In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis–Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which, in turn, increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial “burn-in” step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. This direct-sampling algorithm is shown to accelerate second- and third-order Monte Carlo many-body perturbation calculations by up to 222% and 38%, respectively.

1.
S. Y.
Willow
,
K. S.
Kim
, and
S.
Hirata
,
J. Chem. Phys.
137
,
204122
(
2012
).
2.
S. Y.
Willow
,
K. S.
Kim
, and
S.
Hirata
,
J. Chem. Phys.
138
,
164111
(
2013
).
3.
S. Y.
Willow
,
M. R.
Hermes
,
K. S.
Kim
, and
S.
Hirata
,
J. Chem. Theory Comput.
9
,
4396
(
2013
).
4.
S. Y.
Willow
and
S.
Hirata
,
J. Chem. Phys.
140
,
024111
(
2014
).
5.
S. Y.
Willow
,
K. S.
Kim
, and
S.
Hirata
,
Phys. Rev. B
90
,
201110
(
2014
).
6.
S. Y.
Willow
,
J.
Zhang
,
E. F.
Valeev
, and
S.
Hirata
,
J. Chem. Phys.
140
,
031101
(
2014
).
7.
A. E.
Doran
and
S.
Hirata
,
J. Chem. Theory Comput.
12
,
4821
(
2016
).
8.
C. M.
Johnson
,
A. E.
Doran
,
J.
Zhang
,
E. F.
Valeev
, and
S.
Hirata
,
J. Chem. Phys.
145
,
154115
(
2016
).
9.
C. M.
Johnson
,
A. E.
Doran
,
S. L.
Ten-no
, and
S.
Hirata
,
J. Chem. Phys.
149
,
174112
(
2018
).
10.
A. E.
Doran
and
S.
Hirata
,
J. Chem. Theory Comput.
15
,
6097
(
2019
).
11.
G. H.
Booth
,
A. J. W.
Thom
, and
A.
Alavi
,
J. Chem. Phys.
131
,
054106
(
2009
).
12.
F. R.
Petruzielo
,
A. A.
Holmes
,
H. J.
Changlani
,
M. P.
Nightingale
, and
C. J.
Umrigar
,
Phys. Rev. Lett.
109
,
230201
(
2012
).
13.
D.
Neuhauser
,
E.
Rabani
, and
R.
Baer
,
J. Chem. Theory Comput.
9
,
24
(
2013
).
14.
S.
Ten-no
,
J. Chem. Phys.
138
,
164126
(
2013
).
15.
G. H.
Booth
,
S. D.
Smart
, and
A.
Alavi
,
Mol. Phys.
112
,
1855
(
2014
).
16.
Y.
Cytter
,
D.
Neuhauser
, and
R.
Baer
,
J. Chem. Theory Comput.
10
,
4317
(
2014
).
17.
D.
Neuhauser
,
R.
Baer
, and
D.
Zgid
,
J. Chem. Theory Comput.
13
,
5396
(
2017
).
18.
T. Y.
Takeshita
,
W. A.
de Jong
,
D.
Neuhauser
,
R.
Baer
, and
E.
Rabani
,
J. Chem. Theory Comput.
13
,
4605
(
2017
).
19.
G.
Jeanmairet
,
S.
Sharma
, and
A.
Alavi
,
J. Chem. Phys.
146
,
044107
(
2017
).
20.
Y.
Garniron
,
A.
Scemama
,
P.-F.
Loos
, and
M.
Caffarel
,
J. Chem. Phys.
147
,
034101
(
2017
).
21.
J. E.
Deustua
,
J.
Shen
, and
P.
Piecuch
,
Phys. Rev. Lett.
119
,
223003
(
2017
).
22.
J. S.
Spencer
,
V. A.
Neufeld
,
W. A.
Vigor
,
R. S. T.
Franklin
, and
A. J. W.
Thom
,
J. Chem. Phys.
149
,
204103
(
2018
).
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.
W.
Klopper
,
F. R.
Manby
,
S.
Ten-no
, and
E. F.
Valeev
,
Int. Rev. Phys. Chem.
25
,
427
(
2006
).
36.
T.
Shiozaki
,
E. F.
Valeev
, and
S.
Hirata
,
Annu. Rep. Comput. Chem.
5
,
131
(
2009
).
37.
C.
Hättig
,
W.
Klopper
,
A.
Köhn
, and
D. P.
Tew
,
Chem. Rev.
112
,
4
(
2012
).
38.
S.
Ten-no
and
J.
Noga
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
2
,
114
(
2012
).
39.
L.
Kong
,
F. A.
Bischoff
, and
E. F.
Valeev
,
Chem. Rev.
112
,
75
(
2012
).
40.
A.
Grüneis
,
S.
Hirata
,
Y.-y.
Ohnishi
, and
S.
Ten-no
,
J. Chem. Phys.
146
,
080901
(
2017
).
41.
S.
Obara
and
A.
Saika
,
J. Chem. Phys.
84
,
3963
(
1986
).
42.
N.
Metropolis
,
A. W.
Rosenbluth
,
M. N.
Rosenbluth
,
A. H.
Teller
, and
E.
Teller
,
J. Chem. Phys.
21
,
1087
(
1953
).
43.
W. K.
Hastings
,
Biometrika
57
,
97
(
1970
).
44.
H.
Flyvbjerg
and
H. G.
Petersen
,
J. Chem. Phys.
91
,
461
(
1989
).
45.
M.
Head-Gordon
,
J. A.
Pople
, and
M. J.
Frisch
,
Chem. Phys. Lett.
153
,
503
(
1988
).
46.
S. I. L.
Kokkila Schumacher
,
E. G.
Hohenstein
,
R. M.
Parrish
,
L.-P.
Wang
, and
T. J.
Martínez
,
J. Chem. Theory Comput.
11
,
3042
(
2015
).
47.
C.
Song
and
T. J.
Martínez
,
J. Chem. Phys.
144
,
174111
(
2016
).
48.
C.
Song
and
T. J.
Martínez
,
J. Chem. Phys.
146
,
034104
(
2017
).
49.
Q.
Ge
,
Y.
Gao
,
R.
Baer
,
E.
Rabani
, and
D.
Neuhauser
,
J. Phys. Chem. Lett.
5
,
185
(
2014
).
50.
M.
Kalos
and
P.
Whitlock
,
Monte Carlo Methods
(
Wiley
,
2008
).
51.
A. B.
Owen
, Monte Carlo Theory, Methods and Examples (
2013
).
52.
D.
Kroese
,
T.
Taimre
, and
Z.
Botev
,
Handbook of Monte Carlo Methods
(
Wiley
,
2013
).
53.
J.
Pitman
,
Probability
(
Springer
,
1993
).
54.
J.
Rice
,
Mathematical Statistics and Data Analysis
(
Cengage Learning
,
2006
).
55.
G. E. P.
Box
and
M. E.
Muller
,
Ann. Math. Stat.
29
,
610
(
1958
).
58.
J.
Besag
and
P. J.
Green
,
J. R. Stat. Soc.
55
,
25
(
1993
).
59.
J.
Besag
,
P.
Green
,
D.
Higdon
, and
K.
Mengersen
,
Stat. Sci.
10
,
3
(
1995
).
60.
G. O.
Roberts
,
A.
Gelman
, and
W. R.
Gilks
,
Ann. Appl. Probab.
7
,
110
(
1997
).
61.
P.
Neal
and
G.
Roberts
,
Ann. Appl. Probab.
16
,
475
(
2006
).
62.
M.
Bédard
,
Stochastic Processes Appl.
118
,
2198
(
2008
).
63.
A.
Householder
,
The Numerical Treatment of a Single Nonlinear Equation
(
McGraw-Hill
,
1970
).
64.
T. H.
Dunning
,
J. Chem. Phys.
90
,
1007
(
1989
).

Supplementary Material

You do not currently have access to this content.