We derive a multidimensional instanton theory for calculating ground-state tunneling splittings in Cartesian coordinates for general paths. It is an extension of the method by Mil’nikov and Nakamura [J. Chem. Phys. 115, 6881 (2001)] to include asymmetric paths that are necessary for calculating tunneling splitting patterns in multi-well systems, such as water clusters. The approach avoids multiple expensive matrix diagonalizations to converge the fluctuation prefactor in the ring-polymer instanton (RPI) method, and instead replaces them by an integration of a Riccati differential equation. When combined with the string method for locating instantons, we avoid the need to converge the calculation with respect to the imaginary time period of the semiclassical orbit, thereby reducing the number of convergence parameters of the optimized object to just one: the number of equally spaced system replicas used to represent the instanton path. The entirety of the numerical effort is thus concentrated in optimizing the shape of the path and evaluating hessians along the path, which is a dramatic improvement over RPI. In addition to the standard instanton approximations, we neglect the coupling of vibrational modes to external rotations. The method is tested on the model potential of malonaldehyde and on the water dimer and trimer, giving close agreement with RPI at a much-reduced cost.

1.
V. A.
Benderskii
,
D. E.
Makarov
, and
C. A.
Wight
, “
Chemical dynamics at low temperatures
,” in
Advances in Chemical Physics
(
Wiley
,
New York
,
1994
), Vol. 88.
2.
M. T.
Cvitaš
and
S. C.
Althorpe
,
Phys. Scr.
80
,
048115
(
2009
).
3.
M.
Schröder
,
F.
Gatti
, and
H.-D.
Meyer
,
J. Chem. Phys.
134
,
234307
(
2011
).
4.
T.
Hammer
and
U.
Manthe
,
J. Chem. Phys.
134
,
224305
(
2011
).
5.
H.-G.
Yu
,
H.
Song
, and
M.
Yang
,
J. Chem. Phys.
146
,
224307
(
2017
).
6.
R.
Marquardt
,
K.
Sagui
,
J.
Zheng
,
W.
Thiel
,
D.
Luckhaus
,
S.
Yurchenko
,
F.
Mariotti
, and
M.
Quack
,
J. Phys. Chem. A
117
,
7502
(
2013
).
7.
X.
Huang
,
S.
Carter
, and
J.
Bowman
,
J. Chem. Phys.
118
,
5431
(
2003
).
8.
T.
Kawatsu
and
S.
Miura
,
Chem. Phys. Lett.
634
,
146
(
2015
).
9.
A. R.
Sharma
,
B. J.
Braams
,
S.
Carter
,
B. C.
Shepler
, and
J. M.
Bowman
,
J. Chem. Phys.
130
,
174301
(
2009
).
10.
M. T.
Cvitaš
and
S. C.
Althorpe
,
J. Chem. Theory Comput.
12
,
787
(
2016
).
11.
Y.
Guo
,
T. D.
Sewell
, and
D. L.
Thompson
,
Chem. Phys. Lett.
224
,
470
(
1994
).
12.
T.
Carrington
and
W. H.
Miller
,
J. Chem. Phys.
84
,
4364
(
1986
).
13.
G. V.
Mil’nikov
,
T.
Ishida
, and
H.
Nakamura
,
J. Phys. Chem. A
110
,
5430
(
2006
).
14.
E.
Kamarchik
,
Y.
Wang
, and
J.
Bowman
,
J. Phys. Chem. A
113
,
7556
(
2009
).
15.
D. J.
Nesbitt
and
F.
Dong
,
Phys. Chem. Chem. Phys.
10
,
2113
(
2008
).
16.
Y.
Wang
,
B. J.
Braams
,
J. M.
Bowman
,
S.
Carter
, and
D. P.
Tew
,
J. Chem. Phys.
128
,
224314
(
2008
).
17.
M. D.
Coutinho-Neto
,
A.
Viel
, and
U.
Manthe
,
J. Chem. Phys.
121
,
9207
(
2004
).
18.
C. S.
Tautermann
,
A. F.
Voegele
,
T.
Loerting
, and
K. R.
Liedl
,
J. Chem. Phys.
117
,
1962
(
2002
).
19.
Z.
Smedarchina
,
W.
Siebrand
, and
M. Z.
Zgierski
,
J. Chem. Phys.
103
,
5326
(
1995
).
20.
K.
Liu
,
J. D.
Cruzan
, and
R. J.
Saykally
,
Science
271
,
929
(
1996
).
21.
F. N.
Keutsch
and
R. J.
Saykally
,
Proc. Natl. Acad. Sci. U. S. A.
98
,
10533
(
2001
).
22.
X.-G.
Wang
and
T.
Carrington
,
J. Chem. Phys.
148
,
074108
(
2018
).
23.
C.
Leforestier
,
K.
Szalewicz
, and
A.
van der Avoird
,
J. Chem. Phys.
137
,
014305
(
2012
).
24.
S. C.
Althorpe
and
D. C.
Clary
,
J. Chem. Phys.
102
,
4390
(
1995
).
25.
T. R.
Walsh
and
D. J.
Wales
,
J. Chem. Soc., Faraday Trans.
92
,
2505
(
1996
).
26.
A.
van der Avoird
,
E. H. T.
Olthof
, and
P. E. S.
Wormer
,
J. Chem. Phys.
105
,
8034
(
1996
).
27.
D.
Sabo
,
Z.
Bačić
,
T.
Bürgi
, and
S.
Leutwyler
,
Chem. Phys. Lett.
244
,
283
(
1995
).
28.
D.
Blume
and
K. B.
Whaley
,
J. Chem. Phys.
112
,
2218
(
2000
).
29.
T.
Taketsugu
and
D. J.
Wales
,
Mol. Phys.
100
,
2793
(
2002
).
30.
J. O.
Richardson
,
S. C.
Althorpe
, and
D. J.
Wales
,
J. Chem. Phys.
135
,
124109
(
2011
).
31.
J. K.
Gregory
and
D. C.
Clary
,
J. Chem. Phys.
102
,
7817
(
1995
).
32.
M.
Quack
and
M. A.
Suhm
,
Chem. Phys. Lett.
234
,
71
(
1995
).
33.
M.
Takahashi
,
Y.
Watanabe
,
T.
Taketsugu
, and
D. J.
Wales
,
J. Chem. Phys.
123
,
044302
(
2005
).
34.
L. H.
Coudert
and
J. T.
Hougen
,
J. Mol. Spectrosc.
130
,
86
(
1988
).
35.
J. O.
Richardson
,
C.
Pérez
,
S.
Lobsiger
,
A. A.
Reid
,
B.
Temelso
,
G. C.
Shields
,
Z.
Kisiel
,
D. J.
Wales
,
B. H.
Pate
, and
S. C.
Althorpe
,
Science
351
,
1310
(
2016
).
36.
J. O.
Richardson
,
D. J.
Wales
,
S. C.
Althorpe
,
R. P.
McLaughlin
,
M. R.
Viant
,
O.
Shih
, and
R. J.
Saykally
,
J. Phys. Chem. A
117
,
6960
(
2013
).
37.
M. T.
Cvitaš
and
J. O.
Richardson
,
Phys. Chem. Chem. Phys.
22
,
1035
(
2019
).
38.
E.
Mátyus
,
D. J.
Wales
, and
S. C.
Althorpe
,
J. Chem. Phys.
144
,
114108
(
2016
).
39.
C. L.
Vaillant
,
D. J.
Wales
, and
S. C.
Althorpe
,
J. Chem. Phys.
148
,
234102
(
2018
).
40.
C. L.
Vaillant
,
D. J.
Wales
, and
S. C.
Althorpe
,
J. Phys. Chem. Lett.
10
,
7300
(
2019
).
41.
A. I.
Vainshtein
,
V. I.
Zakharov
,
V. A.
Novikov
, and
M. A.
Shifman
,
Sov. Phys. Uspekhi
25
,
195
(
1982
);
also in
Instantons in Gauge Theories
, edited by
M.
Shifman
(
World Scientific
,
Singapore
,
1994
), p.
468
.
42.
R. P.
Feynman
and
A. R.
Hibbs
,
Quantum Mechanics and Path Integrals
(
McGraw-Hill
,
New York
,
1965
).
43.
G. V.
Mil’nikov
and
H.
Nakamura
,
J. Chem. Phys.
115
,
6881
(
2001
).
44.
H.
Nakamura
and
G.
Mil’nikov
,
Quantum Mechanical Tunneling in Chemical Physics
(
CRC Press
,
Boca Raton, FL
,
2013
).
45.
Z.
Smedarchina
,
W.
Siebrand
, and
A.
Fernández-Ramos
,
J. Chem. Phys.
137
,
224105
(
2012
).
46.
W.
Siebrand
,
Z.
Smedarchina
,
M. Z.
Zgierski
, and
A.
Fernández-Ramos
,
Int. Rev. Phys. Chem.
18
,
5
(
1999
).
47.
J. O.
Richardson
and
S. C.
Althorpe
,
J. Chem. Phys.
134
,
054109
(
2011
).
48.
H.
Kleinert
,
Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets
, 5th ed. (
World Scientific
,
Singapore
,
2009
).
49.
L. S.
Schulman
,
Techniques and Applications of Path Integration
(
Wiley
,
1981
).
50.
T.
Kawatsu
and
S.
Miura
,
J. Chem. Phys.
141
,
024101
(
2014
).
51.
W. H.
Miller
,
J. Chem. Phys.
62
,
1899
(
1975
).
52.
J. O.
Richardson
,
Int. Rev. Phys. Chem.
37
,
171
(
2018
).
53.
J. O.
Richardson
,
Faraday Discuss.
195
,
49
(
2016
).
54.
S. R.
McConnell
,
A.
Löhle
, and
J.
Kästner
,
J. Chem. Phys.
146
,
074105
(
2017
).
55.
J. B.
Rommel
and
J.
Kästner
,
J. Chem. Phys.
134
,
184107
(
2011
).
56.
A.
Löhle
and
J.
Kästner
,
J. Chem. Theory Comput.
14
,
5489
(
2018
).
57.
P.
Winter
and
J. O.
Richardson
,
J. Chem. Theory Comput.
15
,
2816
(
2019
).
58.
C.
Vaillant
and
M. T.
Cvitaš
,
Phys. Chem. Chem. Phys.
20
,
26809
(
2018
).
60.
D. C.
Liu
and
J.
Nocedal
,
Math. Program.
45
,
503
(
1989
).
61.
M. T.
Cvitaš
,
J. Chem. Theory Comput.
14
,
1487
(
2018
).
62.
S. K.
Burger
and
W.
Yang
,
J. Chem. Phys.
124
,
054109
(
2006
).
63.
M. T.
Cvitaš
and
J. O.
Richardson
, in
Molecular Spectroscopy and Quantum Dynamics
, edited by
R.
Marquardt
and
M.
Quack
(
Elsevier
,
2020
), Chap. 10.
64.
D. M.
Einarsdóttir
,
A.
Arnaldsson
,
F.
Óskarsson
, and
H.
Jónsson
, in
Applied Parallel and Scientific Computing: 10th International Conference, PARA 2010
, Lecture Notes in Computer Science Vol. 7134, edited by
K.
Jónasson
(
Springer-Verlag
,
Berlin
,
2012
), pp.
45
55
.
65.
H.
Goldstein
,
C.
Poole
, and
J.
Safko
,
Classical Mechanics
, 3rd ed. (
Addison-Wesley
,
San Francisco
,
2002
).
66.
P. R.
Bunker
and
P.
Jensen
,
Molecular Symmetry and Spectroscopy
, 2nd ed. (
NRC Research Press
,
Ottawa
,
2006
).
67.
W. H.
Press
,
S. A.
Teukolsky
,
W. T.
Vetterling
, and
B. P.
Flannery
,
Numerical Recipes: The Art of Scientific Computing
, 3rd ed. (
Cambridge University Press
,
Cambridge
,
2007
).
68.
T.
Baba
,
T.
Tanaka
,
I.
Morino
,
K. M. T.
Yamada
, and
K.
Tanaka
,
J. Chem. Phys.
110
,
4131
(
1999
).
69.
S. K.
Reddy
,
S. C.
Straight
,
P.
Bajaj
,
C.
Huy Pham
,
M.
Riera
,
D. R.
Moberg
,
M. A.
Morales
,
C.
Knight
,
A. W.
Götz
, and
F.
Paesani
,
J. Chem. Phys.
145
,
194504
(
2016
).
70.
U.
Góra
,
W.
Cencek
,
R.
Podeszwa
,
A.
van der Avoird
, and
K.
Szalewicz
,
J. Chem. Phys.
140
,
194101
(
2014
).
71.
Y.
Wang
,
X.
Huang
,
B. C.
Shepler
,
B. J.
Braams
, and
J. M.
Bowman
,
J. Chem. Phys.
134
,
094509
(
2011
).
72.
Y.
Wang
,
B. C.
Shepler
,
B. J.
Braams
, and
J. M.
Bowman
,
J. Chem. Phys.
131
,
054511
(
2009
).
73.
G. V.
Mil’nikov
and
H.
Nakamura
,
J. Chem. Phys.
122
,
124311
(
2005
).
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