The use of many control variates is proposed as a method to accelerate the second- and third-order Monte Carlo (MC) many-body perturbation (MC-MP2 and MC-MP3) calculations. A control variate is an exactly integrable function that is strongly correlated or anti-correlated with the target function to be integrated by the MC method. Evaluating both integrals and their covariances in the same MC run, one can effect a mutual cancellation of the statistical uncertainties and biases in the MC integrations, thereby accelerating its convergence considerably. Six and thirty-six control variates, whose integrals are known a priori, are generated for MC-MP2 and MC-MP3, respectively, by systematically replacing one or more two-electron-integral vertices of certain configurations by zero-valued overlap-integral vertices in their Goldstone diagrams. The variances and covariances of these control variates are computed at a marginal cost, enhancing the overall efficiency of the MC-MP2 and MC-MP3 calculations by a factor of up to 14 and 20, respectively.
REFERENCES
1.
G. H.
Booth
, A. J.
Thom
, and A.
Alavi
, J. Chem. Phys.
131
, 054106
(2009
).2.
S. Y.
Willow
, K. S.
Kim
, and S.
Hirata
, J. Chem. Phys.
137
, 204122
(2012
).3.
F. R.
Petruzielo
, A. A.
Holmes
, H. J.
Changlani
, M. P.
Nightingale
, and C. J.
Umrigar
, Phys. Rev. Lett.
109
, 230201
(2012
).4.
D.
Neuhauser
, E.
Rabani
, and R.
Baer
, J. Chem. Theory Comput.
9
, 24
(2013
).5.
S. Y.
Willow
, K. S.
Kim
, and S.
Hirata
, J. Chem. Phys.
138
, 164111
(2013
).6.
S. Y.
Willow
, M. R.
Hermes
, K. S.
Kim
, and S.
Hirata
, J. Chem. Theory Comput.
9
, 4396
(2013
).7.
S.
Ten-no
, J. Chem. Phys.
138
, 164126
(2013
).8.
S. Y.
Willow
and S.
Hirata
, J. Chem. Phys.
140
, 024111
(2014
).9.
S. Y.
Willow
, K. S.
Kim
, and S.
Hirata
, Phys. Rev. B
90
, 201110
(2014
).10.
S. Y.
Willow
, J.
Zhang
, E. F.
Valeev
, and S.
Hirata
, J. Chem. Phys.
140
, 031101
(2014
).11.
G. H.
Booth
, S. D.
Smart
, and A.
Alavi
, Mol. Phys.
112
, 1855
(2014
).12.
Y.
Cytter
, D.
Neuhauser
, and R.
Baer
, J. Chem. Theory Comput.
10
, 4317
(2014
).13.
A. E.
Doran
and S.
Hirata
, J. Chem. Theory Comput.
12
, 4821
(2016
).14.
C. M.
Johnson
, A. E.
Doran
, J.
Zhang
, E. F.
Valeev
, and S.
Hirata
, J. Chem. Phys.
145
, 154115
(2016
).15.
D.
Neuhauser
, R.
Baer
, and D.
Zgid
, J. Chem. Theory Comput.
13
, 5396
(2017
).16.
T. Y.
Takeshita
, W. A.
de Jong
, D.
Neuhauser
, R.
Baer
, and E.
Rabani
, J. Chem. Theory Comput.
13
, 4605
(2017
).17.
G.
Jeanmairet
, S.
Sharma
, and A.
Alavi
, J. Chem. Phys.
146
, 044107
(2017
).18.
Y.
Garniron
, A.
Scemama
, P.-F.
Loos
, and M.
Caffarel
, J. Chem. Phys.
147
, 034101
(2017
).19.
J. E.
Deustua
, J.
Shen
, and P.
Piecuch
, Phys. Rev. Lett.
119
, 223003
(2017
).20.
C. M.
Johnson
, A. E.
Doran
, S. L.
Ten-no
, and S.
Hirata
, J. Chem. Phys.
149
, 174112
(2018
).21.
J. S.
Spencer
, V. A.
Neufeld
, W. A.
Vigor
, R. S. T.
Franklin
, and A. J. W.
Thom
, J. Chem. Phys.
149
, 204103
(2018
).22.
A. E.
Doran
and S.
Hirata
, J. Chem. Theory Comput.
15
, 6097
(2019
).23.
M.-A.
Filip
, C. J. C.
Scott
, and A. J. W.
Thom
, J. Chem. Theory Comput.
15
, 6625
(2019
).24.
W.
Dou
, T. Y.
Takeshita
, M.
Chen
, R.
Baer
, D.
Neuhauser
, and E.
Rabani
, J. Chem. Theory Comput.
15
, 6703
(2019
).25.
M.
Caffarel
, J. Chem. Phys.
151
, 064101
(2019
).26.
Z.
Li
, J. Chem. Phys.
151
, 244114
(2019
).27.
D. M.
Ceperley
and B. J.
Alder
, Phys. Rev. Lett.
45
, 566
(1980
).28.
B. L.
Hammond
, W. A.
Lester
, and P. J.
Reynolds
, Monte Carlo Methods in Ab Initio Quantum Chemistry
(World Scientific
, 1994
).29.
A.
Lüchow
and J. B.
Anderson
, Annu. Rev. Phys. Chem.
51
, 501
(2000
).30.
W. M. C.
Foulkes
, L.
Mitas
, R. J.
Needs
, and G.
Rajagopal
, Rev. Mod. Phys.
73
, 33
(2001
).31.
J.
Kolorenč
and L.
Mitas
, Rep. Prog. Phys.
74
, 026502
(2011
).32.
B. M.
Austin
, D. Y.
Zubarev
, and W. A.
Lester
, Chem. Rev.
112
, 263
(2012
).33.
L. K.
Wagner
, Int. J. Quantum Chem.
114
, 94
(2014
).34.
A. A.
Kunitsa
and S.
Hirata
, Phys. Rev. E
101
, 013311
(2020
).35.
M.
Head-Gordon
, J. A.
Pople
, and M. J.
Frisch
, Chem. Phys. Lett.
153
, 503
(1988
).36.
M. J.
Frisch
, M.
Head-Gordon
, and J. A.
Pople
, Chem. Phys. Lett.
166
, 281
(1990
).37.
R. J.
Bartlett
, Annu. Rev. Phys. Chem.
32
, 359
(1981
).38.
R. J.
Bartlett
, H.
Sekino
, and G. D.
Purvis
, Chem. Phys. Lett.
98
, 66
(1983
).39.
S. J.
Cole
and R. J.
Bartlett
, J. Chem. Phys.
86
, 873
(1987
).40.
R. J.
Bartlett
and M.
Musiał
, Rev. Mod. Phys.
79
, 291
(2007
).41.
I.
Shavitt
and R.
Bartlett
, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory
(Cambridge University Press
, 2009
).42.
W.
Klopper
, F. R.
Manby
, S.
Ten-no
, and E. F.
Valeev
, Int. Rev. Phys. Chem.
25
, 427
(2006
).43.
T.
Shiozaki
, E. F.
Valeev
, and S.
Hirata
, Annu. Rep. Comput. Chem.
5
, 131
(2009
).44.
C.
Hättig
, W.
Klopper
, A.
Köhn
, and D. P.
Tew
, Chem. Rev.
112
, 4
(2012
).45.
S.
Ten-no
and J.
Noga
, Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
, 114
(2012
).46.
L.
Kong
, F. A.
Bischoff
, and E. F.
Valeev
, Chem. Rev.
112
, 75
(2012
).47.
A.
Grüneis
, S.
Hirata
, Y.-y.
Ohnishi
, and S.
Ten-no
, J. Chem. Phys.
146
, 080901
(2017
).48.
J.
Almlöf
, Chem. Phys. Lett.
181
, 319
(1991
).49.
J. L.
Whitten
, J. Chem. Phys.
58
, 4496
(1973
).50.
S. A.
Maurer
, D. S.
Lambrecht
, D.
Flaig
, and C.
Ochsenfeld
, J. Chem. Phys.
136
, 144107
(2012
).51.
L. F.
Greengard
, The Rapid Evaluation of Potential Fields in Particle Systems
(MIT Press
, 1988
).52.
C. A.
White
, B. G.
Johnson
, P. M.
Gill
, and M.
Head-Gordon
, Chem. Phys. Lett.
230
, 8
(1994
).53.
L. E.
McMurchie
and E. R.
Davidson
, J. Comput. Phys.
26
, 218
(1978
).54.
S.
Obara
and A.
Saika
, J. Chem. Phys.
84
, 3963
(1986
).55.
S.
Obara
and A.
Saika
, J. Chem. Phys.
89
, 1540
(1988
).56.
A. V.
Titov
, I. S.
Ufimtsev
, N.
Luehr
, and T. J.
Martinez
, J. Chem. Theory Comput.
9
, 213
(2013
).57.
E. F.
Valeev
, Libint: A library for the evaluation of molecular integrals of many-body operators over Gaussian functions, version 2.7.0-beta.6, 2020
, http://libint.valeyev.net/.58.
J.
Zhang
, J. Chem. Theory Comput.
14
, 572
(2018
).59.
I. M. B.
Nielsen
and E. T.
Seidl
, J. Comput. Chem.
16
, 1301
(1995
).60.
M.
Katouda
and T.
Nakajima
, J. Chem. Theory Comput.
9
, 5373
(2013
).61.
D. G.
Tomlinson
, A.
Asadchev
, and M. S.
Gordon
, J. Comput. Chem.
37
, 1274
(2016
).62.
S.
Saebo
and P.
Pulay
, Annu. Rev. Phys. Chem.
44
, 213
(1993
).63.
M.
Schütz
, G.
Hetzer
, and H.-J.
Werner
, J. Chem. Phys.
111
, 5691
(1999
).64.
H.-J.
Werner
, F. R.
Manby
, and P. J.
Knowles
, J. Chem. Phys.
118
, 8149
(2003
).65.
E. G.
Hohenstein
, R. M.
Parrish
, and T. J.
Martínez
, J. Chem. Phys.
137
, 044103
(2012
).66.
67.
A. B.
Owen
, “Monte Carlo theory, methods and examples
,” (unpublished).68.
69.
A. E.
Doran
and S.
Hirata
, “Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling
,” J. Chem. Phys.
(to be published).70.
N.
Metropolis
, A. W.
Rosenbluth
, M. N.
Rosenbluth
, A. H.
Teller
, and E.
Teller
, J. Chem. Phys.
21
, 1087
(1953
).71.
W. K.
Hastings
, Biometrika
57
, 97
(1970
).72.
73.
A.
Szabo
and N. S.
Ostlund
, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory
(Dover
, 1989
).74.
75.
76.
N.
March
, W.
Young
, and S.
Sampanthar
, The Many-Body Problem in Quantum Mechanics
(Dover
, 1995
).77.
78.
A. E.
Doran
and S.
Hirata
, “Stochastic evaluation of fourth-order many-body perturbation energies
” (unpublished) (2020
).© 2020 Author(s).
2020
Author(s)
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