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Theorie de la fonctionnnelle de la densite
avec separation de porteepour les forces de van der Waals
Julien Toulouse1
Iann Gerber2, Georg Jansen3, Andreas Savin1, Janos Angyan4
1 Laboratoire de Chimie Theorique, UPMC Univ Paris 06 et CNRS, Paris, France
2 Universite de Toulouse, INSA-UPS, LPCNO, Toulouse, France3 Fachbereich Chemie, Universitat Duisburg-Essen, Essen, Germany
4 CRM2, Institut Jean Barriol, Universite de Nancy et CNRS, Vandoeuvre-les-Nancy,
France
Email : julien.toulouse@upmc.fr
Page web : www.lct.jussieu.fr/pagesperso/toulouse/
novembre 2008
1 Kohn-Sham DFT and ACFDT approaches
2 Range-separated multideterminant DFT
3 Short-range density functionals
4 Range-separated ACFDT method
5 Some results
1 Kohn-Sham DFT and ACFDT approaches
2 Range-separated multideterminant DFT
3 Short-range density functionals
4 Range-separated ACFDT method
5 Some results
Kohn-Sham DFT
Kohn-Sham (KS) scheme
E = minΦ
{
〈Φ|T + Vne |Φ〉 + EH[nΦ] + Exc [nΦ]}
Φ : single-determinant wave function
Kohn-Sham DFT
Kohn-Sham (KS) scheme
E = minΦ
{
〈Φ|T + Vne |Φ〉 + EH[nΦ] + Exc [nΦ]}
Φ : single-determinant wave function
One problem (among others):Usual approximations for exchange-correlation functional Exc [n](LDA, GGA, ...) do not describe well (long-range) van derWaals dispersion forces
Example: interaction energy curve of Ne2
LDA and PBE functionals, aug-cc-pV5Z basis:
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
5 6 7 8 9 10
Inte
ract
ion
en
ergy (
mH
art
ree)
Interatomic distance (Bohr)
AccurateLDAPBE
Ne2
ACFDT approach to DFT
Starting from the adiabatic connection formula for correlation energy:
Ec =
∫ 1
0
dλ{
〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}
ACFDT approach to DFT
Starting from the adiabatic connection formula for correlation energy:
Ec =
∫ 1
0
dλ{
〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}
or, with a compact notation,
Ec =1
2
∫ 1
0
dλ Tr [wee ∗ Pc,λ]
ACFDT approach to DFT
Starting from the adiabatic connection formula for correlation energy:
Ec =
∫ 1
0
dλ{
〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}
or, with a compact notation,
Ec =1
2
∫ 1
0
dλ Tr [wee ∗ Pc,λ]
and using the fluctuation-dissipation theorem
Pc,λ = −1
π
∫
∞
0
dω [χλ(iω) − χKS(iω)]
ACFDT approach to DFT
Starting from the adiabatic connection formula for correlation energy:
Ec =
∫ 1
0
dλ{
〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}
or, with a compact notation,
Ec =1
2
∫ 1
0
dλ Tr [wee ∗ Pc,λ]
and using the fluctuation-dissipation theorem
Pc,λ = −1
π
∫
∞
0
dω [χλ(iω) − χKS(iω)]
leads to
Ec = −1
2π
∫ 1
0
dλ
∫
∞
0
dω Tr [wee ∗ (χλ(iω) − χKS(iω))]
ACFDT approach to DFT
Starting from the adiabatic connection formula for correlation energy:
Ec =
∫ 1
0
dλ{
〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}
or, with a compact notation,
Ec =1
2
∫ 1
0
dλ Tr [wee ∗ Pc,λ]
and using the fluctuation-dissipation theorem
Pc,λ = −1
π
∫
∞
0
dω [χλ(iω) − χKS(iω)]
leads to
Ec = −1
2π
∫ 1
0
dλ
∫
∞
0
dω Tr [wee ∗ (χλ(iω) − χKS(iω))]
where the response function χλ(iω) is given by
χλ(iω)−1 = χKS(iω)−1 − fHxc,λ(iω)
Random Phase Approximation (RPA)
RPA approximation: fxc,λ = 0
Random Phase Approximation (RPA)
RPA approximation: fxc,λ = 0
So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA
Random Phase Approximation (RPA)
RPA approximation: fxc,λ = 0
So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA
=⇒ increasing interest in the DFT community:Perdew, Dobson, Furche, Gonze, Kresse, Scuseria, ...
Random Phase Approximation (RPA)
RPA approximation: fxc,λ = 0
So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA
=⇒ increasing interest in the DFT community:Perdew, Dobson, Furche, Gonze, Kresse, Scuseria, ...
Encouraging results:
consistent with exact exchange
correct dispersion forces at (very) large separation
good cohesive energies and lattice constants of solids
some improvement in description of bond dissociation
Random Phase Approximation (RPA)
RPA approximation: fxc,λ = 0
So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA
=⇒ increasing interest in the DFT community:Perdew, Dobson, Furche, Gonze, Kresse, Scuseria, ...
Encouraging results:
consistent with exact exchange
correct dispersion forces at (very) large separation
good cohesive energies and lattice constants of solids
some improvement in description of bond dissociation
But several unsatisfactory aspects:
correlation energies far too negative
strong dependence on basis size
bump at intermediate distances in some dissociation curves
dependence on input orbitals
embarrassing results for simple van der Waals dimers!
Example: interaction energy curve of Ne2
RPA (with PBE orbitals), aug-cc-pV5Z basis:
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
5 6 7 8 9 10
Inte
ract
ion
en
ergy (
mH
art
ree)
Interatomic distance (Bohr)
AccurateRPA
Ne2
Example: interaction energy curve of Be2
RPA (with PBE orbitals), cc-pV5Z basis:
-4
-2
0
2
4
4 5 6 7 8 9 10 11
Inte
ract
ion
en
ergy (
mH
art
ree)
Interatomic distance (Bohr)
AccurateRPA
Be2
1 Kohn-Sham DFT and ACFDT approaches
2 Range-separated multideterminant DFT
3 Short-range density functionals
4 Range-separated ACFDT method
5 Some results
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
W lree =
∑
i<j
erf(µrij)
rij: long-range electron-electron interaction
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
W lree =
∑
i<j
erf(µrij)
rij: long-range electron-electron interaction
E srHxc [n] : short-range Hxc density functional
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
W lree =
∑
i<j
erf(µrij)
rij: long-range electron-electron interaction
E srHxc [n] : short-range Hxc density functional
minimizing wave function Ψlr
=∑∑∑
i ciΦi is multi-determinant
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
W lree =
∑
i<j
erf(µrij)
rij: long-range electron-electron interaction
E srHxc [n] : short-range Hxc density functional
minimizing wave function Ψlr
=∑∑∑
i ciΦi is multi-determinant
parameter µ controls the range of separation.Limiting cases:µ = 0 =⇒ KS DFTµ → ∞ =⇒ Standard wave function methods
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
W lree =
∑
i<j
erf(µrij)
rij: long-range electron-electron interaction
E srHxc [n] : short-range Hxc density functional
minimizing wave function Ψlr
=∑∑∑
i ciΦi is multi-determinant
parameter µ controls the range of separation.Limiting cases:µ = 0 =⇒ KS DFTµ → ∞ =⇒ Standard wave function methods
In principle: exact
Range-separated multideterminant DFT
Multideterminant extension of KS scheme with range separation
Ground-state energy:
E = minΨ
{
〈Ψ|T + Vne + W lree |Ψ〉 + E sr
Hxc [nΨ]}
W lree =
∑
i<j
erf(µrij)
rij: long-range electron-electron interaction
E srHxc [n] : short-range Hxc density functional
minimizing wave function Ψlr
=∑∑∑
i ciΦi is multi-determinant
parameter µ controls the range of separation.Limiting cases:µ = 0 =⇒ KS DFTµ → ∞ =⇒ Standard wave function methods
In principle: exact
In practice: approximations are necessary for Ψlr and E sr
xc [n]
Range-separated multideterminant DFT: approximations
Approximations for E srxc [n]
short-range LDA
short-range GEA
short-range GGA
...
Range-separated multideterminant DFT: approximations
Approximations for E srxc [n]
short-range LDA
short-range GEA
short-range GGA
...
Approximations for Ψlr
single-determinant =⇒ HF+DFT method (or RSH method)
MCSCF =⇒ MCSCF+DFT method (for near-degeneracy)
CI =⇒ CI+DFT method
CC =⇒ CC+DFT method
MP2 =⇒ RSH+MP2 method (for van der Waals)
RPA or TDHF =⇒ RSH+TDHF method (for van der Waals)
...
1 Kohn-Sham DFT and ACFDT approaches
2 Range-separated multideterminant DFT
3 Short-range density functionals
4 Range-separated ACFDT method
5 Some results
Short-range exchange energy: LDA
Esr,µx ,LDA[n] =
∫
n(r) εsr,µx ,unif(n(r))dr
Short-range exchange energy: LDA
Esr,µx ,LDA[n] =
∫
n(r) εsr,µx ,unif(n(r))dr
For Be atom:
0 2 4 6 8Μ Ha.u.L
-2.5
-2
-1.5
-1
-0.5
0
Exsr,Μ
Ha.u.L
exactLDA
LDA accuratefor a short-range interaction
Short-range exchange energy: LDA
Esr,µx ,LDA[n] =
∫
n(r) εsr,µx ,unif(n(r))dr
For Be atom:
0 2 4 6 8Μ Ha.u.L
-2.5
-2
-1.5
-1
-0.5
0
Exsr,Μ
Ha.u.L
exactLDA
LDA accuratefor a short-range interaction
Asymptotic expansion for µ → ∞ :
E sr,µx = −
A1
µ2
∫
n(r)2dr +A2
µ4
∫
n(r)
(
|∇n(r)|2
2n(r)+ 4τ(r)
)
dr + · · ·
Short-range correlation energy: LDA
Esr,µc,LDA[n] =
∫
n(r) εsr,µc,unif(n(r))dr
Short-range correlation energy: LDA
Esr,µc,LDA[n] =
∫
n(r) εsr,µc,unif(n(r))dr
For Be atom:
0 2 4 6 8Μ Ha.u.L
-0.2
-0.15
-0.1
-0.05
0
Ecsr,Μ
Ha.u.L
exactLDA
LDA accuratefor a short-range interaction
Short-range correlation energy: LDA
Esr,µc,LDA[n] =
∫
n(r) εsr,µc,unif(n(r))dr
For Be atom:
0 2 4 6 8Μ Ha.u.L
-0.2
-0.15
-0.1
-0.05
0
Ecsr,Μ
Ha.u.L
exactLDA
LDA accuratefor a short-range interaction
Asymptotic expansion for µ → ∞ :
E sr,µc =
B1
µ2
∫
n2,c(r, r)dr +B2
µ3
∫
n2(r, r)dr + · · ·
Short-range exchange energy: GGA
Short-range GGA functional of Heyd, Scuseria and Ernzerhof(2003) based on the PBE exchange hole:
εsr,µx ,GGA(n) =
1
2
∫
nx ,PBE(n, |∇n|, r12)w sr,µee (r12)dr12
For Be atom:
0 1 2 3 4 5 6Μ Ha.u.L
-2.5
-2
-1.5
-1
-0.5
0
Exsr,Μ
Ha.u.L
exactLDAGGA
=⇒ GGA describes well a longer range of interaction
Short-range correlation energy: GGA
Interpolation between PBE at µ = 0 and expansion of LDA for µ → ∞:
εsr,µc,GGA(n, |∇n|) =
εc,PBE(n, |∇n|)
1 + d1(n)µ + d2(n)µ2
For Be atom:
0 1 2 3 4 5 6Μ Ha.u.L
-0.2
-0.15
-0.1
-0.05
0
Ecsr,Μ
Ha.u.LexactLDAGGA
=⇒ GGA describes well a longer range of interaction
1 Kohn-Sham DFT and ACFDT approaches
2 Range-separated multideterminant DFT
3 Short-range density functionals
4 Range-separated ACFDT method
5 Some results
Range-separated hybrid (RSH) scheme
Restriction to single-determinant wave functions Φ:
ERSH = minΦ
{
〈Φ|T + Vne + W lree |Φ〉 + E sr
Hxc [nΦ]}
Range-separated hybrid (RSH) scheme
Restriction to single-determinant wave functions Φ:
ERSH = minΦ
{
〈Φ|T + Vne + W lree |Φ〉 + E sr
Hxc [nΦ]}
The minimizing RSH determinant ΦRSH is given by
(
T + Vne + V lrHx ,HF + V sr
Hxc
)
|ΦRSH〉 = E0|ΦRSH〉,
Range-separated hybrid (RSH) scheme
Restriction to single-determinant wave functions Φ:
ERSH = minΦ
{
〈Φ|T + Vne + W lree |Φ〉 + E sr
Hxc [nΦ]}
The minimizing RSH determinant ΦRSH is given by
(
T + Vne + V lrHx ,HF + V sr
Hxc
)
|ΦRSH〉 = E0|ΦRSH〉,
So the RSH energy is
ERSH = 〈ΦRSH|T+Vne |ΦRSH〉+EH[nΦRSH]+E lr
x ,HF[ΦRSH]+E srxc [nΦRSH
]
Adiabatic connection starting from RSH
Exact energy = RSH energy + long-range correlation energy
E = ERSH + E lrc
Adiabatic connection starting from RSH
Exact energy = RSH energy + long-range correlation energy
E = ERSH + E lrc
Let’s define the following adiabatic connection
Eλ = minΨ
{
〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr
Hxc [nΨ]}
with the long-range perturbation operator
W lr = W lree − V lr
Hx ,HF
Adiabatic connection starting from RSH
Exact energy = RSH energy + long-range correlation energy
E = ERSH + E lrc
Let’s define the following adiabatic connection
Eλ = minΨ
{
〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr
Hxc [nΨ]}
with the long-range perturbation operator
W lr = W lree − V lr
Hx ,HF
minimizing wave function Ψlrλ is multideterminant
Adiabatic connection starting from RSH
Exact energy = RSH energy + long-range correlation energy
E = ERSH + E lrc
Let’s define the following adiabatic connection
Eλ = minΨ
{
〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr
Hxc [nΨ]}
with the long-range perturbation operator
W lr = W lree − V lr
Hx ,HF
minimizing wave function Ψlrλ is multideterminant
Limits:For λ = 0: Ψ
lrλ=0 = ΦRSH
For λ = 1: Ψlrλ=1 = Ψ
lr and Eλ=1 = E
Adiabatic connection starting from RSH
Exact energy = RSH energy + long-range correlation energy
E = ERSH + E lrc
Let’s define the following adiabatic connection
Eλ = minΨ
{
〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr
Hxc [nΨ]}
with the long-range perturbation operator
W lr = W lree − V lr
Hx ,HF
minimizing wave function Ψlrλ is multideterminant
Limits:For λ = 0: Ψ
lrλ=0 = ΦRSH
For λ = 1: Ψlrλ=1 = Ψ
lr and Eλ=1 = E
the density is NOT constant on the adiabatic connection
Long-range correlation energy Elrc
We have the following adiabatic connection formula:
E lrc =
∫ 1
0
dλ{
〈Ψlrλ|W
lr|Ψlrλ〉 − 〈ΦRSH|W
lr|ΦRSH〉}
Long-range correlation energy Elrc
We have the following adiabatic connection formula:
E lrc =
∫ 1
0
dλ{
〈Ψlrλ|W
lr|Ψlrλ〉 − 〈ΦRSH|W
lr|ΦRSH〉}
=1
2
∫ 1
0
dλ Tr[
w lr ∗ P lrc,λ
]
Long-range correlation energy Elrc
We have the following adiabatic connection formula:
E lrc =
∫ 1
0
dλ{
〈Ψlrλ|W
lr|Ψlrλ〉 − 〈ΦRSH|W
lr|ΦRSH〉}
=1
2
∫ 1
0
dλ Tr[
w lr ∗ P lrc,λ
]
and using the fluctuation-dissipation theorem
P lrc,λ = −
1
π
∫
∞
0
dω[
χlrλ(iω) − χRSH(iω)
]
+ ∆lrλ
where ∆lrλ comes from the variation of the density. So
Long-range correlation energy Elrc
We have the following adiabatic connection formula:
E lrc =
∫ 1
0
dλ{
〈Ψlrλ|W
lr|Ψlrλ〉 − 〈ΦRSH|W
lr|ΦRSH〉}
=1
2
∫ 1
0
dλ Tr[
w lr ∗ P lrc,λ
]
and using the fluctuation-dissipation theorem
P lrc,λ = −
1
π
∫
∞
0
dω[
χlrλ(iω) − χRSH(iω)
]
+ ∆lrλ
where ∆lrλ comes from the variation of the density. So
E lrc = −
1
2π
∫ 1
0
dλ
∫
∞
0
dω Tr[
w lr ∗(
χlrλ(iω) − χRSH(iω)
)]
+1
2
∫ 1
0
dλ Tr[
w lr ∗ ∆lrλ
]
Long-range correlation energy Elrc
We have the following adiabatic connection formula:
E lrc =
∫ 1
0
dλ{
〈Ψlrλ|W
lr|Ψlrλ〉 − 〈ΦRSH|W
lr|ΦRSH〉}
=1
2
∫ 1
0
dλ Tr[
w lr ∗ P lrc,λ
]
and using the fluctuation-dissipation theorem
P lrc,λ = −
1
π
∫
∞
0
dω[
χlrλ(iω) − χRSH(iω)
]
+ ∆lrλ
where ∆lrλ comes from the variation of the density. So
E lrc = −
1
2π
∫ 1
0
dλ
∫
∞
0
dω Tr[
w lr ∗(
χlrλ(iω) − χRSH(iω)
)]
+1
2
∫ 1
0
dλ Tr[
w lr ∗ ∆lrλ
]
The long-range response function χlrλ(iω) is given by
χlrλ(iω)−1 = χlr
IP,λ(iω)−1 − f lrHxc,λ(iω)
Approximations for Elrc
Several approximations possible for E lrc :
TDHF approximation: f lrc,λ = 0 =⇒ RSH+TDHF method
Approximations for Elrc
Several approximations possible for E lrc :
TDHF approximation: f lrc,λ = 0 =⇒ RSH+TDHF method
MP2 approximation (2nd order in w lree) =⇒ RSH+MP2 method
Approximations for Elrc
Several approximations possible for E lrc :
TDHF approximation: f lrc,λ = 0 =⇒ RSH+TDHF method
MP2 approximation (2nd order in w lree) =⇒ RSH+MP2 method
Comparison:
RSH+TDHF is an extension of RSH+MP2
RSH+TDHF is expected to supersede RSH+MP2 for systemswith small HOMO-LUMO gap
Implementation of long-range TDHF
Orbital rotation Hessians:
(Aλ − Bλ)iajb = (ǫa − ǫi )δijδab + λ(
〈ij |w lree |ba〉 − 〈ia|w lr
ee |jb〉)
and
(Aλ + Bλ)iajb = (ǫa − ǫi )δijδab + 2λ〈ij |w lree |ab〉
− λ(
〈ij |w lree |ba〉 + 〈ia|w lr
ee |jb〉)
Implementation of long-range TDHF
Orbital rotation Hessians:
(Aλ − Bλ)iajb = (ǫa − ǫi )δijδab + λ(
〈ij |w lree |ba〉 − 〈ia|w lr
ee |jb〉)
and
(Aλ + Bλ)iajb = (ǫa − ǫi )δijδab + 2λ〈ij |w lree |ab〉
− λ(
〈ij |w lree |ba〉 + 〈ia|w lr
ee |jb〉)
Long-range TDHF second-order density matrix
P lrc,TDHF,λ = (Aλ − Bλ)1/2Mλ
−1/2(Aλ − Bλ)1/2 − 1
where Mλ = (Aλ − Bλ)1/2(Aλ + Bλ)(Aλ − Bλ)1/2
Implementation of long-range TDHF
Orbital rotation Hessians:
(Aλ − Bλ)iajb = (ǫa − ǫi )δijδab + λ(
〈ij |w lree |ba〉 − 〈ia|w lr
ee |jb〉)
and
(Aλ + Bλ)iajb = (ǫa − ǫi )δijδab + 2λ〈ij |w lree |ab〉
− λ(
〈ij |w lree |ba〉 + 〈ia|w lr
ee |jb〉)
Long-range TDHF second-order density matrix
P lrc,TDHF,λ = (Aλ − Bλ)1/2Mλ
−1/2(Aλ − Bλ)1/2 − 1
where Mλ = (Aλ − Bλ)1/2(Aλ + Bλ)(Aλ − Bλ)1/2
The TDHF long-range correlation energy is finally
E lrc,TDHF =
1
2
∫ 1
0
dλ∑
iajb
〈ij |w lree |ab〉
(
P lrc,TDHF,λ
)
iajb
1 Kohn-Sham DFT and ACFDT approaches
2 Range-separated multideterminant DFT
3 Short-range density functionals
4 Range-separated ACFDT method
5 Some results
Dependence on basis size: Ne2
Total energy (aug-cc-pVnZ basis, µ = 0.5, sr-PBE functional):
-258.2
-258.1
-258
-257.9
-257.8
-257.7
-257.6
-257.5
3 4 5 6
Tota
l en
ergy (
Hart
ree)
Size of one-particle basis (n in aug-cc-pVnZ)
ExactTDHF
RPARSH+TDHF
Ne2
=⇒ RSH+TDHF has a small basis dependence
Interaction energy curve of Ne2
Interaction energy (aug-cc-pV5Z basis, µ = 0.5, sr-PBE functional):
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
5 6 7 8 9 10
Inte
ract
ion
en
ergy (
mH
art
ree)
Interatomic distance (Bohr)
AccurateTDHF
RPARSH+TDHF
Ne2
Interaction energy curve of Be2
Interaction energy (cc-pV5Z basis, µ = 0.5, sr-PBE functional):
-4
-2
0
2
4
4 5 6 7 8 9 10 11
Inte
ract
ion
en
ergy (
mH
art
ree)
Interatomic distance (Bohr)
AccurateTDHF
RPARSH+TDHF
Be2
Conclusions and perspectives
Conclusions
RSH+TDHF method overcomes some problems of standardRPA
RSH+TDHF method seems well suited for van der Waalssystems
RSH+TDHF method has also problems (e.g., dissociation)
RSH+MP2 can be a cheaper alternative to RSH+TDHF
Perspectives
efficient implementation in quantum chemistry software
application to larger molecular systems (benzene dimer, ...)
application to solids
exploration of other variants of the method
Web page: www.lct.jussieu.fr/pagesperso/toulouse/
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