Density-functional methods for electronic systems at finite temperatures Andreas G¨orling, Max Greiner, Hannes Schulz, Patrick Bleiziffer, and Andreas Heßelmann Lehrstuhl f¨ ur Theoretische Chemie Universit¨ at Erlangen–N¨ urnberg J. Gebhardt, G. Gebhardt, T. Gimon, W. Hieringer, C. Neiß, I. Nikiforidis, K.-G. Warnick, T. W¨ olfle A. G¨ orling (University Erlangen–N¨ urnberg) Los Angeles 2012 1 / 36
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Density-functional methods for electronic systemsat finite temperatures
Andreas Gorling, Max Greiner, Hannes Schulz, Patrick Bleiziffer,and Andreas Heßelmann
J. Gebhardt, G. Gebhardt, T. Gimon, W. Hieringer, C. Neiß, I. Nikiforidis,K.-G. Warnick, T. Wolfle
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 1 / 36
Overview
1 IntroductionElectronic structures at finite temperatures with DFTOrbital-dependent functionals
2 Exact-exchange (EXX) Kohn-Sham methods
3 Finite-temperature EXX Kohn-Sham methodsElectronic structure methods for grand canonical ensemblesKohn-Sham formalism for finite temperaturesFinite-temperature EXX-KS methodExamples for applications
4 Direct RPA and EXXRPA correlation energyFluctuation dissipation theorem for DFT correlation energyEXXRPA methodsPerformance of EXXRPA methods
5 Literature
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 2 / 36
Electronic structures at finitetemperatures with DFT
Finite-temperature density-functional theoryDFT for grand canonical ensembles
Problem: µ- and T -dependent exchange-correlation functionals required
First step: Temperature-dependent exact-exchange formalism,i.e., exchange energy and Kohn-Sham exchange potential are treated exactly
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 3 / 36
Orbital-dependent functionalsHistoric development of DFT
Ground state energy of an electronic systemE0 = Ts + U + Ex + Ec +
∫dr vnuc(r)ρ(r)
Thomas-Fermi-Dirac
E0 = Ts[ρ]+ U [ρ] +Ex[ρ] + Ec[ρ]+∫dr vnuc(r)ρ(r)
δE/δρ(r) = µ
Conventional Kohn-Sham
E0 = Ts[φi]+ U [ρ] +Ex[ρ] + Ec[ρ]+∫dr vnuc(r)ρ(r)
[T + vnuc + vH + vx + vc]φi = εiφi
Kohn-Sham with orbital dependent functionals
E0 = Ts[φi]+U [ρ]+Ex[φi] +Ec[φi]+∫dr vnuc(r)ρ(r)
[T + vnuc + vH + vx + vc]φi = εiφi
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 4 / 36
Exact treatment of KS exchangeOEP equation
Exchange energy
Ex = −1
2
occ.∑
i,j
∫dr dr′
φi(r′)φj(r
′)φj(r)φi(r)
|r′ − r|
Exchange potential vx(~r) =δEx[φi]δρ(~r)∫
dr′ χs(r, r′) vx(r
′) = t(r)
KS response function χs(r, r′) =
δρ(r)
δvs(r′)= 4
occ.∑
i
unocc.∑
a
φi(r)φa(r)φa(r′)φi(r′)
εi − εa
t(r) =δExδvs(r)
= 4
occ.∑
i
unocc.∑
a
φi(r)φa(r)⟨a|vNL
x |i⟩
εi − εa
Plane wave methods for solid numerically stableGaussian basis set methods for molecules numerically demanding
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 5 / 36
Exact-exchange Gaussian basis set KS method
Auxiliary basis set: Electrostatic potential of Gaussian functions
vx(r) =∑
k
vx,k fk(r) with fk(r) =
∫dr′ gk(r′)/|r− r′|
Incorporation of exact conditions to treat asymptotic of vx(r)∫dr ρx(r) = −1 with ρx(r) =
∑
k
vx,k gk(r)
〈φHOMO|vx|φHOMO〉 = 〈φHOMO|vNLx |φHOMO〉
Construction and balancing scheme for auxiliary and orbital basis sets,orbital basis set needs to be converged for given auxiliary basis set,uncontracted orbital basis sets required
JCP 127, 054102 (2007)
By preprocessing of auxiliary basis set and singular value decomposition ofresponse function, EXX calculations with standard contracted orbital basis sets(aug-cc-pCVQZ) possibleA. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 6 / 36
EXX vs. GGA (PBE) orbitals of methane
2 t2u −2.934 eV
3 a1g −4.438 eV
1 t2u −14.724 eV
2 a1g −22.437 eV
-20
-15
-10
-5
0
2 t2u +0.533 eV
3 a1g −0.396 eV
1 t2u −9.448 eV
2 a1g −17.054 eV
contour value 0.032A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 7 / 36
EXX vs. GGA (PBE) orbitals of methane
2 t2u −2.934 eV
3 a1g −4.438 eV
1 t2u −14.724 eV
2 a1g −22.437 eV
-20
-15
-10
-5
0
2 t2u +0.533 eV
3 a1g −0.396 eV
1 t2u −9.448 eV
2 a1g −17.054 eV
contour value 0.013A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 8 / 36
Band gaps of semiconductors
FLAPW vs. PP EXX band gaps
EXX+VWNc Exp.FLAPWa PPb
Si Γ→ Γ 3.21 3.26 3.4Γ→ L 2.28 2.35 2.4Γ→ X 1.44 1.50
SiC Γ→ Γ 7.24 7.37Γ→ L 6.21 6.30Γ→ X 2.44 2.52 2.42
Ge Γ→ Γ 1.21 1.28 1.0Γ→ L 0.94 1.01 0.7Γ→ X 1.28 1.34 1.3
GeAs Γ→ Γ 1.74 1.82 1.63Γ→ L 1.86 1.93Γ→ X 2.12 2.15 2.18
C Γ→ Γ 6.26 6.28 7.3Γ→ L 9.16 9.18Γ→ X 5.33 5.43
aPRB 83, 045105 (2011)bPRB 59, 10031 (1997)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 9 / 36
Summary EXX-KS method
EXX-KS methods solve the problem of Coulomb self-interactions and,in contrast to GGA-KS methods, yield qualitatively correct KS orbitaland eigenvalue spectra.
EXX orbitals and eigenvalues are well-suited as input for TDDFTmethods. Problem of treating excitations with Rydberg character issolved.
Correlation functional supplementing exact treatment of exchangerequired
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 10 / 36
U [ρ] + Ex[ρ] = 〈Φ|Vee|Φ〉U [ρ] + Ex[ρ, T ] = Tr ΓKS[ρ, T ]Vee
Ec[ρ] = F [ρ]− Ts[ρ]− U [ρ]− Ex[ρ]
Ec[ρ, T ] = F [ρ, T ]− Ts[ρ, T ]− U [ρ]− Ex[ρ, T ]
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 13 / 36
Temperature-dependentexact-exchange energy
[T + vs]ΦN,n = EKSN,nΦN,n
ΓKSvs,T,µ=
∑
N
∑
n
exp[(−1/kT )(EKSN,n− µN)]
ZKSvs,T,µ|ΦN,n〉〈ΦN,n|
Ts = Tr ΓKSvs,T,µ[T+k T ln ΓKS
vs,T,µ]=∑
i
[fi〈i|− 1
2~∇2|i〉+ kT [fi lnfi + (1−fi) ln(1−fi)]
]
U + Ex = Tr ΓKSvs,T,µVee =
1
2
∑
i
∑
j
gij [〈ij|ij〉 − 〈ij|ji〉]
gij = fi fj fi =1
1 + e(1/kT )(εi−µ)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 14 / 36
Temperature-dependent exact-exchange potential
OEP equation
∫dr′ Xs(T,N, r, r
′) vx(T,N, r, r′) = tx(T,N, r)
KS response function
Xs(T,N, r, r′) =
δρ(r)
δvs(r′)
=∑
i
fi∑
j 6=i
[φ†i (r)φj(r)φ†j(r
′)φi(r′)
εi − εj+ c.c.
]+∑
i
φ†i(r)φi(r)δfi
δvs(r′)
Right-hand side
tx(T,N, r)=∑
i
fi∑
j 6=i
[〈φi|vNL
x |φj〉φ†j(r)φi(r)
εi − εj+ c.c.
]+∑
i
〈φi| vNLx |φi〉
δfiδvs(r)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 15 / 36
Test application bulk aluminum I
Free energy A = E − TS for bulk Al (fcc lattice, 6×6×6 k-points)
-54
-53
-52
-51
-50
-49
-48
1 2 3 4 5 6 7 8
A[eV
]
Volume [× exp. Vol. (293 K)]
293 K10000 K30000 K50000 K70000 K
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 16 / 36
Test application bulk aluminum II
Contributions to free energy A = E − TS
-54
-53
-52
-51
-50
-49
-48
1 2 3 4 5 6 7 8
A[eV]
Volume [× exp. Vol. (293 K)]
293 K10000 K30000 K50000 K70000 K
-35
-30
-25
-20
-15
-10
-5
1 2 3 4 5 6 7 8
-T*S
[eV]
Volume [× exp. Vol. (293 K)]
293 K10000 K30000 K50000 K70000 K
-55
-50
-45
-40
-35
-30
-25
-20
1 2 3 4 5 6 7 8
E total[eV]
Volume [× exp. Vol. (293 K)]
293 K10000 K30000 K50000 K70000 K
-16
-14
-12
-10
-8
-6
-4
-2
1 2 3 4 5 6 7 8
E X[eV]
Volume [× exp. Vol. (293 K)]
293 K10000 K30000 K50000 K70000 K
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 17 / 36
Test application bulk aluminum III
Band structures of aluminum
exp. Vol. (293 K)
-15
-10
-5
0
5
10
15
20
25
30
35
W L Γ X W K
ε i[eV]
293 K30000 K70000 K
5.1 × exp. Vol. (293 K)
-5
0
5
10
15
20
W L Γ X W K
ε i[eV]
293 K30000 K70000 K
0
0.2
0.4
0.6
0.81
-20
-10
010
2030
f(εi)
ε i−µ
[eV]f
(εi)=
1
1+exp(
1kBT(ε
i−µ))
293
K30
000
K70
000
K
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 18 / 36
Summarytemperature-dependent EXX method
Enables exact treatment of temperature in electronic structure calculationsat exchange level
Correlation functional supplementing exact treatment of exchange required
What can we learn from electronic structure calculations at hightemperatures? (Would inclusion of noncollinear spin, spin-orbitinteractions, or magnetic fields be of interest?)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 19 / 36
Adiabatic-connection fluctuation-dissipationtheorem for DFT correlation energy I
Ec =−1
2π
∫ 1
0
dα
∫drdr′
1
|r− r′|
∫ ∞
0
dω[χα(r, r′, iω) − χ0(r, r′, iω)
]
Integration of response functions along complex frequencies
−1
2π
∫ ∞
0
dω
∫dr dr′g(r, r′) χα(r, r′, iω) =
=
∫dr dr′ g(r, r′)
[ρα2 (r, r′) − 1
2ρ(r)ρ(r′) + ρ(r)δ(r− r′)
]
∞∫
0
dωa
a2 + ω2=π
2later on g(r, r′) =
1
|r− r′|
χα(r, r′, iω) = −2∑
n 6=0
En − E0
(En − E0)2 + ω2〈Ψα
0 |ρ(r)|Ψαn〉 〈Ψα
n|ρ(r′)|Ψα0 〉
Vc(α) =⟨Ψ0(α)|Vee|Ψ0(α)
⟩−⟨Φ0|Vee|Φ0
⟩
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 20 / 36
Adiabatic-connection fluctuation-dissipationtheorem for DFT correlation energy II
Ec =−1
2π
∫ 1
0
dα
∫drdr′
1
|r− r′|
∫ ∞0
dω[χα(r, r′, iω) − χ0(r, r′, iω)
]
Integration along adiabatic connection
Ec =
1∫0
dα Vc(α) with Vc(α) =⟨
Ψ0(α)|Vee|Ψ0(α)⟩−⟨
Φ0|Vee|Φ0
⟩Required input quantities are χ0(r, r′, iω) and χα(r, r′, iω)
KS response function χ0(r, r′, iω)
χ0(r, r′, iω) = −4occ∑i
unocc∑a
εaiε2ai + ω2
ϕi(r)ϕa(r)ϕa(r′)ϕi(r′)
Response functions χα(r, r′, iω) from EXX-TDDFT
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 21 / 36
Exact frequency-dependent exchange kernel
OEP-like equation for sum fHx(ω, r, r′) of Coulomb and EXX kernel∫
– Geometries from test set of Jurecka, P.; Sponer, J.; Cerny, J.; Hobza, P.; Phys. Chem. Chem. Phys. 2006, 8, 1985– Reference: CBS CCSD(T) values from Takatani, T.; Hohenstein, E. G.; Malagoli, M.; Marshall, M. S.; Sherrill, C. D.; J. Chem. Phys. 2010, 132, 144104
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 33 / 36
EXX-RPA vs. HF based RPA
Deviations of reaction energies (RMS) from CCSD(T)
rCCD and SzOst-RPA yield distinctively larger deviations(larger deviations than HF)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 34 / 36
Summary EXX-RPA
EXX-RPA correlation functional combines accuracy at equilibriumgeometries with a correct description of dissociation (static correlation)and a highly accurate treatment of VdW interactions
Promising starting point for further developments, e.g. inclusion ofcorrelation in KS potential or in kernel
Generalization to finite temperatures
Orbital-dependent functionals open up fascinating new possibilities in DFT
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 35 / 36
Literature
EXX for solids
Phys. Rev. Lett. 79, 2089 (1997) Phys. Rev. B 83, 045105 (2011)
finite-temperature EXX
Phys. Rev. B 81, 155119 (2010)
EXX for molecules
Phys. Rev. Lett. 83, 5459 (1999) J. Chem. Phys. 128, 104104 (2008)
TDEXX
Phys. Rev. A 80, 012507 (2009)Phys. Rev. Lett. 102, 233003 (2009)