We present accurate fully quantum calculations of thermal rate constants for a symmetric double well system coupled to a dissipative bath. The calculations are performed using the quasiadiabatic propagator path integral (QUAPI) methodology to evaluate the flux–flux correlation function whose time integral determines the rate coefficient. The discretized path integral converges very rapidly in the QUAPI representation, allowing efficient calculation of quantum correlation functions for sufficiently long times. No ad hoc assumption is introduced and thus these calculations yield the true quantum mechanical rate constants. The results presented in the paper demonstrate the applicability of the QUAPI methodology to practically all regimes of chemical interest, from thermal activation to deep tunneling, and the quantum transmission factor exhibits a Kramers turnover. Our calculations reveal an unusual step structure of the integrated reactive flux in the weak friction regime as well as quantum dynamical enhancement of the rate above the quantum transition state theory value at low temperatures, which is largely due to vibrational coherence effects. The quantum rates are compared to those obtained from classical trajectory simulations. We also use the numerically exact classical and quantum results to establish the degree of accuracy of several analytic and numerical approximations, including classical and quantum Grote–Hynes theories, semiclassical transition state theory (periodic orbit) estimates, classical and quantum turnover theories, and the centroid density approximation.
REFERENCES
1.
For a recent review of reaction rate theories see
P.
Hänggi
, P.
Talkner
, and M.
Borcovec
, Rev. Mod. Phys.
62
, 251
(1990
).2.
3.
4.
Various aspects of transition state theory have been reviewed in the following articles:
W. H.
Miller
, Acc. Chem. Res.
9
, 306
(1976
);R. B.
Walker
and J. C.
Light
, Annu. Rev. Phys. Chem.
31
, 401
(1980
); , Annu. Rev. Phys. Chem.
E. Pollak, in Theory of Chemical Reaction Dynamics, edited by M. Baer (CRC, Boca Raton, 1985), Vol. 3, p. 123.
5.
6.
A. O.
Caldeira
and A. J.
Leggett
, Ann. Phys. (N.Y.)
149
, 374
(1983
).7.
8.
9.
10.
11.
12.
13.
14.
B. J.
Gertner
, K. R.
Wilson
, and J. T.
Hynes
, J. Chem. Phys.
90
, 3537
(1993
).15.
B. J. Berne, in Activated Barrier Crossing, edited by G. R. Fleming and P. Hänggi (World Scientific, Singapore, 1993), p. 82.
16.
17.
E.
Pollak
, H.
Grabert
, and P.
Hänggi
, J. Chem. Phys.
91
, 4073
(1989
).18.
U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993).
19.
20.
21.
V. A. Benderskii, D. E. Makarov, and C. A. Wight, Chemical Dynamics at Low Temperatures, Adv. Chem. Phys. LXXXVIII (1994).
22.
H.
Grabert
, U.
Weiss
, and P.
Hänggi
, Phys. Rev. Lett.
52
, 2193
(1984
).23.
24.
25.
26.
P. S.
Riseborough
, P.
Hänggi
, and E.
Freidkin
, Phys. Rev. A
32
, 489
(1985
).27.
28.
29.
30.
C. H. Bennett, in Algorithms for Chemical Computation, ACS Symposium Series No. 46, edited by R. E. Christoffersen (Academic Chem. Soc., Washington, DC, 1977), p. 63.
31.
32.
J. A.
Montgomery
, D.
Chandler
, and B. J.
Berne
, J. Chem. Phys.
70
, 4056
(1979
).33.
B. J. Berne, in Multiple Time Scales, edited by J. U. Brackbill and B. I. Cohen (Academic, New York, 1985), p. 419.
34.
35.
W. H.
Miller
, S. D.
Schwartz
, and J. W.
Tromp
, J. Chem. Phys.
79
, 4889
(1983
).36.
37.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
38.
R. P. Feynman, Statistical Mechanics (Addison-Wesley, New York, 1972).
39.
L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
40.
N.
Metropolis
, A. W.
Rosenbluth
, M. N.
Rosenbluth
, H.
Teller
, and E.
Teller
, J. Chem. Phys.
21
, 1087
(1953
).41.
J. P. Valleau and S. G. Whittington, in Modern Theoretical Chemistry, edited by B. J. Berne (Plenum, New York, 1977), Vol. 5, pp. 137–168.
42.
K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics (Springer-Verlag, New York, 1988).
43.
For a review describing the use of quantum Monte Carlo methods in chemical physics see
B. J.
Berne
and D.
Thirumalai
, Annu. Rev. Phys. Chem.
37
, 401
(1986
).44.
For reviews of equilibrium path integral techniques see
B. J.
Berne
and D.
Thirumalai
, Annu. Rev. Phys. Chem.
37
, 401
(1987
);Quantum Simulations of Condensed Matter Phenomena, edited by J. D. Doll and J. E. Gubernatis (World Scientific, Singapore, 1990).
45.
46.
E. C.
Behrman
, G. A.
Jongeward
, and P. G.
Wolynes
, J. Chem. Phys.
79
, 6277
(1983
);E. C.
Behrman
and P. G.
Wolynes
, J. Chem. Phys.
83
, 5863
(1985
); , J. Chem. Phys.
R. E.
Cline
,Jr. and P. G.
Wolynes
, J. Chem. Phys.
88
, 4334
(1988
); , J. Chem. Phys.
B. A.
Mason
, K.
Hess
, R. E.
Cline
, and P. G.
Wolynes
, Superlatt. Microstruct.
3
, 421
(1987
).47.
J. D.
Doll
, R. D.
Coalson
, and D. L.
Freeman
, J. Chem. Phys.
87
, 1641
(1987
).48.
49.
50.
51.
J. D.
Doll
, D. L.
Freeman
, and M. J.
Gillan
, Chem. Phys. Lett.
143
, 277
(1988
);J. D.
Doll
, T. L.
Beck
, and D. L.
Freeman
, J. Chem. Phys.
89
, 5753
(1988
);T. L.
Beck
, J. D.
Doll
, and D. L.
Freeman
, J. Chem. Phys.
90
, 3181
(1989
)., J. Chem. Phys.
52.
53.
J. D.
Doll
, D. L.
Freeman
, and T. L.
Beck
, Adv. Chem. Phys.
78
, 61
(1990
).54.
55.
56.
O. A.
Sharafeddin
, D. J.
Kouri
, N.
Nayar
, and D.
Hoffman
, J. Chem. Phys.
95
, 3224
(1991
);D. K.
Hoffman
, N.
Nayar
, O. A.
Sharafeddin
, and D. J.
Kouri
, J. Phys. Chem.
95
, 8299
(1991
);D. K.
Hoffman
and D. J.
Kouri
, J. Phys. Chem.
96
, 1179
(1992
)., J. Phys. Chem.
57.
58.
59.
60.
M.
Winterstetter
and W.
Domcke
, Phys. Rev. A
47
, 2838
(1993
);S.
Krempl
, M.
Winterstetter
, H.
Plohn
, and W.
Domcke
, J. Chem. Phys.
100
, 926
(1994
).61.
N. Makri, in Time-Dependent Quantum Molecular Dynamics, edited by J. Broeckhove and L. Lathouwers (Plenum, New York, 1992).
62.
63.
64.
65.
66.
J. Chem. Phys. (submitted).
67.
D. Ilk and N. Makri, J. Chem. Phys, (in press).
68.
S.
Champan
, B. C.
Garrett
, and W. H.
Miller
, J. Chem. Phys.
63
, 2710
(1975
)., J. Chem. Phys.
69.
70.
B. C.
Garrett
, D. G.
Truhlar
, R. S.
Grev
, and A. W.
Magnuson
, J. Phys. Chem.
84
, 1730
(1980
).71.
72.
73.
74.
N. F. Hansen and H. C. Andersen (unpublished).
75.
G. A.
Voth
, D.
Chandler
, and W. H.
Miller
, J. Chem. Phys.
91
, 7749
(1989
).76.
77.
78.
79.
J. B.
Straus
, J. M. G.
Liorente
, and G. A.
Voth
, J. Chem. Phys.
98
, 4082
(1993
); , J. Chem. Phys.
G. A.
Voth
and E. V.
O’Gorman
, J. Chem. Phys.
94
, 7342
(1991
); , J. Chem. Phys.
80.
M.
Messina
, G. K.
Schenter
, and B. C.
Garrett
, J. Chem. Phys.
98
, 8525
(1993
).81.
K. S.
Schweitzer
, R. M.
Stratt
, D.
Chandler
, and P. G.
Wolynes
, J. Chem. Phys.
75
, 1347
(1981
).82.
83.
84.
85.
R. A.
Chiles
, G. A.
Jongeward
, M. A.
Bolton
, and P. G.
Wolynes
, J. Chem. Phys.
81
, 2039
(1984
).86.
D. O.
Harris
, G. G.
Engerholm
, and W. D.
Gwinn
, J. Chem. Phys.
43
, 1515
(1965
).87.
88.
J. V.
Lill
, G. A.
Parker
, and J. C.
Light
, Chem. Phys. Lett.
89
, 483
(1982
);J. C.
Light
, I. P.
Hamilton
, and J. V.
Lill
, J. Chem. Phys.
82
, 1400
(1985
);J. V.
Lill
, G. A.
Parker
, and J. C.
Light
, J. Chem. Phys.
85
, 900
(1986
)., J. Chem. Phys.
89.
Z.
Bacic
and J. C.
Light
, J. Chem. Phys.
85
, 4594
(1986
); , J. Chem. Phys.
90.
S. M.
Auerbach
and W. H.
Miller
, J. Chem. Phys.
98
, 6917
(1993
)., J. Chem. Phys.
91.
J. A.
Bentley
, R. E.
Wyatt
, M.
Menou
, and C.
Leforestier
, J. Chem. Phys.
97
, 4255
(1992
)., J. Chem. Phys.
92.
C. J.
Williams
, J.
Qian
, and D. J.
Tannor
, J. Chem. Phys.
95
, 1721
(1991
)., J. Chem. Phys.
93.
94.
95.
R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics (Springer-Verlag, New York, 1985).
96.
D. Chandler, Introduction to Modern Statistical Mechanics (Oxford University Press, New York, 1987).
97.
A. J.
Leggett
, S.
Chakravarty
, A. T.
Dorsey
, M. P.
Fisher
, A.
Garg
, and W.
Zwerger
, Rev. Mod. Phys.
59
, 1
(1987
).98.
R.
Jaquet
and W. H.
Miller
, J. Chem. Phys.
89
, 2139
(1985
); , J. Chem. Phys.
J. W.
Tromp
and W. H.
Miller
, Faraday Discuss. Chem. Soc.
84
, 441
(1987
).99.
G.
Wahnström
, B.
Carmelli
, and H.
Metiu
, J. Chem. Phys.
88
, 2478
(1988
);100.
D.
Brown
and J. C.
Light
, J. Chem. Phys.
97
, 5465
(1992
)., J. Chem. Phys.
101.
102.
103.
104.
M.
Thachuk
, H. R.
Mayne
, and T. C.
Schatz
, J. Chem. Phys.
99
, 3516
(1993
)., J. Chem. Phys.
105.
R. P.
McRae
, G. K.
Schenter
, B. C.
Garrett
, G. R.
Haynes
, G. A.
Voth
, and G. C.
Schatz
, J. Chem. Phys.
97
, 7392
(1992
).106.
T. N.
Truong
, J. A.
McCammon
, D. J.
Kouri
, and D. K.
Hoffman
, J. Chem. Phys.
96
, 8136
(1992
).107.
R.
Lefebvre
, V.
Ryaboy
, and N.
Moiseyev
, J. Chem. Phys.
98
, 8601
(1993
);V.
Ryaboy
and R.
Lefebvre
, J. Chem. Phys.
99
, 9547
(1993
)., J. Chem. Phys.
108.
D. Secrest, in Molecular Collision Theory: A Guide for the Experimentalists, edited by R. B. Bernstein (Plenum, New York, 1979).
109.
110.
111.
G. A.
Parker
, T. G.
Schmalz
, and J. C.
Light
, J. Chem. Phys.
73
, 1757
(1980
).112.
A.
Kuppermann
, G. C.
Schatz
, and M.
Baer
, J. Chem. Phys.
65
, 4596
(1976
);G. C.
Schatz
and A.
Kuppermann
, J. Chem. Phys.
65
, 4642
(1976
)., J. Chem. Phys.
113.
114.
115.
116.
343
(1959
)., Ann. Phys.
117.
R.
Silbey
and R. W.
Munn
, J. Chem. Phys.
72
, 2763
(1980
); , J. Chem. Phys.
B.
Jackson
and R.
Silbey
, J. Chem. Phys.
78
, 4193
(1983
)., J. Chem. Phys.
118.
119.
120.
121.
122.
D.
Thirumalai
, E. J.
Bruskin
, and B. J.
Berne
, J. Chem. Phys.
79
, 5063
(1983
);D.
Thirumalai
and B. J.
Berne
, J. Chem. Phys.
81
, 2512
(1984
); , J. Chem. Phys.
123.
124.
R. A.
Harris
and R.
Silbey
, J. Chem. Phys.
83
, 1069
(1985
); , J. Chem. Phys.
R.
Silbey
and R. A.
Harris
, J. Chem. Phys.
93
, 7062
(1989
)., J. Chem. Phys.
125.
126.
127.
128.
129.
130.
131.
A. M.
Berezhkovskii
, E.
Pollak
, and V. Y.
Zitserman
, J. Chem. Phys.
97
, 2422
(1992
).132.
A. M.
Levine
, W.
Hontscha
, and E.
Pollak
, Phys. Rev. B
40
, 2138
(1989
).133.
134.
A. M.
Levine
, M.
Shapiro
, and E.
Pollak
, J. Chem. Phys. B
88
, 1959
(1988
).135.
136.
137.
P.
Hanggi
, H.
Grabert
, G. L.
Ingold
, and U.
Weiss
, Phys. Rev. Lett.
55
, 761
(1985
).138.
The estimate of the crossover temperature in the absence of dissipation was first obtained by
V. I.
Goldanskii
, Dokl. Akad. Nauk SSSR
124
, 1261
(1959
);139.
140.
141.
C.
Zheng
, J. A.
McCammon
, and P. G.
Wolynes
, Chem. Phys.
158
, 261
(1991
).142.
143.
144.
145.
146.
Yu. Georgievskii and E. Pollak (unpublished).
147.
See the discussion preceding Eq. (9.55) in Ref. 1. See also P. Hanggi, E. Pollak, and H. Grabert, The Quantum Kramers Problem (University of Augsburg, FRG, 1989), Report No. 215.
148.
149.
150.
151.
P.
Hanggi
and W.
Hontcha
, Ber. Bunsenges. Phys. Chem.
95
, 379
(1991
).152.
153.
H.
Grabert
, P.
Olschowski
, and U.
Weiss
, Phys. Rev. B
36
, 1931
(1987
).
This content is only available via PDF.
© 1994 American Institute of Physics.
1994
American Institute of Physics
You do not currently have access to this content.
