8),( 4 )(

3/1exp.dens.

− →

−−−−

−−=≈ ∞→

∫πρ

π rr

],,,[ 2 τρρρ ∇∇= xx

)()()(

)( rrr

r respslatx

x vvE

v +==δρ

δr

dnvab

babasslat

)()( − →′

′−−= ∑ ∫ r

rrrr

ρ

)(2

3)( rr LDA

x

UEG

slat vv =

Meta-GGA:

1e-self interaction freer

vr

xc

1)( − → ∞→

r)(rxcEX

Orbital-Dependent Functionals

}])[{φxcE

Energy Functional of the Rung 4-5 are orbital dependent

functional (implicit density functional of the form)

Exact Exchange

∑ ∫∫ ′′−

′′−=

occ.

,

)()()()(

2}][{

ba

babasx dd

nE rr

rr

rrrr φφφφφ

Pag. 23

∫∫ ′−,2 ba rr

How to obtain KS equations with a local xc-potential ?

)(

}])[{)(

rr

δρ

φδ xcxc

Ev =

EXX: self-interaction-free, exactly !

Chain-Rule: Density Response

( )

∫ ′′

′=

′

rr

r

r

rrr

dv

E

sxc

s

v

xc

321321,')(

)(

)(

)(

χ

δ

δρ

δρ

δ

Density response

43421)(

)(

}][{

r

r

xcd

s

xc

v

E

δ

φδ

∑≠ −

′′=

′ all

ij ji

ij

i

s

i

v εε

φφφ

δ

δφ )()()(

)(

)( rrr

r

r)()()()(

2

1 2 rrrr iiisi v φεφφ =+∇−

Only virtual

Green’s

functions

Pag. 24

∑ ∑∑−

′′===′

. . . )()()()(2

)(

)()(2

)(

)()(

occ

a

occ

a

virt

s sa

sasas

s

s

s

s nv

nv εε

φφφφ

δ

δφφ

δ

δρχ

rrrr

r

rr

r

rrr,

Integral equation for the exchange-correlation potential

Density response

)()(),( rrrrr xcxcs ddv =′′′∫ χ

Only virtual

Chain Rule: RHS term

)()(

}][{)()(

)(

)(

)(

}][{

)()(

rr

rr

rr

r

rrr

j

i

xc

i

all

aj ji

ji

s

i

i i

xc

s

xcxc

E

dv

E

v

Ed

φδφ

φδ

εε

φφ

δ

δφ

δφ

φδ

δ

δ

′−=

=′′

′==

∑∑

∫∑

≠

∫ ∫∫ =′′′= 0)(),()( rrrrrrr ddvdd xcsxc χNote that

is an induced density)(rd

Pag. 25

is an induced density)(rxcd

∑∑∫ −==′′′

. .

ˆ)()(

2)(

)()(occ

a

virt

s

s

NL

xa

sa

sas

s

EXXEXX

xs vnv

Edv φφ

εε

φφ

δ

δχ

rr

rrrrr,

Integral equation for the EXX potential

∑′−

′−=′

occ. )()(),(

a

aaNL

xvrr

rrrr

φφNon-local Exchange

Exact-Exchange (EXX)

•1976 (Talman) EXX for atoms (GRID) 850 cit. => ok! •1997 (Gorling) EXX for solids (PW) 272 cit. => ok!•1999 (Gorling; Ivanov) EXX in GTO 243;217 cit. => bad potential !•2002 (Yang) OEP iterative minimiz. in GTO 179 cit. => more stable•2003 (Kummel) EXX on GRID 127cit. => iterative•2007 (Yang) GTO: good potential with penalty function 43 cit.

∑ ∑∫ −==′′′

. . )()(ˆ2

)()()(

occ

a

virt

s sa

sas

NL

xas

s

EXXEXX

xs vnv

Edv

εε

φφφφ

δ

δχ

rr

rrrrr,

Pag. 26

•2007 (Yang) GTO: good potential with penalty function 43 cit.•2007 (Hesselman) GTO: “balanced” basis set

EXX is local-basis set is numerically tricky���� more stable/efficient methods required for large systems

Total Energy deviation from HF (eV)

HF<EXX<LHF<KLI

44444 344444 21444444 3444444 21Term Response

ˆ)(

)(

)(2PotentialSlater

)()(

)(

)()()(

occ.

2occ.

3

jNL

LHF

x

Hi

i

i

s

EXX

x

ab

babas

KLI

x vvndnv φφρ

φ

ε

φφ

ρ

φφ−+

=

′′−

′′−= ∑∑ ∫

≠ r

r

r

rrr

rr

r

rrr

Effective EXX

The (HOMO) term is removed so thatvx vanishes asymptotically

Slater Potential goes asymptoticallyas -1/r (self-interaction free)

(KLI) Krieger, Li, Iafrate 1992 491 cit.

Pag. 27

444444 3444444 21Term Correction

ˆ)(

)()()()(

occ.

),(

jNL

LHF

x

NNij

i

ji

ssla

LHF

x vvnvv φφρ

φφ−+= ∑

≠ r

rrrr

vx vanishes asymptoticallyas -1/r (self-interaction free)

Della Sala, Gorling 2001 207 cit.

CEDA: Gritsenko, Baerends 2001

ELF: Staroverov, Scuseria, Davidson 2006

Localized Hartree-Fock (LHF) method

LHF potential functional

of the density matrix

LHF potential

Different asymptotic behavior

Atomic Shell structures

-1/r

re

α−

Pag. 28

No structure around

hydrogens

Section III

Hybrid Functionals

Pag. 29

Hybrid Functionals

Generalized Kohn-Sham

Non-interacting Kohn-Sham system Interacting MB system

)()( rr exactKS ρρ =)()( rr

exactMB ρρ =

( )[ ]∑=

+++=N

i

ixciext

KSvuVTH

1

)(ˆˆˆ rreeext

MBVVTH ˆˆˆˆ ++=

Single determinant Φ ; MB wavefunction Ψ

Partially-interacting Generalized Kohn-Sham(GKS) system

Local potential

ΨΨ= MBVHE ext ˆ

∫ +++= ][][)()(][][ 3 ρρρρρ xcJextni

VEEdvTE ext rrr

Pag. 30

Generalized Kohn-Sham(GKS) system

( )[ ]∑=

++++=N

i

ixc

NL

xciext

KSrwuVTH

1

ˆ)(ˆˆˆ rr

Single determinant Φ ; )()( rrexactKS ρρ =

Non-Local operator

We are not forced to solve only the very

simple KS non-interacting system.

We can solve also easily solve Hartree-Fock

][][

][)()(][][ 3

ρ

ρρ

ww

Jextwi

V

RE

EdvTE ext

+Φ+

++Φ=Φ ∫ rrra

NL

xc

a

as

w wn

E φφ ˆ2

][ ∑=Φ ][ρwR

[ ] )(ˆ2)(

rr

a

NL

xcs

a

w wnE

φδφ

δ= ( )

)(

][

rr

δρ

ρδ ww

Rr =

Interaction included

in GKS equations

ResidualInteraction

(local potential)

KS-DFT and beyond

)()(V̂)()(2

ext

22

rrrr iiiuvm

φεφ =

+++∇−

h

Exchange-Correlation OperatorClassical Electrostatic potential

∑′−

′−=′=

occ. )()(),(ˆ

a

aaNL

xvVrr

rrrr

φφHartree-Fock (exchange-only)

Kohn-Sham: )(ˆ rxcvV = (Local operator)

Pag. 31

Self-Energy (Quasi-particle)

′−a rr

),,(ˆixcV εrr ′Σ= (Σ=GW)

Generalized KS :

)(),(ˆ rrr xc

NL

xc rwV +′=

)(),( ˆ rrr xc

NL

x rvV +′= α(global. hybrid)

(range-sep.)

Brueckner Orbitals : ),(),(ˆ rrrr ′+′= BO

c

NL

x vvV

)(),(ˆ rrr c

NL

x vvV +′= (HFKS)

Global Hybrid

( ) 0,ˆ =′rrNL

xcw ( ) )(rr GGA

xcxc vr =KS

( ) )()()1( rrrGGA

xx

LDA

xxxc vbvar +−=Hybrid ( ) rr ′

′),(γ

( )rr

rrrr

′−

′−=′

),(,ˆ

γx

NL

xc aw ( ) )()()1( rrr GGA

c

GGA

xxxc vvar +−=Hybrid 1-param

(PBE0,α=1/4)

HF

Fra

ction

Pag. 32

( )

)(

)()()1(

r

rrr

GGA

c

xxxxxc

cv

vbvar

+

+−=Hybrid

3-param

(B3LYP, α=0.2)

( ) ∑′−

′−=′

occ

a

aaNL

xcwrr

rrrr

)()(,ˆ

φφHFKS ( ) )(rr GGA

cxc vr =

( )rr

rrrr

′−

′−=′

),(,ˆ

γx

NL

xc aw

B3LYP : Becke 1983 33811cit. (most cited)

HF

Fra

ction

M05-2X a=0.56 Zhao Thrular 2006 670 cit.

Hybrid performance

Hybrid methods improve bond-lengths of molecules

Pag. 33

Transition metals

GGA better than hybrids

for transition metals

PBE PBE0 B3YLP

Lattice

const

0.044(+) 0.024(+) 0.053(+)

Bulk

Moduli

12.3(-) 0.1 13.7(-)

B3LYP fails for solids: PBE0 much better

Range-separated Hybrid

( ) ),()(

,ˆ rrrr

rrrr γ

ω

′−

′−−=′

erfcaw

NL

xc

Screened

hybrid( ) ),(

)exp(,ˆ rr

rr

rrrr γ

ω

′−

′−−−=′ aw

NL

xc

Short-range=>a*HF

Long-range=>GGA Efficient for solids

Pag. 34

Heyd,Scuseria,Enzerhof (HSE) 2003 548 cit.

As PBE0, but more efficient

Long-range corrected Hybrid

( ) ),()(

,ˆ rrrr

rrrr ′

′−

′−−=′ γ

ωerfw

NL

xc

LC-hybrid

( ) ),()exp(1

,ˆ rrrr

rrrr ′

′−

′−−−=′ γ

ωNL

xcw

Short-range=>GGA

Long-range=>HF

For molecules

(charge-transfer in TD-DFT)

Pag. 35

Yanai, Tew, Handy 2004 (CAM-B3LYP) 578 cit.

Tawada et al. (LC-BYLP) 293 cit.

Hybrid improve electronic

properties (dipole, polarizability)

Summary & Conclusions

Pag. 36

Summary & Conclusions

OEP or Hybrid ?

Both OEP and hybrid methods include EXX in the total energy

However the eigenvalue spectrum is different

Pag. 37

Hybrid (GKS) larger gap than OEPx

KS vs GKS energy gap

Many-Body

Energy-Gap

δδ −+ ∂

∂−

∂

∂=

NN

deriv

gapN

E

N

EE

( ) ( ) AINENENENEE

IA

MB

gap −=−−−−+=

−−

444 3444 21444 3444 21]1[][][]1[

For an Exact Functional

Affinity>0 (ion is more stable) Ionization>0

MB

gap

KSderiv

gap EE <<∆= εLDA/GGAJanak 1978

Pag. 38

xc

KS

LLxc

NL

xcL

KS

L

KS

HHxc

NL

xcH

KS

H

vN

EA

vN

EI

∆+=−Σ+=∂

∂=−

=−Σ+=∂

∂=−

+

−

εφφε

εφφε44 344 21

0

xc

KSderiv

gapE ∆+∆= ε

GKSderiv

gapE ε∆=

OEP

LDA HF EXX MCY3

GKS

MCY3

OEP

Exp

-homo 9.5 18.1 15.9 15.2 14.6 I=15.7

-lumo 6.1 -2.3 10.3 1.4 10.6 A=1.3

-lumo-

∆xc

-2.3 1.4

Gap 3.4 20.4 5.6 13.7 4.0 14.4GKS

Casida 1999

OEP: good gap for neutral excitation; GKS good MB gap

Functionals (non-local, non-linear)

{ }

( )

( )

′

′′−

′−=

∇=

=

⇔

⇔⇔∫

∑

∑

)(

),(

2)(

2)(

)(

)(

)()(

3

2

2

2

rrrr

rrr

rr

rr

r

rr

KS

sEXX

x

occ

a

as

occ

a

as

occ

a

KSext

dn

n

n

vvγ

ρ

γε

φτ

φρ

φ

][ρxcE

Explicit Density

Functional

Meta-GGA

}][{ KS

ixcE φ(Occupied)

orbital-functional (EXX)

Hyper-GGA(local-hybrid)

Pag. 39

′⇔⇔

d)(unexplore others

),()()(

rrrr

KSKSext vv

γ

Potential

Functional

][ extxc vE

(local-hybrid)

][ KS

xcE γ

KS Density Matrix

Functional (EXX)

{ }ALL

i )(rφ

{ } { } ),,(,)( ωχεφ rrr ′↔ s

ALLKS

i

ALLKS

i

][ KS

xcE χKS Density Response

Functional

General orbital/eigenvalue

functional (GLPT; double hybrid)

References

Kummel, Kronik, Rev. Mod. Phys. 80 (2008) 3

Della Sala, Chem. Modell. 7 (2010) 115

OEP/EXX DFT Development

Scuseria, Staroverov in Theory and

Applications of Computational Chemistry,

(2005) pag. 669

Perdew, Kurth A Primer in Density

Functional Theory, Springer, Vol. 620,

(2003) pag. 1

Pag. 40

http://www.tddft.org/programs/octopus/wiki/index.php/Libxc:manual

List of functionals (libxc)

Baerends, Gritsenko, J. Phys. Chem. A

101 (1997) 5383

Postdoc, Ph.D. Positions available

fabio.dellasala@unisalento.it