Closed-shell ring coupled cluster doubles
theory with range separation applied on weak
intermolecular interactions
Julien Toulouse1
Wuming Zhu2 , Andreas Savin1 , Georg Jansen3 , János Ángyán4
1
Laboratoire de Chimie Théorique, UPMC and CNRS, Paris, France
2
Department of Physics, Hangzhou Normal University, HangZhou XiaSha, ZheJiang, China
Fakultät für Chemie, Universität Duisburg-Essen, Essen, Germany
CRM2, Institut Jean Barriol, Université de Nancy and CNRS, Vandoeuvre-lès-Nancy, France
3
4
Email: julien.toulouse@upmc.fr
Web page: www.lct.jussieu.fr/pagesperso/toulouse/
August 2011
Abstract
We explore different variants of the random phase approximation
(RPA) to the correlation energy derived from closed-shell
ring-diagram approximations to coupled cluster doubles theory. We
implement these variants in range-separated density-functional
theory, i.e., by combining the long-range random phase
approximations with short-range density-functional approximations.
We perform tests on the rare-gas dimers He2 , Ne2 , and Ar2 , and
on the weakly interacting molecular complexes of the S22 set of
Jurečka et al. [Phys. Chem. Chem. Phys. 8, 1985 (2006)]. The
two best variants correspond to the ones originally proposed by
Szabo and Ostlund [J. Chem. Phys. 67, 4351 (1977)].
Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, to appear
Range-separated density-functional theory
Multideterminant extension of KS scheme with range separation
n
o
lr
sr
E = min hΨ|T̂ + V̂ne + Ŵee
|Ψi + EHxc
[nΨ ]
Ψ
lr
Ŵee
=
X erf(µrij )
i<j
rij
: long-range electron-electron interaction
sr [n] : short-range Hxc density functional
EHxc
P
minimizing wave function Ψlr = i ci Φi is multi-determinant
parameter µ controls the range of separation.
In principle: exact
sr [n]
In practice: approximations are necessary for Ψlr and Exc
Random phase approximation in range-separated DFT
Start by a self-consistent range-separated hybrid (RSH)
calculation by restriction to a single-determinant wave
function Φ:
o
n
sr
lr
|Φi + EHxc
[nΦ ]
ERSH = min hΦ|T̂ + V̂ne + Ŵee
Φ
This is just a hybrid DFT with Hartree-Fock exchange at long
range.
Add the long-range correlation energy calculated from RPA:
lr
E = ERSH + Ec,RPA
But there are many possible variants of RPA!
RPA variants based on the adiabatic-connection formula
Adiabatic connection formula for correlation energy:
Z
1 1
dλ tr 1 K 1 Pc,λ
Ec =
2 0
with 1 Kia,jb = 2hab|iji and 1 Pc,λ is the spin-singlet correlation
two-particle density matrix.
Pc,λ from fluctuation-dissipation theorem
Z ∞
dω
Pc,λ = −
[χλ (iω) − χ0 (iω)]
−∞ 2π
where the response function χλ (iω) is given by
χλ (iω)−1 = χ0 (iω)−1 − fHxc,λ (iω)
Possible variants:
o fxc,λ = 0 =⇒ direct RPA (dRPA) variant
o fx,λ = λfxHF and fc,λ = 0 =⇒ RPAx-I variant (Toulouse et al.
2009) [1-3]
RPA variants based on ring CCD without exchange (dRPA)
ring CCD equations without exchange for the spin-singlet
amplitudes:
1
K + 1 L 1 TdRPA + 1 TdRPA 1 L + 1 TdRPA 1 K 1 TdRPA = 0
with 1 Kia,jb = 2hab|iji and 1 Lia,jb = (ǫa − ǫi )δij δab + 1 Kia,jb
Two alternative correlation energy expressions:
o direct RPA:
Ec,dRPA =
1 1 1
tr K TdRPA
2
o Second-order screened exchange (SOSEX) (Grüneis et al.
2009) [4]:
1 Ec,SOSEX = tr 1 B 1 TdRPA
2
with 1 Bia,jb = 2hab|iji − hab|jii
RPA variants based on ring CCD with exchange (RPAx)
ring CCD equations with exchange for the spin-singlet and
triplet amplitudes:
1
B + 1 A 1 TRPAx + 1 TRPAx 1 A + 1 TRPAx 1 B 1 TRPAx = 0
3
B + 3 A 3 TRPAx + 3 TRPAx 3 A + 3 TRPAx 3 B 3 TRPAx = 0
with 1 Aia,jb = (ǫa − ǫi )δij δab + 2hib|aji − hib|jai,
3A
3
ia,jb = (ǫa − ǫi )δij δab − hib|jai, and Bia,jb = −hab|jii
Three alternative correlation energy expressions:
o RPAx-II variant (McLachlan-Ball 1964) [5]
1 Ec,RPAx-II = tr 1 B 1 TRPAx + 33 B 3 TRPAx
4
o first Szabo-Ostlund 1977 variant (SO1) [6]
1 Ec,RPAx-SO1 = tr 1 B 1 TRPAx − 3 TRPAx
2
o second Szabo-Ostlund 1977 variant (SO2) [6]
1 Ec,RPAx-SO2 = tr 1 K 1 TRPAx
2
Computational details
implemented in MOLPRO
short-range PBE exchange-correlation density functional of
Goll et al. 2006
range-separation parameter fixed at µ = 0.5 bohr−1
Core electrons frozen in RPA calculations
BSSE removed by counterpoise method
In present implementation, all RPA variants scale as Nv3 No3 .
For comparison, CCD scales as Nv4 No2
Interaction energy curve of He2
with aug-cc-pV6Z basis
Without range separation:
With range separation:
0.01
He2
Interaction energy (mhartree)
Interaction energy (mhartree)
0.01
0
-0.01
Accurate
dRPA
SOSEX
RPAx-I
RPAx-II
RPAx-SO1
RPAx-SO2
-0.02
-0.03
-0.04
5
6
7
8
9
10
11
Internuclear distance (bohr)
He2
0
-0.01
Accurate
dRPA
SOSEX
RPAx-I
RPAx-II
RPAx-SO1
RPAx-SO2
-0.02
-0.03
-0.04
12
5
6
7
8
9
10
11
Internuclear distance (bohr)
=⇒ range separation improves all the RPA variants
12
Interaction energy curve of Ne2
with aug-cc-pV6Z basis
Without range separation:
With range separation:
0.05
Ne2
Interaction energy (mhartree)
Interaction energy (mhartree)
0.05
0
-0.05
Accurate
dRPA
SOSEX
RPAx-I
RPAx-II
RPAx-SO1
RPAx-SO2
-0.1
-0.15
5
6
7
8
9
10
11
Internuclear distance (bohr)
Ne2
0
-0.05
Accurate
dRPA
SOSEX
RPAx-I
RPAx-II
RPAx-SO1
RPAx-SO2
-0.1
-0.15
12
5
6
7
8
9
10
11
Internuclear distance (bohr)
=⇒ range separation improves all the RPA variants
12
Interaction energy curve of Ar2
with aug-cc-pV6Z basis
Without range separation:
With range separation:
0.1
Ar2
0
Interaction energy (mhartree)
Interaction energy (mhartree)
0.1
-0.1
-0.2
-0.3
Accurate
dRPA
SOSEX
RPAx-I
RPAx-II
RPAx-SO1
RPAx-SO2
-0.4
-0.5
-0.6
6
7
8
9
10
11
12
Internuclear distance (bohr)
Ar2
0
-0.1
-0.2
-0.3
Accurate
dRPA
SOSEX
RPAx-I
RPAx-II
RPAx-SO1
RPAx-SO2
-0.4
-0.5
-0.6
13
6
7
8
9
10
11
12
Internuclear distance (bohr)
=⇒ range separation improves all the RPA variants
13
Equilibrium interaction energies of the S22 set
Set of 22 weakly-interacting molecular systems from
water dimer to DNA base pairs (Jurečka et al. 2006)
mean absolute relative error (%)
with range separation and aug-cc-pVDZ basis:
20
15
10
5
0
D
C
C
2
O
1
O
S
x-
PA
R
S
x-
PA
R
II
x-
PA
R
X
I
x-
PA
R
SE
SO
PA
dR
Summary and Conclusions
range separation improves all the RPA variants
dRPA and SOSEX tend to underestimate interaction energies
RPAx-II tends to overestimate interaction energies
RPAx-I, RPAx-SO1 and RPAx-SO2 perform best
RPAx-SO2 has the avantage of not using any adiabatic
connection and of involving only singlet excitations
References:
[1] Toulouse, Gerber, Jansen, Savin, Ángyán, PRL 102, 096404 (2009)
[2] Toulouse, Zhu, Ángyán, Savin, PRA 82, 032502 (2010)
[3] Zhu, Toulouse, Savin, Ángyán, Savin, JCP 132, 244108 (2010)
[4] Grüneis, Marsman, Harl, Schimka, Kresse, JCP 131, 154115 (2009)
[5] McLachlan, Ball, RMP 36, 844 (1964)
[6] Szabo, Ostlund, JCP 67, 4351 (1977)
[7] Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, to appear