An atomic-orbital reformulation of the Laplace-transformed scaled opposite-spin (SOS) coupled cluster singles and doubles (CC2) model within the resolution of the identity (RI) approximation (SOS-RI-CC2) is presented that extends its applicability to molecules with several hundreds of atoms and triple-zeta basis sets. We exploit sparse linear algebra and an attenuated Coulomb metric to decrease the disk space demands and the computational efforts. In this way, an effective sub-quadratic computational scaling is achieved with our ω-SOS-CDD-RI-CC2 model. Moreover, Cholesky decomposition of the ground-state one-electron density matrix reduces the prefactor, allowing for an early crossover with the molecular orbital formulation. The accuracy and performance of the presented method are investigated for various molecular systems.

1.
J.
Čížek
and
J.
Paldus
, “
Correlation problems in atomic and molecular systems III. Rederivation of the coupled-pair many-electron theory using the traditional quantum chemical methods
,”
Int. J. Quantum Chem.
5
,
359
379
(
1971
).
2.
J.
Čížek
, “
On the use of the cluster expansion and the technique of diagrams in calculations of correlation effects in atoms and molecules
,”
Adv. Chem. Phys.
14
35
89
(
1969
).
3.
R. J.
Bartlett
, “
The coupled-cluster revolution
,”
Mol. Phys.
108
,
2905
2920
(
2010
).
4.
T. D.
Crawford
and
H. F.
Schaefer
, “
An introduction to coupled cluster theory for computational chemists
,”
Rev. Comput. Chem.
14
,
33
136
(
2000
).
5.
G. D.
Purvis
III
and
R. J.
Bartlett
, “
A full coupled-cluster singles and doubles model: The inclusion of disconnected triples
,”
J. Chem. Phys.
76
,
1910
1918
(
1982
).
6.
H.
Koch
,
A.
Sánchez de Merás
,
T.
Helgaker
, and
O.
Christiansen
, “
The integral-direct coupled cluster singles and doubles model
,”
J. Chem. Phys.
104
,
4157
4165
(
1996
).
7.
O.
Christiansen
,
H.
Koch
, and
P.
Jørgensen
, “
The second-order approximate coupled cluster singles and doubles model CC2
,”
Chem. Phys. Lett.
243
,
409
418
(
1995
).
8.
Y.
Jung
,
R. C.
Lochan
,
A. D.
Dutoi
, and
M.
Head-Gordon
, “
Scaled opposite-spin second order Møller-Plesset correlation energy: An economical electronic structure method
,”
J. Chem. Phys.
121
,
9793
9802
(
2004
).
9.
Y.
Jung
,
Y.
Shao
, and
M.
Head-Gordon
, “
Fast evaluation of scaled opposite spin second-order Møller-Plesset correlation energies using auxiliary basis expansions and exploiting sparsity
,”
J. Comput. Chem.
28
,
1953
1964
(
2007
).
10.
P.
Pulay
, “
Localizability of dynamic electron correlation
,”
Chem. Phys. Lett.
100
,
151
154
(
1983
).
11.
P.
Pulay
and
S.
Saebø
, “
Orbital-invariant formulation and second-order gradient evaluation in Møller-Plesset perturbation theory
,”
Theor. Chim. Acta
69
,
357
368
(
1986
).
12.
S.
Saebø
and
P.
Pulay
, “
A low-scaling method for second order Møller-Plesset calculations
,”
J. Chem. Phys.
115
,
3975
3983
(
2001
).
13.
H.-J.
Werner
,
F. R.
Manby
, and
P. J.
Knowles
, “
Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations
,”
J. Chem. Phys.
118
,
8149
8160
(
2003
).
14.
S. A.
Maurer
,
L.
Clin
, and
C.
Ochsenfeld
, “
Cholesky-decomposed density MP2 with density fitting: Accurate MP2 and double-hybrid DFT energies for large systems
,”
J. Chem. Phys.
140
,
224112
(
2014
).
15.
P. Y.
Ayala
and
G. E.
Scuseria
, “
Linear scaling second-order Moller-Plesset theory in the atomic orbital basis for large molecular systems
,”
J. Chem. Phys.
110
,
3660
3671
(
1999
).
16.
S. A.
Maurer
,
D. S.
Lambrecht
,
J.
Kussmann
, and
C.
Ochsenfeld
, “
Efficient distance-including integral screening in linear-scaling Moller-Plesset perturbation theory
,”
J. Chem. Phys.
138
,
014101
(
2013
).
17.
S.
Schweizer
,
B.
Doser
, and
C.
Ochsenfeld
, “
An atomic orbital-based reformulation of energy gradients in second-order Møller-Plesset perturbation theory
,”
J. Chem. Phys.
128
,
154101
(
2008
).
18.
M.
Glasbrenner
,
D.
Graf
, and
C.
Ochsenfeld
, “
Efficient reduced-scaling second-order Møller-Plesset perturbation theory with Cholesky-decomposed densities and an attenuated Coulomb metric
,”
J. Chem. Theory Comput.
16
,
6856
6868
(
2020
).
19.
R. A.
Kendall
and
H. A.
Früchtl
, “
The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development
,”
Theor. Chem. Acc.
97
,
158
163
(
1997
).
20.
M.
Feyereisen
,
G.
Fitzgerald
, and
A.
Komornicki
, “
Use of approximate integrals in ab initio theory. An application in MP2 energy calculations
,”
Chem. Phys. Lett.
208
,
359
363
(
1993
).
21.
C.
Hättig
and
F.
Weigend
, “
CC2 excitation energy calculations on large molecules using the resolution of the identity approximation
,”
J. Chem. Phys.
113
,
5154
5161
(
2000
).
22.
H.
Koch
,
A.
Sánchez de Merás
, and
T. B.
Pedersen
, “
Reduced scaling in electronic structure calculations using Cholesky decompositions
,”
J. Chem. Phys.
118
,
9481
9484
(
2003
).
23.
J.
Boström
,
M.
Pitoňák
,
F.
Aquilante
,
P.
Neogrády
,
T. B.
Pedersen
, and
R.
Lindh
, “
Coupled cluster and Møller–Plesset perturbation theory calculations of noncovalent intermolecular interactions using density fitting with auxiliary basis sets from Cholesky decompositions
,”
J. Chem. Theory Comput.
8
,
1921
1928
(
2012
).
24.
P.
Baudin
,
J. S.
Marín
,
I. G.
Cuesta
, and
A. M. J.
Sánchez de Merás
, “
Calculation of excitation energies from the CC2 linear response theory using Cholesky decomposition
,”
J. Chem. Phys.
140
,
104111
(
2014
).
25.
S. D.
Folkestad
,
E. F.
Kjønstad
,
L.
Goletto
, and
H.
Koch
, “
Multilevel CC2 and CCSD in reduced orbital spaces: Electronic excitations in large molecular systems
,”
J. Chem. Theory Comput.
17
,
714
726
(
2021
).
26.
S.
Grimme
, “
Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel-and antiparallel-spin pair correlation energies
,”
J. Chem. Phys.
118
,
9095
9102
(
2003
).
27.
M.
Häser
, “
Møller-Plesset (MP2) perturbation theory for large molecules
,”
Theor. Chim. Acta
87
,
147
173
(
1993
).
28.
J.
Almlöf
, “
Elimination of energy denominators in Møller-Plesset perturbation theory by a Laplace transform approach
,”
Chem. Phys. Lett.
181
,
319
320
(
1991
).
29.
M.
Häser
and
J.
Almlöf
, “
Laplace transform techniques in Møller-Plesset perturbation theory
,”
J. Chem. Phys.
96
,
489
494
(
1992
).
30.
N. O. C.
Winter
and
C.
Hättig
, “
Scaled opposite-spin CC2 for ground and excited states with fourth order scaling computational costs
,”
J. Chem. Phys.
134
,
184101
(
2011
).
31.
D.
Kats
,
T.
Korona
, and
M.
Schütz
, “
Local CC2 electronic excitation energies for large molecules with density fitting
,”
J. Chem. Phys.
125
,
104106
(
2006
).
32.
D.
Kats
and
M.
Schütz
, “
A multistate local coupled cluster CC2 response method based on the Laplace transform
,”
J. Chem. Phys.
131
,
124117
(
2009
).
33.
K.
Ledermüller
and
M.
Schütz
, “
Local CC2 response method based on the Laplace transform: Analytic energy gradients for ground and excited states
,”
J. Chem. Phys.
140
,
164113
(
2014
).
34.
P.
Baudin
and
K.
Kristensen
, “
LoFEx—A local framework for calculating excitation energies: Illustrations using RI-CC2 linear response theory
,”
J. Chem. Phys.
144
,
224106
(
2016
).
35.
F.
Neese
,
A.
Hansen
, and
D. G.
Liakos
, “
Efficient and accurate approximations to the local coupled cluster singles doubles method using a truncated pair natural orbital basis
,”
J. Chem. Phys.
131
,
064103
(
2009
).
36.
F.
Neese
,
F.
Wennmohs
, and
A.
Hansen
, “
Efficient and accurate local approximations to coupled-electron pair approaches: An attempt to revive the pair natural orbital method
,”
J. Chem. Phys.
130
,
114108
(
2009
).
37.
B.
Helmich
and
C.
Hättig
, “
A pair natural orbital implementation of the coupled cluster model CC2 for excitation energies
,”
J. Chem. Phys.
139
,
084114
(
2013
).
38.
F.
Weigend
and
M.
Häser
, “
RI-MP2: First derivatives and global consistency
,”
Theor. Chem. Acc.
97
,
331
340
(
1997
).
39.
R. C.
Lochan
,
Y.
Shao
, and
M.
Head-Gordon
, “
Quartic-scaling analytical energy gradient of scaled opposite-spin second-order Møller- Plesset perturbation theory
,”
J. Chem. Theory Comput.
3
,
988
1003
(
2007
).
40.
G. E.
Scuseria
and
P. Y.
Ayala
, “
Linear scaling coupled cluster and perturbation theories in the atomic orbital basis
,”
J. Chem. Phys.
111
,
8330
8343
(
1999
).
41.
M.
Beer
and
C.
Ochsenfeld
, “
Efficient linear-scaling calculation of response properties: Density matrix-based Laplace-transformed coupled-perturbed self-consistent field theory
,”
2008
.
42.
M.
Beuerle
,
D.
Graf
,
H. F.
Schurkus
, and
C.
Ochsenfeld
, “
Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism
,”
J. Chem. Phys.
148
,
204104
(
2018
).
43.
A.
Luenser
,
H. F.
Schurkus
, and
C.
Ochsenfeld
, “
Vanishing-overhead linear-scaling random phase approximation by Cholesky decomposition and an attenuated Coulomb-metric
,”
J. Chem. Theory Comput.
13
,
1647
1655
(
2017
).
44.
D.
Graf
,
M.
Beuerle
, and
C.
Ochsenfeld
, “
Low-scaling self-consistent minimization of a density matrix based random phase approximation method in the atomic orbital space
,”
J. Chem. Theory Comput.
15
,
4468
4477
(
2019
).
45.
Y.
Jung
,
A.
Sodt
,
P. M. W.
Gill
, and
M.
Head-Gordon
, “
Auxiliary basis expansions for large-scale electronic structure calculations
,”
Proc. Natl. Acad. Sci. U. S. A.
102
,
6692
6697
(
2005
).
46.
T. D.
Crawford
,
A.
Kumar
,
A. P.
Bazanté
, and
R.
Di Remigio
, “
Reduced-scaling coupled cluster response theory: Challenges and opportunities
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
9
,
e1406
(
2019
).
47.
R. J.
Bartlett
and
G. D.
Purvis
, “
Many-body perturbation theory, coupled-pair many-electron theory, and the importance of quadruple excitations for the correlation problem
,”
Int. J. Quantum Chem.
14
,
561
581
(
1978
).
48.
P.
Pulay
, “
Improved SCF convergence acceleration
,”
J. Comput. Chem.
3
,
556
560
(
1982
).
49.
A.
Takatsuka
,
S.
Ten-No
, and
W.
Hackbusch
, “
Minimax approximation for the decomposition of energy denominators in Laplace-transformed Møller-Plesset perturbation theories
,”
J. Chem. Phys.
129
,
044112
(
2008
).
50.
B.
Helmich-Paris
and
L.
Visscher
, “
Improvements on the minimax algorithm for the Laplace transformation of orbital energy denominators
,”
J. Comput. Phys.
321
,
927
931
(
2016
).
51.
J.
Kussmann
and
C.
Ochsenfeld
, “
Pre-selective screening for matrix elements in linear-scaling exact exchange calculations
,”
J. Chem. Phys.
138
,
134114
(
2013
).
52.
J.
Kussmann
and
C.
Ochsenfeld
, “
Preselective screening for linear-scaling exact exchange-gradient calculations for graphics processing units and general strong-scaling massively parallel calculations
,”
J. Chem. Theory Comput.
11
,
918
922
(
2015
).
53.
J.
Kussmann
and
C.
Ochsenfeld
, “
Hybrid CPU/GPu integral engine for strong-scaling ab initio methods
,”
J. Chem. Theory Comput.
13
,
3153
3159
(
2017
).
54.
N. J.
Higham
, “
Cholesky factorization
,”
Wiley Interdiscip. Rev.: Comput. Stat.
1
,
251
254
(
2009
).
55.
H.
Harbrecht
,
M.
Peters
, and
R.
Schneider
, “
On the low-rank approximation by the pivoted Cholesky decomposition
,”
Appl. Numer. Math.
62
,
428
440
(
2012
).
56.
S.
Schweizer
,
J.
Kussmann
,
B.
Doser
, and
C.
Ochsenfeld
, “
Linear-scaling Cholesky decomposition
,”
J. Comput. Chem.
29
,
1004
1010
(
2008
).
57.
F.
Aquilante
,
T.
Bondo Pedersen
,
A.
Sánchez de Merás
, and
H.
Koch
, “
Fast noniterative orbital localization for large molecules
,”
J. Chem. Phys.
125
,
174101
(
2006
).
58.
V.
Drontschenko
,
D.
Graf
,
H.
Laqua
, and
C.
Ochsenfeld
, “
Lagrangian-based minimal-overhead batching scheme for the efficient integral-direct evaluation of the RPA correlation energy
,”
J. Chem. Theory Comput.
17
,
5623
5634
(
2021
).
59.
V.
Turbomole
, 7.3, TURBOMOLE GmbH 2018, TURBOMOLE is a development of University of Karlsruhe and Forschungszentrum Karlsruhe 2007,
1989
.
60.
J.
Kussmann
,
H.
Laqua
, and
C.
Ochsenfeld
, “
Highly efficient resolution-of-identity density functional theory calculations on central and graphics processing units
,”
J. Chem. Theory Comput.
17
,
1512
1521
(
2021
).
61.
H.
Laqua
,
T. H.
Thompson
,
J.
Kussmann
, and
C.
Ochsenfeld
, “
Highly efficient, linear-scaling seminumerical exact-exchange method for graphic processing units
,”
J. Chem. Theory Comput.
16
,
1456
1468
(
2020
).
62.
H.
Laqua
,
J.
Kussmann
, and
C.
Ochsenfeld
, “
Accelerating seminumerical Fock-exchange calculations using mixed single-and double-precision arithmethic
,”
J. Chem. Phys.
154
,
214116
(
2021
).
63.
F.
Weigend
,
F.
Furche
, and
R.
Ahlrichs
, “
Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr
,”
J. Chem. Phys.
119
,
12753
12762
(
2003
).
64.
F.
Weigend
and
R.
Ahlrichs
, “
Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
,”
Phys. Chem. Chem. Phys.
7
,
3297
3305
(
2005
).
65.
F.
Weigend
,
M.
Häser
,
H.
Patzelt
, and
R.
Ahlrichs
, “
RI-MP2: Optimized auxiliary basis sets and demonstration of efficiency
,”
Chem. Phys. Lett.
294
,
143
152
(
1998
).
66.
P.
Jurečka
,
J.
Šponer
,
J.
Černỳ
, and
P.
Hobza
, “
Benchmark database of accurate (MP2 and CCSD (T) complete basis set limit) interaction energies of small model complexes, DNA base pairs, and amino acid pairs
,”
Phys. Chem. Chem. Phys.
8
,
1985
1993
(
2006
).
67.
R.
Sedlak
,
T.
Janowski
,
M.
Pitoňák
,
J.
Řezáč
,
P.
Pulay
, and
P.
Hobza
, “
Accuracy of quantum chemical methods for large noncovalent complexes
,”
J. Chem. Theory Comput.
9
,
3364
3374
(
2013
).
68.
Structures available online from http://www.cup.lmu.de/pc/ochsenfeld/.
69.
P. E.
Maslen
,
C.
Ochsenfeld
,
C. A.
White
,
M. S.
Lee
, and
M.
Head-Gordon
, “
Locality and sparsity of ab initio one-particle density matrices and localized orbitals
,”
J. Phys. Chem. A
102
,
2215
2222
(
1998
).
70.
C.
Ochsenfeld
,
J.
Kussmann
, and
D. S.
Lambrecht
, “
Linear-scaling methods in quantum chemistry
,”
Rev. Comput. Chem.
23
,
1
(
2007
).
71.
D.
Yu
,
Y.
Zhu
,
T.
Jiao
,
T.
Wu
,
X.
Xiao
,
B.
Qin
,
Y.
Hu
,
H.
Chong
,
X.
Lei
,
L.
Ren
et al, “
Structure-based design and characterization of novel fusion-inhibitory lipopeptides against SARS-CoV-2 and emerging variants
,”
Emerging Microbes Infect.
10
,
1227
1240
(
2021
).
72.
C.
Seuring
,
J.
Verasdonck
,
J.
Gath
,
D.
Ghosh
,
N.
Nespovitaya
,
M. A.
Wälti
,
S. K.
Maji
,
R.
Cadalbert
,
P.
Güntert
,
B. H.
Meier
, and
R.
Riek
, “
The three-dimensional structure of human β-endorphin amyloid fibrils
,”
Nat. Struct. Mol. Biol.
27
,
1178
1184
(
2020
).
73.
Structures available online from http://www.petachem.com/products.html.
74.
C. J.
Williams
,
J. J.
Headd
,
N. W.
Moriarty
,
M. G.
Prisant
,
L. L.
Videau
,
L. N.
Deis
,
V.
Verma
,
D. A.
Keedy
,
B. J.
Hintze
,
V. B.
Chen
et al, “
Molprobity: More and better reference data for improved all-atom structure validation
,”
Protein Sci.
27
,
293
315
(
2018
).

Supplementary Material

You do not currently have access to this content.