This paper reports the release of PathSum, a new software suite of state-of-the-art path integral methods for studying the dynamics of single or extended systems coupled to harmonic environments. The package includes two modules, suitable for system–bath problems and extended systems comprising many coupled system–bath units, and is offered in C++ and Fortran implementations. The system–bath module offers the recently developed small matrix path integral (SMatPI) and the well-established iterative quasi-adiabatic propagator path integral (i-QuAPI) method for iteration of the reduced density matrix of the system. In the SMatPI module, the dynamics within the entanglement interval can be computed using QuAPI, the blip sum, time evolving matrix product operators, or the quantum–classical path integral method. These methods have distinct convergence characteristics and their combination allows a user to access a variety of regimes. The extended system module provides the user with two algorithms of the modular path integral method, applicable to quantum spin chains or excitonic molecular aggregates. An overview of the methods and code structure is provided, along with guidance on method selection and representative examples.

1.
A. G.
Redfield
,
IBM J. Res. Dev.
1
,
19
31
(
1957
).
2.
W. T.
Pollard
,
A. K.
Felts
, and
R. A.
Friesner
, in
Advances in Chemical Physics
(
Wiley
,
1996
), Vol.
93
, pp.
77
134
.
3.
J. C.
Tully
and
R. K.
Preston
,
J. Chem. Phys.
55
,
562
572
(
1971
).
4.
J. C.
Tully
,
J. Chem. Phys.
93
,
1061
1071
(
1990
).
5.
J. C.
Tully
,
J. Chem. Phys.
137
(
22
),
22A301
(
2012
).
6.
S.
Hammes-Schiffer
,
Acc. Chem. Res.
42
,
1881
1889
(
2009
).
8.
E. J.
Heller
and
R. C.
Brown
,
J. Chem. Phys.
75
,
1048
1050
(
1981
).
9.
H.
Wang
,
X.
Sun
, and
W. H.
Miller
,
J. Chem. Phys.
108
,
9726
9736
(
1998
).
10.
X.
Sun
,
H.
Wang
, and
W. H.
Miller
,
J. Chem. Phys.
109
,
7064
7074
(
1998
).
11.
W. H.
Miller
,
J. Phys.Chem.
103
,
9384
9387
(
1999
).
12.
J. A.
Poulsen
,
G.
Nyman
, and
P. J.
Rossky
,
J. Chem. Phys.
119
(
23
),
12179
12193
(
2003
).
13.
J.
Shao
and
N.
Makri
,
J. Phys. Chem. A
103
,
7753
7756
(
1999
).
14.
N.
Makri
,
A.
Nakayama
, and
N. J.
Wright
,
J. Theor. Comput. Chem.
03
,
391
417
(
2004
).
15.
16.
J.
Liu
and
Z.
Zhang
,
J. Chem. Phys.
144
,
034307
(
2016
).
17.
H.-D.
Meyer
,
U.
Manthe
, and
L. S.
Cederbaum
,
Chem. Phys. Lett.
165
,
73
78
(
1990
).
18.
M.
Beck
,
A.
Jäckle
,
G. A.
Worth
, and
H.-D.
Meyer
,
Phys. Rep.
324
,
1
105
(
2000
).
19.
G. A.
Worth
,
H.-D.
Meyer
,
H.
Köppel
,
L. S.
Cederbaum
, and
I.
Burghardt
,
Int. Rev. Phys. Chem.
27
(
3
),
569
606
(
2008
).
20.
H.
Wang
and
M.
Thoss
,
J. Chem. Phys.
119
(
3
),
1289
1299
(
2003
).
22.
J.
Ren
,
Z.
Shuai
, and
G.
Kin-Lic Chan
,
J. Chem. Theory Comput.
14
(
10
),
5027
5039
(
2018
).
24.
U.
Schollwöck
,
Ann. Phys.
326
,
96
192
(
2011
).
25.
R. P.
Feynman
,
Rev. Mod. Phys.
20
,
367
387
(
1948
).
26.
R. P.
Feynman
and
A. R.
Hibbs
,
Quantum Mechanics and Path Integrals
(
McGraw-Hill
,
New York
,
1965
).
27.
L. S.
Schulman
,
Techniques and Applications of Path Integration
(
John Wiley and Sons
,
New York
,
1981
).
28.
J. D.
Doll
,
D. L.
Freeman
, and
T. L.
Beck
,
Adv. Chem. Phys.
78
,
61
127
(
1990
).
29.
N.
Makri
,
Comput. Phys. Commun.
63
,
389
414
(
1991
).
30.
R. P.
Feynman
,
Statistical Mechanics
(
Addison-Wesley
,
Redwood City
,
1972
).
31.
D.
Chandler
and
P. G.
Wolynes
,
J. Chem. Phys.
74
,
4078
4095
(
1981
).
32.
K.
Binder
and
D. W.
Heermann
,
Monte Carlo Simulation in Statistical Physics
(
Springer-Verlag
,
1988
).
33.
D. M.
Ceperley
,
Rev. Mod. Phys.
67
,
279
355
(
1995
).
34.
T. E.
Markland
and
M.
Ceriotti
,
Nat. Rev. Chem.
2
(
3
),
0109
(
2018
).
35.
G. A.
Voth
,
Adv. Chem. Phys.
93
,
135
218
(
1996
).
36.
S.
Habershon
,
D. E.
Manolopoulos
,
T. E.
Markland
, and
T. F.
Miller
,
Annu. Rev. Phys. Chem.
64
,
387
413
(
2013
).
37.
38.
T. J. H.
Hele
,
M. J.
Willatt
,
A.
Muolo
, and
S. C.
Althorpe
,
J. Chem. Phys.
142
,
191101
(
2015
).
40.
R. A.
Marcus
and
N.
Sutin
,
Biochim. Biophys. Acta
811
,
265
322
(
1985
).
41.
D.
Chandler
,
Introduction to Modern Statistical Mechanics
(
Oxford University Press
,
New York
,
1987
).
42.
N.
Makri
,
J. Phys. Chem.
103
,
2823
2829
(
1999
).
43.
R. P.
Feynman
and
F. L.
Vernon
,
Ann. Phys.
24
,
118
173
(
1963
).
44.
R.
Zwanzig
,
J. Stat. Phys.
9
,
215
220
(
1973
).
45.
D.
Thirumalai
,
E. J.
Bruskin
, and
B. J.
Berne
,
J. Chem. Phys.
79
,
5063
5069
(
1983
).
46.
B. J.
Berne
and
D.
Thirumalai
,
Annu. Rev. Phys. Chem.
37
,
401
424
(
1986
).
47.
A.
Ishizaki
and
Y.
Tanimura
,
J. Phys. Soc. Jpn.
74
(
12
),
3131
3134
(
2005
).
48.
H.
Liu
,
L.
Zhu
,
S.
Bai
, and
Q.
Shi
,
J. Chem. Phys.
140
(
13
),
134106
(
2014
).
49.
N.
Makri
,
Chem. Phys. Lett.
193
,
435
444
(
1992
).
50.
M.
Topaler
and
N.
Makri
,
Chem. Phys. Lett.
210
,
448
(
1993
).
51.
D. E.
Makarov
and
N.
Makri
,
Chem. Phys. Lett.
221
,
482
491
(
1994
).
52.
N.
Makri
and
D. E.
Makarov
,
J. Chem. Phys.
102
,
4600
4610
(
1995
).
53.
N.
Makri
and
D. E.
Makarov
,
J. Chem. Phys.
102
,
4611
4618
(
1995
).
54.
N.
Makri
,
J. Math. Phys.
36
,
2430
2456
(
1995
).
55.
N.
Makri
,
J. Phys. Chem.
97
,
2417
2424
(
1993
).
56.
E.
Sim
and
N.
Makri
,
Chem. Phys. Lett.
249
,
224
230
(
1996
).
57.
E.
Sim
and
N.
Makri
,
Comput. Phys. Commun.
99
,
335
354
(
1997
).
58.
A. A.
Golosov
,
R. A.
Friesner
, and
P.
Pechukas
,
J. Chem. Phys.
110
,
138
146
(
1999
).
59.
A. A.
Golosov
,
R. A.
Friesner
, and
P.
Pechukas
,
J. Chem. Phys.
112
,
2095
2105
(
2000
).
60.
E.
Sim
,
J. Chem. Phys.
115
,
4450
4456
(
2001
).
61.
D.
Segal
,
D. R.
Reichman
, and
A. J.
Millis
,
Phys. Rev. B
76
,
195316
(
2007
).
62.
P.
Nalbach
,
J.
Eckel
, and
M.
Thorwart
,
New J. Phys.
12
,
065043
(
2010
).
63.
R.
Lambert
and
N.
Makri
,
Mol. Phys.
110
,
1967
1975
(
2012
).
64.
N. S.
Dattani
,
AIP Adv.
2
,
012121
(
2012
).
66.
P. L.
Walters
,
T.
Banerjee
, and
N.
Makri
,
J. Chem. Phys.
143
,
074112
(
2015
).
67.
H.-G.
Duan
,
A. G.
Dijkstra
,
P.
Nalbach
, and
M.
Thorwart
,
Phys. Rev. E
92
(
4
),
042708
(
2015
).
68.
M.
Richter
and
B. P.
Fingerhut
,
J. Chem. Phys.
146
,
214101
(
2017
).
69.
Y.
Sato
,
J. Chem. Phys.
150
(
22
),
224108
(
2019
).
70.
71.
T. P.
Straatsma
,
B. W.
Lovett
, and
P.
Kirton
,
New J. Phys.
19
,
093009
(
2017
).
72.
A.
Strathearn
,
P.
Kirton
,
D.
Kilda
,
J.
Keeling
, and
B. W.
Lovett
,
Nat. Commun.
9
,
3322
(
2018
).
73.
M. R.
Jørgensen
and
F. A.
Pollock
,
Phys. Rev. Lett.
123
(
24
),
240602
(
2019
).
74.
D.
Gribben
,
A.
Strathearn
,
J.
Iles-Smith
,
D.
Kilda
,
A.
Nazir
,
B. W.
Lovett
, and
P.
Kirton
,
Phys. Rev. Res.
2
(
1
),
013265
(
2020
).
75.
A.
Bose
and
P. L.
Walters
,
J. Chem. Phys.
156
(
2
),
024101
(
2022
).
76.
77.
N.
Makri
,
J. Chem. Theory Comput.
16
,
4038
4049
(
2020
).
78.
N.
Makri
,
J. Chem. Theory Comput.
17
,
1
6
(
2021
).
79.
R.
Lambert
and
N.
Makri
,
J. Chem. Phys.
137
,
22A553
(
2012
).
80.
R.
Lambert
and
N.
Makri
,
J. Chem. Phys.
137
,
22A552
(
2012
).
81.
N.
Makri
,
Int. J. Quantum Chem.
115
,
1209
1214
(
2015
).
82.
83.
84.
S.
Kundu
and
N.
Makri
,
J. Chem. Phys.
151
,
074110
(
2019
).
85.
S.
Kundu
and
N.
Makri
,
J. Chem. Phys.
153
,
044124
(
2020
).
86.
S.
Kundu
and
N.
Makri
,
Mol. Phys.
119
,
e1797200
(
2021
).
87.
R.
Dani
and
N.
Makri
,
J. Chem. Phys.
155
,
234705
(
2021
).
88.
K. B.
Arnardottir
,
A. J.
Moilanen
,
A.
Strashko
,
P.
Törmä
, and
J.
Keeling
,
Phys. Rev. Lett.
125
(
23
),
233603
(
2020
).
89.
G. E.
Fux
,
E. P.
Butler
,
P. R.
Eastham
,
B. W.
Lovett
, and
J.
Keeling
,
Phys. Rev. Lett.
126
(
20
),
200401
(
2021
).
90.
S.
Kundu
and
N.
Makri
,
Annu. Rev. Phys. Chem.
73
(
1
),
349
375
(
2022
).
91.
R.
Dani
and
N.
Makri
,
J. Phys. Chem. C
126
(
25
),
10309
10319
(
2022
).
92.
H.-G.
Duan
,
V. I.
Prokhorenko
,
R. J.
Cogdell
,
K.
Ashraf
,
A. L.
Stevens
,
M.
Thorwart
, and
R. J. D.
Miller
,
Proc. Natl. Acad. Sci. U. S. A.
114
(32),
8493
8498
(
2017
).
93.
H.-G.
Duan
,
P.
Nalbach
,
R. J. D.
Miller
, and
M.
Thorwart
,
Photosynth. Res.
144
(
2
),
137
145
(
2020
).
94.
S.
Kundu
and
N.
Makri
,
J. Phys. Chem. Lett.
11
,
8783
8789
(
2020
).
95.
A.
Bose
and
P. L.
Walters
,
J. Chem. Theory Comput.
18
(
7
),
4095
4108
(
2022
).
96.
S.
Kundu
,
R.
Dani
, and
N.
Makri
,
J. Chem. Phys.
157
,
015101
(
2022
).
97.
S.
Kundu
,
R.
Dani
, and
N.
Makri
,
Sci. Adv.
8
(
43
),
eadd0023
(
2022
).
98.
Z.
Bacic
and
J. C.
Light
,
Annu. Rev. Phys. Chem.
40
,
469
498
(
1989
).
99.
J.
Echave
and
D. C.
Clary
,
J. Chem. Phys.
190
,
225
230
(
1992
).
100.
N.
Makri
,
J. Chem. Phys.
58
,
144107
(
2023
).
101.
A. O.
Caldeira
and
A. J.
Leggett
,
Physica A
121
,
587
616
(
1983
).
103.
V.
May
and
O.
Kühn
,
Charge and Energy Transfer Dynamics in Molecular Systems
, 3rd ed. (
Wiley
,
2011
).
104.
N.
Makri
,
J. Phys. A: Math. Theor.
56
,
144001
(
2023
).
105.
N.
Makri
,
J. Phys. Chem.
102
,
4414
4427
(
1998
).
106.
J.
Shao
and
N.
Makri
,
J. Chem. Phys.
116
,
507
514
(
2002
).
107.
A. J.
Leggett
,
S.
Chakravarty
,
A. T.
Dorsey
,
M. P. A.
Fisher
,
A.
Garg
, and
W.
Zwerger
,
Rev. Mod. Phys.
59
,
1
85
(
1987
).
108.
109.
N.
Makri
,
Faraday Discuss.
195
,
81
92
(
2016
).
110.
T.
Banerjee
and
N.
Makri
,
J. Phys. Chem.
117
,
13357
13366
(
2013
).
111.
F.
Wang
and
N.
Makri
,
J. Chem. Phys.
150
,
184102
(
2019
).
112.
N.
Makri
,
Chem. Phys. Lett.
593
,
93
103
(
2014
).
113.
P. L.
Walters
and
N.
Makri
,
J. Chem. Phys.
144
,
044108
(
2016
).
114.
S.
Kundu
and
N.
Makri
,
J. Phys. Chem. Lett.
13
,
3492
3498
(
2022
).
115.
S.
Chatterjee
and
N.
Makri
,
J. Phys. Chem. B
23
,
10470
(
2019
).
116.
P. L.
Walters
,
T. C.
Allen
, and
N.
Makri
,
J. Comput. Chem.
38
,
110
115
(
2017
).
117.
N.
Makri
,
J. Phys. Chem. A
125
(
48
),
10500
10506
(
2021
).
118.
N. S.
Dattani
,
Comput. Phys. Commun.
184
,
2828
2833
(
2013
).
119.
M.
Fishman
,
S. R.
White
, and
E. M.
Stoudenmire
,
SciPost Phys. Codebases
(published online
2022
).
120.
S.
Paeckel
,
T.
Köhler
,
A.
Swoboda
,
S. R.
Manmana
,
U.
Schollwöck
, and
C.
Hubig
,
Ann. Phys.
411
,
167998
(
2019
).
121.
N.
Makri
,
J. Chem. Theory Comput.
17
,
3825
3829
(
2021
).
122.
N.
Makri
,
Phys. Chem. Chem. Phys.
23
,
12537
12540
(
2021
).
123.
R.
Dani
and
N.
Makri
,
J. Phys. Chem. B
126
(
45
),
9361
9375
(
2022
).
124.
R.
Dani
and
N.
Makri
,
J. Phys. Chem. Lett.
13
(
34
),
8141
8149
(
2022
).
125.
R.
Dani
,
S.
Kundu
, and
N.
Makri
,
J. Phys. Chem. Lett.
14
,
3835
3843
(
2023
).
You do not currently have access to this content.