Exact Exchange Within The FP-LAPW Method

Motivation and TheorySemiconductors and insulators

Magnetic metalsConclusions and outlook

Exact Exchange Within The FP-LAPW Method

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

Institut fur PhysikKarl-Franzens-Universitat Graz

Universitatsplatz 5A-8010 Graz, Austria

14th March 2005

S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method



MotivationDensity functional theoryPracticalities

Motivation

Potential problems with pseudopotentials

does not react properly to the solid state environment: norelaxation of core states

non-local matrix elements are for pseudo-wavefunctions

possible issues with non-locality





Motivation

Full-potential linearised augmented planewaves (FP-LAPW)

includes core orbitals

potential is fully described

space divided into muffin-tin and interstitial regions

most precise method available

MT

I

II

I





Density functional theory

In principle

|Ψ〉 v B

n, non-degenerate X X 0

n, degenerate × X 0

(n,m), non-degenerate X × ×

(n,m), degenerate X? × ×

In the absence of B and spin-orbit coupling, if the system isspin-degenerate then |Ψ〉 is not a functional of n.

However, |Ψ〉 is a functional of (n,m).

An energy functional with n, m independent, E[n,m], impliesB 6= 0 in general.





Density functional theory

We write

E[n,m] =

∫

v(r)n(r)dr +

∫

B(r) · m(r)dr + F [n,m]

whereF [n,m] ≡ Ts[n,m] + U [n] + Exc[n,m]

and

Ts[n,m] ≡ minφ→(n,m)

∑

ik

∫

φ†ik(r) · (−1

2∇2)φik(r)dr

(φik are 2-spinors)





Exact exchange

Use Fock exchange for Exc:

Ex[n,m] =

minφ→(n,m)

∑

ik,jq

∫

φ†ik(r1) · φjk+q(r1) φ

†jk+q(r2) · φik(r2)

|r1 − r2|dr1dr2

We require

δE[n,m]

δn(r)

∣

∣

∣

∣

N,m

= 0,

δE[n,m]

δm(r)

∣

∣

∣

∣

n

= 0.





Kohn-Sham equations

This leads us to(

−1

2∇2 + vs(r) + σ · Bs(r)

)

φik(r) = εikφik(r),

where

vs(r) = v(r) + vH(r) +δEx[n,m]

δn(r)

and

Bs(r) = B(r) +δEx[n,m]

δm(r).





Functional derivatives

δEx

δn(r)=∑

ik

∫

δEx

δφik(r1)

δφik(r1)

δvs(r2)

δvs(r2)

δn(r3)+

δφik(r1)

δBs(r2)·δBs(r2)

δn(r3)

+c.c.

δEx

δm(r)=∑

ik

∫

δEx

δφik(r1)

δφik(r1)

δvs(r2)

δvs(r2)

δm(r3)+

δφik(r1)

δBs(r2)

(

δBs(r2)

δm(r3)

)

+c.c.

χ(r, r′) ≡δn(r)

δvs(r′)

=∑

k

occ∑

i

unocc∑

j

φ†ik(r) · φjk(r) φ

†jk(r′) · φik(r′)

εik − εjk+ c.c.

etc.S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method




Practicalities

Essentials for exact exchange

efficient basis set for inverting χ

Poisson solver for complex densities: 〈φik|vNLx |φjk〉





Practicalities

The long range coulomb term of the NL matrix elements

〈φik|vNLx |φjk〉LR =

occ∑

lq

wq

4πΩ

q2ρ∗il(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 equivalentto that of the BZ

〈φik|vNLx |φjk〉LR ' 2

(

6Ω5

π

)1/3 occ∑

lq

wqρ∗il(q)ρlj(q).





Basis for expanding and inverting the response

χ(r, r′) =∑

k

occ∑

i

unocc∑

j

φ∗ik(r)φjk(r)φ∗

jk(r′)φik(r′)

εik − εjk+ c.c.

Choose the overlap densities

ρα(r) ≡ φ∗ik(r)φjk(r),

and complex conjugates, where α ≡ (ik, jk).

Diagonalise

Oαβ ≡

∫

dr ρ∗α(r)ρβ(r),

and eliminate eigenvectors with small eigenvalues.






Find transformation matrix C such that if

ρα(r) =∑

β

Cαβ

∑

γ

vβγ ργ(r)

then∫

dr ρ∗α(r)ρβ(r) = δαβ .

The matrix equation

CC† =(

v†Ov)−1

is solved by Cholesky decomposition.






1 By construction∫

dr ρα(r) = 0, so ρα form an ideal basisfor inversion of χ

2 Highly efficient: e.g. Si 70 basis set elements needed forconverged results.




Band-gapBand-gap problemd-band position

Semiconductors and insulators: band-gaps

Ge GaAs CdS Si ZnS-2

-1

0

1

2

Eg

KS

- Eg

(eV

)

PP-EXXLMTO-ASA-EXXFP-LDA

C BN Ar Ne Kr Xe

-6

-4

-2

0

ASA: Kotani PRL 74 2989 PP-EXX : Städele et al. PRB 59 10031, Magyar et al. PRB 69 045111





The fundamental band-gap for materials:

Eg = EKSg + ∆xc

1 Why EKSg is so close to experiments?

2 Why is EXX inconsistent in its treatment of semiconductors andinsulators?

Magyar et al. PRB 69 045111







-1

0

1

2

Eg

KS

- Eg

(eV

)

FP-EXXPP-EXXLMTO-ASA-EXXFP-LDA

C BN Ar Ne Kr Xe

-6

-4

-2

0







-1

0

1

2

Eg

KS

- Eg

(eV

)

FP-EXXPP-EXXLMTO-ASA-EXXFP-EXX-ncvFP-LDA

C BN Ar Ne Kr Xe

-6

-4

-2

0

http://arxiv.org/abs/cond-mat/0501353





Semiconductors and insulators: d-band position

Ge GaAs InP ZnS CdS

-4

-2

0

2

4

EdK

S -

Ed(e

V)

PP-EXXSIRCPP-SICFP-LDA

PP-SIC: Vogle et al. PRB 54 5495 PP-EXX: Rinke et al. /cond-mat/0502404






Ge GaAs InP ZnS CdS

-4

-2

0

2

4

EdK

S -

Ed(e

V)

PP-EXXGW-FPLMTOGW-LMTO-ASAGW-PP-EXXSIRCPP-SICFP-LDA

GW LMTO-ASA: Aryasetiawan et al. PRB 54 17564 FPLMTO: Kotani et al. SSC 121 461






Ge GaAs InP ZnS CdS

-4

-2

0

2

4

EdK

S -

Ed(e

V)

FP-EXXPP-EXXGW-FPLMTOGW-LMTO-ASAGW-PP-EXXSIRCPP-SICFP-EXX-ncvFP-LDA

http://arxiv.org/abs/cond-mat/0501353S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method



DOSPotentialCe

Magnetic metals: introduction to the problem

Magnetic moment in Bohr magneton

Compound FP-LDA Experiment

FeAl 0.71 0.0

1 P. Mohn et al. Phys. Rev. Lett. 87 196401 (2001): LDA+U

2 Petukhov et al. Phys. Rev. B 67 153106 (2003): LDA+U+DMFT




DOSPotentialCe

Magnetic metals: FeAl

-6 -4 -2 0 2 4 6Energy [eV]

0

1

2

3

DO

S (s

tate

s/eV

/spi

n)

LDAExpt




DOSPotentialCe


-6 -4 -2 0 2 4 6Energy [eV]

0

1

2

3

DO

S (s

tate

s/eV

/spi

n)

LDAEXXExpt




DOSPotentialCe





DOSPotentialCe


-6 -4 -2 0 2 4 6Energy [eV]

0

1

2

3

4

DO

S (s

tate

s/eV

/spi

n)

LDAEXX-sphEXX




DOSPotentialCe

α − γ Ce

140 160 180 200 220 240

Volume (au3)

-8651

-8650

-8649

-8648

Tota

l ene

rgy

(Ha)

α − γ Ce




ConclusionsOutlook

Conclusions

1 EXX within an all electron full potential method isimplemented within EXC!TING code.

2 A new and one of the most optimal basis is proposed forcalculating and inverting the response. This basis may beuseful for future TD-DFT and GW calculations.

3 Magnetic metals: Asymmetry in exchange potential is veryimportant to get the correct ground-state.

4 Semiconductors and insulators: Core-valence interaction iscrucial for correct treatment of the EXX. Lack of which couldlead to spurious agreement with experiments.




ConclusionsOutlook

Outlook

1 EXX can be generalised to handle non-collinear magnetism.

2 Future inclusion of exact correlation may be possible(multiconfiguration approach or adiabatic fluctuationdissipation).

3 LDA+U , SIC, RPA correlations and some sort of hybridfunctionals.




ConclusionsOutlook

Acknowledgements

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

Lars Nordstrom

Clas Persson

Sam Shallcross




ConclusionsOutlook

Magnetic metals: FeAl, Ni3Ga and Ni3Al


Compound FP-LDA FP-EXX Experiment

FeAl 0.71 0.0 0.0

Ni3Ga 0.79 0.0 0.0

Ni3Al 0.70 0.20 0.23

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




ConclusionsOutlook

Magnetic moment: Fe, Ni


Metal Expt FP-EXX LMTO-EXX FP-LDA HF

Fe 2.12 2.05 3.40 2.18 2.90

Ni 0.56 0.59 0.68 0.63 0.76

LMTO-EXX: Kotani JPCM 10 9241 (1998)




ConclusionsOutlook

DOS for Ni

-10 -8 -6 -4 -2 0 2 4 6Energy (eV)

-3

-2

-1

0

1

2

3

DO

S (s

tate

s/eV

/spi

n)

EXXLDA




ConclusionsOutlook

DOS for Fe

-10 -5 0 5 10Energy(eV)

-1

0

1

2

3

DO

S (s

tate

s/eV

/spi

n)

EXXLDA




ConclusionsOutlook

Convergence-Fe

0 5 10 15 20 25 30Empty states

2.2

2.4

2.6

2.8

3

Mom

ent (

µ b)

Fe




ConclusionsOutlook

Convergence-Si

0 10 20 30 40Empty states

3.5

3.6

3.7

3.8

3.9

4

Eg(e

V)

Si




ConclusionsOutlook

Exchange discontinuity

Magnetic moment in Bohr magnetonCompound Eg (eV) ∆x

Ge 1.42 3.61

GaAs 2.42 2.37

CdS 4.08 2.75

Si 3.58 4.67

ZnS 3.80 5.19

C 6.87 7.57

BN 12.19 4.36

Ne 16.3 10.19

Ar 11.95 5.63

Kr 10.59 5.57

Xe 10.47 4.84


Exact Exchange Within The FP-LAPW Method

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