exact exchange within the fp-lapw method
TRANSCRIPT
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
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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)
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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).
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
Practicalities
Essentials for exact exchange
efficient basis set for inverting χ
Poisson solver for complex densities: 〈φik|vNLx |φjk〉
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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).
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
Basis for expanding and inverting the response
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
Basis for expanding and inverting the response
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
Basis for expanding and inverting the response
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
MotivationDensity functional theoryPracticalities
Basis for expanding and inverting the response
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
Band-gapBand-gap problemd-band position
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
Band-gapBand-gap problemd-band position
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
)
FP-EXXPP-EXXLMTO-ASA-EXXFP-LDA
C BN Ar Ne Kr Xe
-6
-4
-2
0
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
)
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
Band-gapBand-gap problemd-band position
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
Band-gapBand-gap problemd-band position
Semiconductors and insulators: d-band position
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
Band-gapBand-gap problemd-band position
Semiconductors and insulators: d-band position
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
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
DOSPotentialCe
Magnetic metals: FeAl
-6 -4 -2 0 2 4 6Energy [eV]
0
1
2
3
DO
S (s
tate
s/eV
/spi
n)
LDAEXXExpt
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
DOSPotentialCe
Magnetic metals: FeAl
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
DOSPotentialCe
Magnetic metals: FeAl
-6 -4 -2 0 2 4 6Energy [eV]
0
1
2
3
4
DO
S (s
tate
s/eV
/spi
n)
LDAEXX-sphEXX
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
DOSPotentialCe
α − γ Ce
140 160 180 200 220 240
Volume (au3)
-8651
-8650
-8649
-8648
Tota
l ene
rgy
(Ha)
α − γ Ce
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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.
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
ConclusionsOutlook
Acknowledgements
Austrian Science Fund (project P16227)EXCITING network funded by the EU (HPRN-CT-2002-00317)
Lars Nordstrom
Clas Persson
Sam Shallcross
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
ConclusionsOutlook
Magnetic metals: FeAl, Ni3Ga and Ni3Al
Magnetic moment in Bohr magneton
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
ConclusionsOutlook
Magnetic moment: Fe, Ni
Magnetic moment in Bohr magneton
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)
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
ConclusionsOutlook
DOS for Fe
-10 -5 0 5 10Energy(eV)
-1
0
1
2
3
DO
S (s
tate
s/eV
/spi
n)
EXXLDA
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
ConclusionsOutlook
Convergence-Si
0 10 20 30 40Empty states
3.5
3.6
3.7
3.8
3.9
4
Eg(e
V)
Si
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method
Motivation and TheorySemiconductors and insulators
Magnetic metalsConclusions and outlook
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
S. Sharma, J. K. Dewhurst and C. Ambrosch-Draxl Exact Exchange Within The FP-LAPW Method