exact exchange in density functional theory

Theory and motivation Magnetic metals

Semiconductors and insulators Conclusions and outlook

Exact Exchange in Density Functional Theory

S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl

Institut fur PhysikKarl-Franzens-Universit ta Graz

Universita tsplatz 5A-8010 Graz, Austria

19th January 2005

S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory

Density functional theory Kohn-Sham equationsExchange-correlation functionalsPracticalities

Density functional theory

Many-body theory involves solving for functions in 3N coordinates: Ψ(r1, r2, . . . , rN )

Density functional theory (DFT) requires solving for functions in 3 coordinates: n(r)

Many-body theory involves solving for functions in 3N coordinates: Ψ(r1, r2, . . . , rN )

Density functional theory (DFT) requires solving for functions in 3 coordinates: n(r)

Hohenberg-Kohn Theorems (1964)1

External potential v(r) is uniquely determined by n(r) The variational principle holds

E0 = Ev0 [n0] < Ev0 [n]3

Ev0 [n] = F [n] + ¸ dr

v0(r)n(r)

E0 = Ev0 [n0] < Ev0 [n]3

Ev0 [n] = F [n] + ¸ dr

v0(r)n(r)

E0 = Ev0 [n0] < Ev0 [n]3

Ev0 [n] = F [n] + ¸ dr

v0(r)n(r)

DFT is an exact theory for interacting systems in the ground state

Kohn-Sham equations

Find set of auxilliary single particle orbitals such that

1 2. .

− 2 ∇ + vs(r) φi(r) = siφi(r)

and occ

n(r) = . |φi(r)|2.i

If1 ¸

F [n] = Ts[n] + 2 dr drt

n(r)n(rt)|r − rt| + Exc[n]

Kohn-Sham equations

Find set of auxilliary single particle orbitals such that

1 2. .

− 2 ∇ + vs(r) φi(r) = siφi(r)

and occ

n(r) = . |φi(r)|2.i

If1 ¸

F [n] = Ts[n] + 2 dr drt

n(r)n(rt)|r − rt| + Exc[n]

Kohn-Sham equations

¸vs[n](r) = v(r) + drt n(rt)

|r − rt| + vxc[n](r)

where vxc[n](r) = δExc[n] .

δn(r)

Many sins hidden in Exc[n] !

Kohn-Sham equations

¸vs[n](r) = v(r) + drt n(rt)

|r − rt| + vxc[n](r)

where vxc[n](r) = δExc[n] .

δn(r)

Many sins hidden in Exc[n] !

Exchange-correlation functionals

First generation : Local density approximation (LDA)

E LDAxc

¸unif[n] = dr n(r)exc (n(r))

Second generation : Generalised gradient approximations

EGGAxc

¸[n] = dr f (n(r), ∇n(r))

EMeta−GGAxc

¸[n] = dr g(n(r), ∇n(r), τ (r))

E LDAxc

EGGAxc

¸[n] = dr f (n(r), ∇n(r))

EMeta−GGAxc

¸[n] = dr g(n(r), ∇n(r), τ (r))

E LDAxc

EGGAxc

¸[n] = dr f (n(r), ∇n(r))

EMeta−GGAxc

¸[n] = dr g(n(r), ∇n(r), τ (r))

Development of new functionals leads not only to improved accuracy but also correct qualitative features

Third generation: Exact exchange (EXX)

Neglect correlation and use the Hartree-Fock exchange energy

Ex[n] = −

1 occ ¸dr drt φ∗ ∗

ti (r)φj (r )φj (r)φi(r )

|r − rt|

To solve the Kohn-Sham system we require

δEx[n]vx[n](r) =

δn(r)

Third generation: Exact exchange (EXX)

Neglect correlation and use the Hartree-Fock exchange energy

Ex[n] = −

1 occ ¸dr drt φ∗ ∗

ti (r)φj (r )φj (r)φi(r )

|r − rt|

To solve the Kohn-Sham system we require

δEx[n]vx[n](r) =

δn(r)

Using the functional derivative chain rule:

occ ¸vx[n](r) = . dr drtt

.δEx δφi(rtt) +

iδφi(rtt ) δvs(rt) δφ∗(rtt ) δvs(rt)

δEx δφ∗(rtt ) . δvs(rt)δn(r)

occ unocc

(φ |vˆ |φ ) j iφ∗(rt)φ (rt)si − sj

+ c.c.δvs(rt) δn(r) ,

ik NLx(φ |vˆ

φ ) = wjk kt

¸dr drt tik lktφ∗ (r)φlk (r)φ∗ (rt)φjk(rt)

|r − rt|.

Using the linear-response operatortχ(r, r ) ≡ δn(r)

s tδv (r )

=occ unocc

∗i j i∗ t

tφ (r)φ (r)φ (r )φ (r )j

si − sj+ c.c.

we have

δn(r)δvs(rt) = χ˜−1(r, rt).basis

Exact exchange is NOT

Hartree-Focka mean-field theoryan attempt to solve the quasi-particle equation a parameterised correction to LDAeasy to implement

Practicalities

Pseudopotentials (PP)atomic core is frozen and represented by a non-local potentialdoes not react properly to the solid state environment: no relaxation of core statesplanewaves are used as the basis

Practicalities

Atomic sphere approximation (ASA)

includes core orbitalspotential is spherically symmetric muffin-tins are space-filling

Practicalities

Full-potential linearised augmented planewaves (FP-LAPW) includes core orbitals

potential is fully describedspace divided into muffin-tin and interstitial regions

most precise method available

Practicalities

Basis setAn efficient basis set is required so that the response

tχ(r, r ) ≡ δn(r)s tδv (r )

can be invertedBasis set should not contain constant functions

response

Practicalities

Basis setAn efficient basis set is required so that the response

tχ(r, r ) ≡ δn(r)s tδv (r )

can be invertedBasis set should not contain constant functions

response

Practicalities

Choose the overlap densities

ikρα(r) ≡ φ∗ (r)φjk(r),

and complex conjugates, where α ≡ (ik,

jk). Diagonalise ¸

αOαβ ≡ dr ρ∗ (r)ρβ (r),

and eliminate eigenvectors with small eigenvalues.

Practicalities

Choose the overlap densities

ikρα(r) ≡ φ∗ (r)φjk(r),

and complex conjugates, where α ≡ (ik,

jk). Diagonalise ¸

αOαβ ≡ dr ρ∗ (r)ρβ (r),

and eliminate eigenvectors with small eigenvalues.

Practicalities

Find transformation matrix C such that if

ρ˜ (r) = Cβ. .

αβ γ γv ρ (r)

αdr ρ˜∗ (r)ρ˜β (r) = δαβ .The matrix equationCC† =

is solved by Cholesky decomposition.

Practicalities

Find transformation matrix C such that if

ρ˜ (r) = Cβ. .

αβ γ γv ρ (r)

αdr ρ˜∗ (r)ρ˜β (r) = δαβ .The matrix equationCC† =

is solved by Cholesky decomposition.

Practicalities

By construction ¸ dr ρ˜α(r) = 0, so ρ˜α form an ideal basis

for inversion of χ

Practicalities

NL Lastly, given q ≡ k − kt, the long range coulomb term of theNL matrix

elementsNL φ

ik x jk LR

lq(φ |vˆ | ) = wq

4πΩq2

ilρ∗ (q)ρlj (q),

where ρil(q) and ρlj (q) are the pseudo-charge densities.

Poor convergence with respect to the number of q-points.

Approximate it by an integral over a sphere of volume equivalent to that of the BZ

NLik x jk LR

(φ |vˆ |φ ) c 2 π

. 6Ω5 . 1 /3 occ.

lqq ∗

il l jw ρ (q)ρ (q).

Practicalities

elementsNL φ

lq(φ |vˆ | ) = w

ik x j k LR q4πΩq2

NLik x

(φ |vˆ|

jk LRφ ) c 2 π

. 6Ω5 . 1 /3 occ.

lqq ∗

Practicalities

elementsNL φ

ik x jk LR

lq(φ |vˆ | ) = wq

4πΩq2

NLik x

(φ |vˆ|

jk LRφ ) c 2 π

. 6Ω5 . 1 /3 occ.

lqq ∗

Applications

EXX Applied to1

Magnetic metals Semiconductors and insulators

DOSPotential

Magnetic metals: introduction to the problem

Magnetic moment in Bohr magnetonCompoun

dFP-LDA Experimen

tFeAl 0.71 0.0

P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U +DMFT

DOSPotential

Magnetic metals: introduction to the problem

dFP-LDA Experimen

tFeAl 0.71 0.0

P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U +DMFT

DOSPotential

Magnetic metals: FeAl

-6 -4 -2 2 4 60Energy [eV]

LDAExpt

DOSPotential

-6 -4 -2 2 4 60Energy [eV]

LDA EXXExpt

DOSPotential

LDA EXX

FeAl AlS. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange in Density Functional Theory

DOSPotential

-6 -4 -2 2 4 60Energy [eV]

LDAEXX-sphEXX

DOSPotential

Magnetic metals: stringent tests for EXX

dFP-LDA FP-EXX Experimen

tFeAl 0.71 0.0 0.0

Ni3Ga 0.79 ? 0.0Ni3Al 0.70 ? 0.23

http://arxiv.org/abs/cond-mat/0501258

DOSPotential

Magnetic metals: FeAl, Ni3Ga and Ni3Al

dFP-LDA FP-EXX Experimen

tFeAl 0.71 0.0 0.0

Ni3Ga 0.79 0.0 0.0Ni3Al 0.70 0.20 0.23

Band-gapBand-gap problemd-band position

-2Ge GaAs CdS

E g- E

PP-EXX FP-LDA

Si ZnS C BN Ar Ne Kr Xe

Semiconductors and insulators: band-gapsPP-EXX : Städele et al. PRB 59 10031 PP-EXX: Magyar et al. PRB 69 045111

The fundamental band-gap for materials:

Eg = A− I = (EN +1 −EN )−(EN −EN −1)

δE Eg = δn+ −

δn−

E[n] = T [n] + Us[n] + Exc[n]

δT δT

Eg = ( δn+ −

δn− ) + ( δn+ −

δn− )

gEg = EKS + ∆xc

Eg = A− I = (EN +1 −EN )−(EN −EN −1)

δE Eg = δn+ −

δn−

E[n] = T [n] + Us[n] + Exc[n]

δT δT

Eg = ( δn+ −

δn− ) + ( δn+ −

δn− )

gEg = EKS + ∆xc

Eg = A− I = (EN +1 −EN )−(EN −EN −1)

δE Eg = δn+ −

δn−

E[n] = T [n] + Us[n] + Exc[n]

δT δT

Eg = ( δn+ −

δn− ) + ( δn+ −

δn− )

gEg = EKS + ∆xc

Eg = A− I = (EN +1 −EN )−(EN −EN −1)

δE Eg = δn+ −

δn−

E[n] = T [n] + Us[n] + Exc[n]

δT δT

Eg = ( δn+ −

δn− ) + ( δn+ −

δn− )

gEg = EKS + ∆xc

Eg = A− I = (EN +1 −EN )−(EN −EN −1)

δE Eg = δn+ −

δn−

E[n] = T [n] + Us[n] + Exc[n]

δT δT

Eg = ( δn+ −

δn− ) + ( δn+ −

δn− )

gEg = EKS + ∆xc

Semiconductors and insulators: band-gaps

E g- E

PP-EXXLMTO-ASA-EXX FP-LDA

Kotani PRL 74 2989

-2Ge GaAs CdS Si ZnS C BN Ar Ne Kr Xe

E g- E

FP-EXX PP-EXXLMTO-ASA-EXXFP-LDA

E g- E

FP-EXX PP-EXXLMTO-ASA-EXXFP-EXX-ncv FP-LDA

Semiconductors and insulators: d-band position

Ge GaAs InP ZnS CdS

PP-EXX SIRC PP-SIC FP-LDA

SIC: Vogle et al. PRB 54 5495 PP-EXX: Rinke et al. (unpublished)

Ge GaAs InP ZnS CdS

PP-EXXGW-FPLMTO GW-LMTO-ASA GW-PP-EXX SIRCPP-SIC FP-LDA

GW LMTO-ASA: Aryasetiawan et al. PRB 54 17564 FPLMTO: Kotani et al. SSC 121

Ge GaAs InP ZnS CdS

FP-EXX PP-EXXGW-FPLMTOGW-LMTO-ASA GW-PP-EXX SIRCPP-SICFP-EXX-ncv FP-LDAhttp://arxiv.org/abs/cond-mat/0501353

Conclusions Outlook

Conclusions 1

EXX within an all electron full potential method is implemented within EXC!TING code. Right now this is the only FP code to be able to do EXX.A new and one of the most optimal basis is proposed for calculating and inverting the response. This basis may be useful for future TD-DFT and GW calculations.Magnetic metals: Asymmetry in exchange potential is very important to get the correct ground-state.Semiconductors and insulators: Core-valence interaction is crucial for correct treatment of the EXX.Lack of Asymmetry and/or core-valence interaction could lead to spurious agreement with experiments.

Conclusions Outlook

Conclusions 1

Conclusions Outlook

Conclusions 1

Conclusions Outlook

Conclusions 1

Conclusions Outlook

Conclusions 1

Conclusions Outlook

Outlook 1

EXX can be generalised to handle non-collinear magnetism (derivatives w.r.t. nσσt (r) ).Future inclusion of exact correlation may be possible (multiconfiguration approach or adiabatic fluctuation dissipation)

Conclusions Outlook

Outlook 1

EXX can be generalised to handle non-collinear magnetism (derivatives w.r.t. nσσt (r) ).Future inclusion of exact correlation may be possible (multiconfiguration approach or adiabatic fluctuation dissipation)

Conclusions Outlook

Acknowledgements

Prof. P. MohnMagnetic metals work done in collaboration with Dr. C. Persson.

Austrian Science Fund (project P16227)EXCITING network funded by the EU (HPRN-CT-2002-00317)

Code available at:ht tp : / /phys i k . k fun ig raz .ac .a t /∼kde / sec re t garden/exc i t ing .html