Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
Density-functional methods for electronic systemsat finite temperatures
Andreas Gorling, Max Greiner, Hannes Schulz, Patrick Bleiziffer,and Andreas Heßelmann
Lehrstuhl fur Theoretische ChemieUniversitat Erlangen–Nurnberg
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
Approach: Orbital-dependent, finite-temperature exchange-correlation functionals
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
Electronic structure methods forgrand canonical ensembles
Electronic Schrodinger equation: [T + Vee + v]ΨN,n = EN,nΨN,n
T = 0 finite T
EN,0 = minΨ→N
E[Ψ] Ωv,T,µ = minΓ
Ω[Γ]
E[Ψ] = 〈Ψ|H|Ψ〉 Ω[Γ] = Tr Γ[H + kT ln Γ− µN
]
EN,0 = E[Ψ0] Ωv,T,µ = Ω[Γv,T,µ]
HΨ0 = EN,0Ψ0 Γv,T,µ =∑
N
∑
n
exp[(−1/kT )(EN,n − µN)]
Zv,T,µ|ΨN,n〉〈ΨN,n|
Zv,T,µ =∑
N
∑
n
exp[(−1/kT )(EN,n − µN)]
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 11 / 36
Finite-temperature Kohn-Sham formalism
T = 0
Φ0 ←→ T + vs ←→ ρ0 ←→ T + Vee + v ←→ Ψ0
vS(r) = v(r) + vH ([ρ0]; r) + vxc ([ρ0]; r)
finite T
ΓKSvS ,T,µ ←→ T + vs ←→ ρv,T,µ ←→ T + Vee + v ←→ Γv,T,µ
vs(r) = v(r) + vH ([ρv,T,µ]; r) + vxc ([ρv,T,µ], T, r)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 12 / 36
Basic density-functionals forgrand canonical ensembles
T = 0 finite T
F [ρ] = minΨ→ρ
〈Ψ|T + Vee|Ψ〉
→ Ψ[ρ]
F [ρ, T ] = minΓ→ρ
Tr Γ [T + Vee + kT ln Γ]
→ Γ[ρ, T ]
Ts[ρ] = minΨ→ρ
〈Ψ|T |Ψ〉
→ Φ[ρ]
Ts[ρ, T ] = minΓ→ρ
Tr Γ [T + kT ln Γ]
→ ΓKS[ρ, T ]
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∫
dr′′∫dr′′′ X0(r, r′′, ω) fHx(ω, r
′′, r′′′) X0(r′′′, r′, ω) = hHx(ω, r, r′)
withhHx(ω, r, r
′) = 14Yᵀ(r)λ(ω) [A + B + ∆]λ(ω)Y(r′)
+ ω2 14Yᵀ(r)λ(ω)ε−1 [A + B + ∆] ε−1λ(ω)Y(r)
+∑i
∑j
∑a
Yia(r)λia(ω)
⟨a|vNL
x − vx|j⟩
εa − εjφi(r)φj(r) + · · ·
+∑a
∑b
∑i
Yia(r)λia(ω)
⟨b|vNL
x − vx|i⟩
εb − εiφa(r)φb(r) + · · ·
Aia,jb = 2(ai|jb)− (ab|ji) Bia,jb = 2(ai|bj)− (aj|bi)∆ia,jb = δij 〈ϕa|vNL
x − vx|ϕb〉 − δab 〈ϕi|vNLx − vx|ϕj〉
λia,jb = δia,jb−4εiaε2ia+ω2 εia,jb = δia,jb εia = δia,jb(εa − εi)
Yia(r) = φi(r)φa(a)
hH(r, r′, ω) = Yᵀ(r)λ(ω) C λ(ω)Y(r) with Cia,jb = (ia|jb)A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 22 / 36
RI-EXXRPA method I
Ec =−1
2π
∫ ∞0
dω
∫ 1
0
dα
∫drdr′
1
|r− r′|[χα(r, r′, iω) − χ0(r, r′, iω)
]Representation of χα(iω), hx(iω) in RI basis set with respect to Coulomb norm
Xα = [1− αX0FHx]−1 X0 ⇒ Xα = X0 [X0 − αHHc]
−1 X0
(using FHx = X−10 HHcX
−10 )
Ec =−1
2π
∫ ∞0
dω
∫ 1
0
dα Tr[S−1 X0 [X0 − αH]−1X0 − S−1X0
]
Orthonormalize RI basis set, i.e. make S = E, and use Xα = −(−Xα)12 (−Xα)
12
Ec =−1
2π
∫ ∞0
dω
∫ 1
0
dα Tr
[(−X0)
12
[−1− α(−X0)−
12 H(−X0)−
12
]−1
(−X0)12 −X0
]with
X0(iω) = Dᵀλ(iω)D with Dia,h = (ϕiϕa|fh)Coul and λia,jb = δia,jb−4εiaε2ia + ω2
and
H(iω) = 14
DTλ(iω) [A+B+∆]λ(iω)D + (iω)2 14
DTλ(iω) ε−1[A+B+∆] ε−1 λ(iω)D
+ W1(iω) + WT1 (iω) + W2(iω) + WT
2 (iω)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 23 / 36
RI-EXXRPA method II
Ec =−1
2π
∫ ∞0
dω
∫ 1
0
dα Tr
[(−X0)
12
[1− α(−X0)−
12 H(−X0)−
12
]−1
(−X0)12 − X0
]
Analytic integration over coupling constant
Ec =−1
2π
∫ ∞0
dω Tr[(−X0(iω))
12 U(iω)
(−τ−1(iω) ln[|1 + τ (iω)|]+1
)U(iω)T (−X0(iω))
12
]with
(−X0(iω))12 H(iω)(−X0(iω))
12 = U(iω) τ (iω) Uᵀ(iω)
Complete exchange kernel can be treated (RI-EXXRPA+)
N5 scaling
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 24 / 36
RI2-dRPA method
RI-EXXRPA
Ec =−1
2π
∫ ∞0
dω
∫ 1
0
dα Tr
[(−X0)
12
[1− α(−X0)−
12 H(−X0)−
12
]−1
(−X0)12 − X0
]
For dRPA, with second RI approximation and S = E, H simplifies to
H = X0X0
With spectral representation X0(iω) = −V(iω)σ(iω) V>(iω)
Ec =1
2π
∫ ∞0
dω Tr [ln[1 + σ(iω)]− σ(iω)]
Only KS response function X0(iω) required
N4 scaling
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 25 / 36
Results EXXRPA correlation methods I
Deviations of total energies from CCSD(T) energies, ∆E = EMethod − ECCSD(T)
Orbital basis: aug-cc-pVQZ RI basis: aug-cc-pVQZ
OEP (EXX) balanced uncontracted basis sets
-5
0
5
10
15
20
25
30
35
H2
H2 O
H2 O
2
CO
CO
2
NH
3
CH
4
C2 H
2
C2 H
4
C2 H
6
CH
3 OH
C2 H
5 OH
HC
HO
HN
CO
HC
OO
H
C2 H
4 O
CH
3 CH
O
H2 C
CO
HC
ON
H2
HC
OO
CH
3
NH
2 CO
NH
2
∆ E
[kca
l/m
ol]
CCSDMP2
EXXRPARI-EXXRPA
RI-EXXRPA+
0
10
20
30
40
50
60
70
80
90
100
110
120
CCSD
MP2
B3LYP
EXXRPA
RI-EXXR
PA
RI-EXXR
PA+
RI-dR
PA
RM
S [
kca
l/m
ol]
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 26 / 36
Results EXXRPA correlation methods II
Deviations of total energies from CCSD(T) energies, ∆E = EMethod − ECCSD(T)
Orbital basis: aug-cc-pCVQZ RI basis: aug-cc-pV5Z
OEP (EXX) basis sets equal to RPA basis sets
-5
0
5
10
15
20
25
30
35
H2
H2 O
H2 O
2
CO
CO
2
NH
3
CH
4
C2 H
2
C2 H
4
C2 H
6
CH
3 OH
C2 H
5 OH
HC
HO
HN
CO
HC
OO
H
C2 H
4 O
CH
3 CH
O
H2 C
CO
HC
ON
H2
HC
OO
CH
3
NH
2 CO
NH
2
∆ E
[kca
l/m
ol]
CCSDMP2
RI-EXXRPA+RI-EXXRPA
RI-EXX*RPA+
0 10 20 30 40 50 60 70 80 90
100 110 120
CCSD
MP2
RI-EXXR
PA
RI-EXXR
PA+
RI-EXX*R
PA+
RM
S [kcal/m
ol]
? OEP (EXX) balanced uncontracted basis sets
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 27 / 36
Results EXXRPA correlation methods III
Deviations of reaction energies from CCSD(T) reaction energies
Orbital basis: aug-cc-pVQZRI basis: aug-cc-pVQZ
OEP (EXX) balanced uncontracted basis sets
0
0.5
1
1.5
2
2.5
3
3.5
CCSD
MP2
B3LYP
EXXRPA
RI-EXXR
PA
RI-EXXR
PA+
RI-dR
PA
RM
S [
kca
l/m
ol]
Orbital basis: aug-cc-pCVQZRI basis: aug-cc-pV5Z
OEP (EXX) basis sets equal to RPA basis sets
0
0.5
1
1.5
2
2.5
3
3.5
CCSD
MP2
RI-EXXR
PA
RI-EXXR
PA+
RI-EXX *R
PA+
RM
S [
kca
l/m
ol]
? OEP (EXX) balanced uncontracted basis sets
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 28 / 36
Dissociation of H2
basis set: aug-cc-pVQZ
0 2 4 6 8 10 12 14 16 18 20r(H-H) [a0]
-1.1
-1
-0.9
-0.8
-0.7
-0.6en
ergy
[a.u
.]HF/EXXCIHF-RPAEXX-RPA
Dissociation limit of other molecules (CO, N2, etc.) is also treated correctly
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 29 / 36
Coupling constant integrand
Ec =1∫0
dα Vc(α)
-0.25
-0.2
-0.15
-0.1
-0.05
0
Vc(
α) [a
.u.]
r=1.4 a0r=6.0 a0r=10.0 a0r=20.0 a0r=30.0 a0
0 0.2 0.4 0.6 0.8 1coupling strength
-0.25
-0.2
-0.15
-0.1
-0.05
0
Vc(
α) [a
.u.]
HF-RPA
EXX-RPA
Vc(α)
-0.25
-0.2
-0.15
-0.1
-0.05
0
VcH
L (α)
[a.u
.]
r=1.4 a0r=6.0 a0r=10.0 a0r=20.0 a0r=30.0 a0
0 0.2 0.4 0.6 0.8 1coupling strength
-0.25
-0.2
-0.15
-0.1
-0.05
0
VcH
L (α)
[a.u
.]
HF-RPA
EXX-RPA
−(ia|ia)
V HLc (α)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 30 / 36
Dissociation of N2
-109.4
-109.2
-109
-108.8
-108.6
-108.4
-108.2
-108
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Energ
y [hart
ree]
r [bohr]
Orbital basis: avtz RI basis: avtz
HFEXX
CCSDRI-EXX-RPA
Special treatment of singularity in ω-Integrand ( −τ−1 ln [ |1 + τ (iω)| ] + 1 )
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 31 / 36
Twisting of ethene
Orbital basis: avqz RI basis: avqz
-78.6
-78.5
-78.4
-78.3
-78.2
-78.1
-78
-77.9
-77.8
0 20 40 60 80 100 120 140 160 180
En
erg
y [
Ha
rtre
e]
Torsion angle [degree]
HFEXX
CCSD
B3LYPRI-EXXRPA+
MCSCF
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 32 / 36
RI-EXXRPA for noncovalent bonds
Binding energies in kcal/mol
Complex Basis Reference RI-EXXRPA RI-EXXRPA+ RI-dRPA TPSS-RI-dRPA MP2 F12-CCSD
(NH3)2 3 2.54 2.76 2.33 3.004 2.61 2.83 2.46 3.11
CBS 3.15 2.70 2.92 2.58 2.74 3.18 2.88(H2O)2 3 4.29 4.55 3.73 4.72
4 4.60 4.86 4.24 5.02CBS 5.07 4.81 5.06 4.59 4.52 5.21 4.74
(HCONH2)2 3 14.87 15.27 13.43 15.084 15.42 15.84 14.23 15.57
CBS 16.11 15.77 16.21 15.28 15.42 15.86 15.28(HCOOH)2 3 17.40 17.90 15.25 17.61
4 18.06 18.46 16.24 18.23CBS 18.81 18.50 18.81 16.91 17.91 18.61 17.92
(C2H4)2 3 1.04 1.14 0.93 1.474 1.13 1.24 1.04 1.55
CBS 1.48 1.19 1.33 1.13 1.20 1.60 1.14C2H4···C2H2 3 1.34 1.45 1.21 1.59
4 1.37 1.46 1.26 1.64CBS 1.50 1.41 1.49 1.31 1.27 1.67 1.31
(CH4)2 3 0.30 0.37 0.29 0.464 0.34 0.36 0.33 0.48
CBS 0.53 0.36 0.34 0.35 0.40 0.49 0.41RMS 3 0.83 0.55 1.80 0.61RMS 4 0.50 0.25 1.29 0.31RMS CBS 0.29 0.13 0.94 0.52 0.15 0.51
– 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)
HFdRPA
SOSEX
HF-RPA
EXX-RPA
0
5
10
15
20
25
30
RM
S [k
cal/m
ol]
C2H2+H2 → C2H4
C2H4+H2 → C2H6
C2H6+H2 → 2CH4
CO+H2 → H2COH2CO+H2 → CH3OHH2O2+H2 → 2H2OC2H2+H2O → CH3CHOC2H4+H2O → C2H5OHCH3CHO+H2 → C2H5OHCO+NH3 → HCONH2
CO+H2O → CO2+H2
HNCO+NH3 → NH2CONH2
CH3OH+CO → HCOOCH3
CO+H2O2 → CO2+H2O
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)
Int. J. Quantum. Chem. 110, 2202 (2010)J. Chem. Phys. 134, 034120 (2011)
EXX-RPA
Mol. Phys. 108, 359 (2010)Mol. Phys. 109, 2473 (2011) Review
Phys. Rev. Lett. 106, 093001 (2011)J. Chem. Phys. 136, 134102 (2012)
A. Gorling (University Erlangen–Nurnberg) Los Angeles 2012 36 / 36