Instanton-based rate theory is a powerful tool that is used to explore tunneling in many-dimensional systems. Yet, it diverges at the so-called “crossover temperature.” Using the uniform semiclassical transmission probability of Kemble [Phys. Rev. 48, 549 (1935)], we showed recently that in one dimension, one might derive a uniform semiclassical instanton rate theory, which has no divergence. In this paper, we generalize this uniform theory to many-dimensional systems. The resulting theory uses the same input as in the previous instanton theory, yet does not suffer from the divergence. The application of the uniform theory to dissipative systems is considered and used to revise Wolynes’ well-known analytical expression for the rate [P. G. Wolynes, Phys. Rev. Lett. 47, 968 (1981)] so that it does not diverge at the “crossover temperature.”

Quantum effects in reaction rate theory have been challenging. Two central effects are quantum tunneling and quantization of vibrotational levels. The central approximations used, which are also amenable to on-the-fly computations, are semiclassical theory,1–4 classical Wigner dynamics,5–7 centroid,8 ring polymer molecular dynamics (RPMD),9–11 and vibrational perturbation theory.12,13 Each one of these has advantages and disadvantages. Semiclassical theories, especially those based on initial value representations (IVRs) of the propagator, are derived from first principles, and, with perturbation theory, can lead to the exact result;14 however, the cost at this point is too high.15 The number of trajectories needed to converge such a theory is too large, due to the phase averaging involved. Classical Wigner dynamics, also suffers from a phase problem, computation of the numerically exact thermal flux operator in phase space demands averaging over phases. The time evolution in this theory is purely classical and so much less expensive than the semiclassical IVR method. Although it is the first term in a Wigner-Moyal expansion for the exact rate,6 the convergence is slow. The method correctly accounts for quantization; however, its accuracy deteriorates, especially at low temperatures and for asymmetric reactions.6 

Vibrational perturbation theory accounts to a certain extent for the quantization of anharmonically coupled degrees of freedom and is especially useful at relatively high temperatures, where it gives the exact tunneling factor in one dimension when expanding the rate to first order in 2.16 However, it does not give accurate deep tunneling rates, especially when considering asymmetric reactions.17 

The centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) approaches are less expensive to implement than IVR methods, and both include quantization of energy levels due to the numerically exact treatment of the thermal operator. In both methods, there are no problematic phases and this is perhaps the reason why especially the RPMD methodology has become so popular. Progress has been achieved in providing a formal basis to the method;18,19 however, implementation on-the-fly is expensive due to the number of trajectories needed for convergence.20 

A semiclassical-based theory, which is derived from first principles and does account for the quantization of energy levels in a harmonic sense, is the semiclassical instanton theory.21–28 Its main advantage is that only the periodic orbit on the upside-down multidimensional potential energy surface and its near vicinity, expressed in terms of the stability frequencies of the instanton orbit is needed. There is no averaging involved, and therefore, it is computationally inexpensive to implement it on-the-fly.26 The instanton is an unstable periodic orbit; however, numerical methods have been devised which make its computation rather straightforward.24,28,29 Currently, it is one of the central tools used to study tunneling in chemically reactive systems, using on-the-fly methodology.30,31

The instanton method, up until very recently, yet had one central drawback. It gave a good approximation for deep tunneling but extending it efficiently to high temperatures was challenging due to its divergence when ω/kBT=2π (ω is the barrier frequency and T is the temperature).23,32–35 Originally, two different formulae were used to derive the rate. Different prefactors were used when deriving the rate using the imaginary free energy (ImF) method36–38 above and below the “crossover” temperature.32 Methods have been devised to overcome this difficulty. A divergence occurs when the instanton energy is the barrier energy. Expanding the instanton action to second order about the barrier energy allows for a smooth transition,32,33 and the input required remains as before, the instanton action and its first and second derivatives. However, two different expressions are used for the two temperature regimes, and the formal result is not valid at lower temperatures, leading to difficulty in the transition to lower temperatures. A different approach is to use an RPMD-based computation at the relevant temperatures, as in Ref. 34, however, one then loses the main advantage of the instanton method which relies on a single trajectory. A more recent approach is to use a microcanonical theory and then perform the thermal average numerically27 but then one needs to compute the instanton over a range of energies and devise methods for obtaining the microcanonical rate at the above barrier energies.

This “crossover” temperature is not an unimportant limitation. Consider a barrier frequency of n · 100 cm−1 with n = 1, 2, 3, …. The corresponding “crossover” temperature is Tnn · 143 K, and above this temperature, the instanton as an object with a finite period on the upside-down potential energy surface ceases to exist. This implies that for many chemical reactions, even those with heavy atom tunneling, the instanton theory is not applicable at room temperature and one must resort to other methods.

We have shown in a recent paper,39 using Kemble’s40 uniform semiclassical expression for the energy-dependent transmission probability, that in one dimension, one might derive the instanton approximation using a uniform semiclassical theory, for which there is no divergence. When considering the dynamics above the barrier energy, as already mentioned, the instanton ceases to exist as a periodic orbit on the upside-down potential. However, with the uniform theory, this occurs at a much higher temperature. In the uniform theory, the instanton energy equals the barrier energy when ω/kBT=π so that Tnn · 286. The factor of two significantly extends the range of temperatures for which one can implement the instanton theory. Then, it is, of interest, to extend the uniform semiclassical rate theory to multidimensional systems. This is the central theme of this paper.

A class of multidimensional systems, which has served as a model for the influence of an environment on tunneling, is dissipative systems, in which the “system coordinate” is coupled bilinearly to a continuum of harmonic bath modes.41 In a classical paper, Caldeira and Leggett,42 using the ImF method, derived the instanton expression for the rate when T = 0,42 and this has been extended to somewhat higher temperatures in Refs. 43 and 44. The high-temperature limit, based on a dissipative parabolic barrier, was presented by Wolynes a few years earlier.45 However, both the low-temperature instanton expression and the high-temperature expression diverge when λ/kBT=2π (λ is known as the Kramers–Grote–Hynes renormalized barrier frequency46,47). Methods have been devised to overcome this divergence.32,33 However, when using the multidimensional uniform instanton method developed in this paper, this divergence naturally disappears, leading to an expression for the tunneling rate in dissipative systems which smoothly covers the whole temperature range.

Multidimensional uniform instanton semiclassical rate theory is developed in Sec. II. It is then applied to dissipative systems in Sec. III, paying special attention to the separable approximation which provides a natural extension of Wolynes’ expression to temperatures below λ/kBT=2π. We end with a discussion of the advantages, drawbacks, and future avenues of research of the multidimensional uniform semiclassical rate theory.

We consider a Hamiltonian with N + 1 degrees of freedom, where the N + 1th degree of freedom corresponds to the unstable mode of the multidimensional potential energy surface,
(2.1)
where q denotes the N + 1 dimensional vector of configuration modes q1, q2, …, qN.qF, and momenta and masses are correspondingly pj and Mj. We assume that the potential V has a barrier with height V, which separates between reactants and products. At the total energy H = E, we assume that there exists an orbit on the upside down potential energy surface—the instanton—with period τE=dSEdE and Euclidean action,
(2.2)
Based on Gutzwiller’s semiclassical analysis of Green’s function,48 Miller derived21 the following expression for the microcanonical cumulative transmission probability:
(2.3)
Here, ujE,j=1,,N are the stability parameters associated with the instanton orbit.
Then, Miller uses the expansion of the sinh function,
(2.4)
to rewrite the cumulative transmission probability as
(2.5)
where
(2.6)
so that one may explicitly evaluate the summation over the index k in Eq. (2.5) to find
(2.7)
The thermal semiclassical transmission probability at energy E is then
(2.8)
with
(2.9)
The stability parameters may always be rewritten as
(2.10)
and this defines the N stability frequencies ωjE. Following Miller, we use the resummation
(2.11)
and this would limit the energy from below such that
(2.12)
Changing the energy variable to
(2.13)
allows us to rewrite the transmission coefficient as
(2.14)
It is at this point that we introduce the uniform semiclassical instanton approximation. Defining
(2.15)
we define the steepest descent energy ɛβ such that
(2.16)
where τβεβ is the period of the instanton at the steepest descent energy ɛβ. The steepest descent approximation to Pnβ will then be
(2.17)
where Φ2εβ is the second derivative of the function Φε at the instanton energy ɛβ, and we assumed that the stability frequencies are slowly varying functions of the energy. Following the discussion of Ref. 39, we correct for the high-energy exponential decay of the integrand in Eq. (2.8) to rewrite
(2.18)
Inserting this result into Eq. (2.8), and performing the summation over the indices n as in Eq. (2.4) gives
(2.19)
and this is the central result of this paper.

As in the one-dimensional uniform semiclassical instanton rate theory, the important difference between this expression and the “standard” multidimensional instanton expression comes from the steepest descent value of the energy as determined by Eq. (2.16), and the fact that the second derivative Φ2εβ does not vanish even when the second derivative of the action does vanish. As a result, the thermal transmission probability Pβ, as given in Eq. (2.19), is well defined and does not lead to any divergences. The action Sεβ will vanish when ɛβ = V, and this implies from Eq. (2.16) that at this steepest descent energy ℏβω = π. If one defines a crossover temperature between tunneling and thermal activation as the temperature at which ɛβ = V, then, in the uniform theory, this crossover occurs at a temperature that is twice that of the “primitive crossover temperature” defined by ℏβω = 2π.

From a practical point of view, the computation of the probability as in Eq. (2.19) is not more expensive than in the “primitive” instanton theory, since all that is needed is the instanton action, period, and stability frequencies at a given total energy E, and these are the same as before, only the relation between the instanton energy and the temperature, as determined from the steepest descent condition [Eq. (2.16)], is different. In the “old” instanton theory, the steepest descent condition implied that at a given (inverse) temperature β, the period of the instanton is known, it is ℏβ. In the uniform instanton theory, this simple relation no longer holds. To know the period, one must also know the instanton action, and, in principle, if one wants to determine the rate at (inverse) temperature β, one would have to compute the instanton over a range of energies. This may still be advantageous as compared to the microcanonical theory of Ref. 27 for which one needs to establish the microcanonical transmission probability also above the barrier energy. The uniform instanton method is local, and therefore, one does not need to determine the above barrier transmission probability.

Second, the number of instanton energies needed will typically be limited. The period of the instanton is a smooth function of the energy. A practical algorithm would then be to find an instanton with some energy and, thus, some period, and this will correspond to some (inverse) temperature β*. Then, from knowledge of the first and second derivatives of the action with respect to the energy, one may use the steepest descent condition [Eq. (2.16)] and a linear expansion in the energy,
(2.20)
to determine approximately the relation of the instanton energy to the temperature in the vicinity of β*. (The prime denotes the derivative with respect to the argument). With a reasonable initial guess, one would not need more than a handful of instanton computations (without the stability frequencies) to obtain the instanton related to the desired temperature.
The classical Hamiltonian equivalent that leads in the continuum limit to the classical Generalized Langevin Equation (GLE), whereby a particle with mass M moves along a system coordinate q with momentum pq under the influence of a potential Vq, whose motion is coupled bilinearly to a harmonic bath is
(3.1)
where xj and pxj are the mass-weighted coordinate and momentum of the jth bath oscillator, respectively, whose frequency is ωj, which is coupled to the system via the coupling constant cj. It is well known49 that one may recast the equation of motion for the system as a GLE,
(3.2)
with the identification of the friction function,
(3.3)
and the Gaussian random force Ft, whose mean vanishes and whose correlation function is proportional to the friction function (fluctuation–dissipation relation41).

In principle, it is straightforward to apply the uniform instanton theory by discretization of the bath. Given a continuum form for the friction function, following Ref. 50, one may discretize it for N bath degrees of freedom to obtain the relevant N coupling coefficients cj and frequencies ωj. Then, one undertakes a numerical search for an instanton trajectory at some energy E, for example, by following the algorithms of Refs. 24 and 29. Having located the instanton at a given energy, one finds the associated temperature through the steepest descent relation Eq. (2.16). Afterward, one computes the stability frequencies as described, for example, in Ref. 51. This process may then be repeated with a larger number of bath modes until one converges to the required accuracy. Finally, the process is repeated for a range of energies, leading to the desired range of temperatures.

1. The dissipative parabolic barrier limit

Although a general analytic expression for the quantum rate in dissipative systems is not available, it is known in the high-temperature limit where a parabolic barrier potential is a good approximation. Specifically, if the potential is
(3.4)
then the dissipative Hamiltonian of Eq. (3.1) has a quadratic form so that it may be diagonalized using a normal mode transformation,53,54
(3.5)
where pρ and ρ denote the momentum and coordinate, respectively, of the unstable normal mode, pyj and yj denote the jth stable mass-weighted normal mode momentum and coordinate, respectively, and λj is the associated normal mode frequency. The frequency λ is known as the Kramers–Grote–Hynes46,47 barrier frequency, and it is related to the Laplace transform of the time-dependent friction [denoted as γ̂s] through the relation,
(3.6)
In this separable parabolic barrier model, the instanton action is
(3.7)
and the thermal transmission probability is exactly given by (no need for a steepest descent estimate)
(3.8)
Then, the rate expression is obtained by making a harmonic approximation for the reactant state. Specifically, the potential Vq is assumed to be harmonic around q0 with frequency ω0 and parabolic around q = 0. The Hamiltonian in the region of the well is quadratic and so may also be diagonalized to give the N + 1 stable frequencies λj, j = 0, …, N. Then, the rate Γβ is given by the transition state theory expression and, after some manipulation, is shown to be52 
(3.9)
where Ξw is the “Wolynes factor,”
(3.10)
This factor diverges when ℏβλ = 2π since it is based on a purely parabolic barrier transmission probability.

2. Overcoming the divergence

Comparison of Eqs. (2.19) and (3.8) points to one way of extending the Wolynes factor to lower temperatures without any divergence. Using normal modes and following the perturbation theory analysis of Ref. 55, one may write down a leading order separable approximation for the dissipative Hamiltonian as
(3.11)
With this approximation, one may rewrite the Wolynes factor as
(3.12)
with
(3.13)
The action SE is obtained from the separable Hamiltonian for the unstable mode motion,
(3.14)
The remaining task is to rewrite the revised Wolynes factor in the continuum limit, as a function of the time-dependent friction, rather than as a discretized expression, using a finite number of degrees of freedom. For this purpose, we note that the continuum limit expression for the Wolynes factor is52 
(3.15)
where νk=2πkβ is the kth Matsubara frequency. Dividing by βλ2/sinβλ2, using the identity
(3.16)
multiplying by Iβ, and using the Kramers–Grote–Hynes relation [Eq. (3.6)], one readily finds that
(3.17)
and there is no divergence.
There are previous attempts to bridge the divergence of the Wolynes factor at ℏβλ = 2π. Some of them32,33 are based on assuming that the energy action relation is quadratic in the vicinity of the temperature ℏβλ = 2π,
(3.18)
where S2 is the absolute value of the second derivative of the action at the barrier energy (ɛ = V). Typically, this second derivative may be rather small so that the resulting estimate for the rate which goes as 1/S2 will be finite but large. In the uniform theory presented, the action is determined by the potential VλMωρ, there is no need for a quadratic expansion, and the estimate for the rate is quite reasonable also at the barrier energy.

The central theme of this paper is a generalization of the uniform instanton theory presented in Ref. 39 to more complex many-dimensional systems. The resulting expression [Eq. (2.19)] is a straightforward generalization of the “standard” multidimensional instanton expression with the fundamental change that it is valid in principle for any temperature and does not exhibit any divergence. The main advantage of the thermal instanton theory is that for a given temperature, one needs to compute only a single trajectory, and methods have been developed for doing this on-the-fly.

From a practical point of view, the instanton is a periodic orbit on the upside-down potential, which exists for all energies ranging from the threshold to the barrier height, leading to an instanton rate theory that is valid for the temperature range spanning 0 to the temperature at which ℏβω = π. Above this temperature, the instanton is a periodic orbit bouncing between two imaginary turning points. For one-dimensional systems, it is straightforward to obtain the instanton action at any energy so that the rate is well defined at any temperature. In multidimensional systems, this becomes a rather complex process, which, at present, is not amenable to on-the-fly computations. Richardson has suggested ways of overcoming the problem by using a harmonic approximation for energies above the barrier.27,56 Although reasonable, it would be much more satisfactory if one could get the correct “action” on-the-fly also for energies above the barrier energy.

Another issue has to do with the microcanonical rate. The present derivation of the uniform thermal instanton rate starts from Eq. (2.3). The steepest descent integral is performed by assuming that the stability frequencies are slowly varying with energy. This is a good assumption, especially in the high-temperature region 0 ≤ ℏβω ≤ ∼2π, but not necessarily for lower temperature, as discussed in Ref. 56. One may somewhat alleviate the difficulty by expanding the stability frequency about the steepest descent energy and then carrying out the Gaussian integration over the expansion terms. However, this implies computing the second-order energy derivatives of the stability frequencies, which numerically would be quite demanding, especially for on-the-fly computations. In addition, as discussed at length in Ref. 56, even though the thermal instanton approximation turns out to be rather accurate, there are difficulties with the microcanonical transmission probability, when using the resummation approximation as in Eq. (2.11).

The multidimensional uniform instanton method is, in principle, readily applicable to dissipative systems, as discussed in Sec. III. Furthermore, as noted in Ref. 41, following Ref. 57, for dissipative systems with bilinear system bath coupling one may replace the prefactor of the product of stability frequencies with the eigenvalue spectrum of the instanton periodic orbit. However, from a practical point of view, this leads numerically to a truncation and a finite number of eigenvalues, so it will not necessarily lead to cheaper numerical algorithms. It is tempting to adapt the vibrational perturbation theory—VPT2 to dissipative systems, since this is not a linear theory and so may shed light on the importance of nonlinearities beyond the present steepest descent approximation involving stability frequencies. However, one should keep in mind that the VPT2 theory is best suited for symmetric exchange reactions and temperatures such that 0 ≤ ℏβω ≤ ∼2π. It is known that it is not very accurate for much lower temperatures. However, as discussed, it is especially in this high-temperature region that one has a problem due to the analytic continuation of the action for above barrier energies, which is overcome analytically within the VPT2 theory and, thus, may be complementary to the low-temperature instanton results.

In summary, the uniform semiclassical instanton rate theory can be generalized to multidimensional systems, however, this is not yet the last word. Theoretically, there are many remaining questions, and benchmark computations are needed to further understand the range of validity of the theory.

This work is dedicated to Professor Philip Pechukas who was my teacher, mentor, and lifelong friend, and always insisted “first look at the simplest case.” This work was generously funded by the Israel Science Foundation.

The author has no conflicts to disclose.

Eli Pollak: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available within the article.

1.
W. H.
Miller
,
J. Phys. Chem. A
105
,
2942
(
2001
).
2.
J.
Tatchen
and
E.
Pollak
,
J. Chem. Phys.
130
,
041103
(
2009
).
3.
M.
Ceotto
,
S.
Atahan
,
S.
Shim
,
G. F.
Tantardini
, and
A.
Aspuru-Guzik
,
Phys. Chem. Chem. Phys.
11
,
3861
3867
(
2009
).
4.
M.
Wehrle
,
M.
Sulc
, and
J.
Vanicek
,
J. Chem. Phys.
140
,
244114
(
2014
).
5.
E. J.
Heller
J. Chem. Phys.
62
,
1544
(
1975
).
6.
J.
Shao
,
J.-L.
Liao
, and
E.
Pollak
,
J. Chem. Phys.
108
,
9711
(
1998
).
7.
J.
Liu
,
Int. J. Quantum Chem.
115
,
657
670
(
2015
).
8.
J.
Cao
and
G. A.
Voth
,
J. Chem. Phys.
100
,
5106
5117
(
1994
).
9.
S.
Habershon
,
D. E.
Manolopoulos
,
T. E.
Markland
, and
T. F.
Miller
III
,
Annu. Rev. Phys. Chem.
64
,
387
413
(
2013
).
10.
Y. V.
Suleimanov
,
F. J.
Aoiz
, and
H.
Guo
,
J. Phys. Chem. A
120
(
43
),
8488
8502
(
2016
).
11.
J. E.
Lawrence
and
D. E.
Manolopoulos
,
Faraday Discuss.
221
,
9
(
2020
).
12.
W. H.
Miller
,
R.
Hernandez
,
N. C.
Handy
,
D.
Jayatilaka
, and
A.
Willetts
,
Chem. Phys. Lett.
172
,
62
68
(
1990
).
13.
T. L.
Nguyen
,
J. F.
Stanton
, and
J. R.
Barker
,
J. Phys. Chem. A
115
,
5118
5126
(
2011
).
14.
S.
Zhang
and
E.
Pollak
,
Phys. Rev. Lett.
91
,
190201
(
2003
).
15.
E.
Martin-Fierro
and
E.
Pollak
,
J. Chem. Phys.
126
,
164108
(
2007
).
16.
E.
Pollak
and
J.
Cao
,
J. Chem. Phys.
157
,
074109
(
2022
).
17.
P.
Goel
and
J. F.
Stanton
,
J. Chem. Phys.
149
,
134109
(
2018
).
18.
T. J. H.
Hele
and
S. C.
Althorpe
,
J. Chem. Phys.
138
,
084108
(
2013
).
19.
S. C.
Althorpe
and
T. J. H.
Hele
,
J. Chem. Phys.
139
,
084115
(
2013
).
20.
Y.
Takahashi
,
Y.
Hashimoto
,
K.
Saito
, and
T.
Takayanagi
,
Molecules
26
,
7250
(
2021
).
21.
W. H.
Miller
,
J. Chem. Phys.
62
,
1899
1906
(
1975
).
22.
S.
Coleman
,
Phys. Rev. D
15
,
2929
2936
(
1977
).
23.
M.
Kryvohuz
,
J. Chem. Phys.
134
,
114103
(
2011
).
24.
M.
Kryvohuz
,
J. Chem. Phys.
137
,
234304
(
2012
).
25.
J.
Meisner
and
J.
Kästner
,
Angew. Chem., Int. Ed.
55
,
5400
5413
(
2016
).
26.
A. N.
Beyer
,
J. O.
Richardson
,
P. J.
Knowles
,
J.
Rommel
, and
S. C.
Althorpe
,
J. Phys. Chem. Lett.
7
,
4374
4379
(
2016
).
27.
J. O.
Richardson
,
Faraday Discuss.
195
,
49
(
2016
).
28.
J. O.
Richardson
,
Int. Rev. Phys. Chem.
37
,
171
216
(
2018
).
29.
J. B.
Rommel
,
T. P. M.
Goumans
, and
J.
Kästner
,
J. Chem. Theory Comput.
7
,
690
698
(
2011
).
30.
T.
Lamberts
,
P. K.
Samanta
,
A.
Köhn
, and
J.
Kästner
,
Phys. Chem. Chem. Phys.
18
,
33021
33030
(
2016
).
31.
E.
Han
,
W.
Fang
,
M.
Stamatakis
,
J. O.
Richardson
, and
J.
Chen
,
J. Phys. Chem. Lett.
13
,
3173
3181
(
2022
).
32.
I.
Affleck
,
Phys. Rev. Lett.
46
,
388
391
(
1981
).
33.
P.
Hänggi
and
W.
Hontscha
,
J. Chem. Phys.
88
,
4094
(
1988
).
34.
Y.
Zhang
,
J. B.
Rommel
,
M. T.
Cvitaš
, and
S. C.
Althorpe
,
Phys. Chem. Chem. Phys.
16
,
24292
24300
(
2014
).
35.
S. R.
McConnell
and
J.
Kästner
,
J. Comput. Chem.
40
,
866
874
(
2019
).
38.
C. G.
Callan
and
S.
Coleman
,
Phys. Rev. D
16
,
1762
(
1977
).
39.
S.
Upadhyayula
and
E.
Pollak
,
J. Phys. Chem. Lett.
14
,
9892
(
2023
).
41.
P.
Hänggi
,
P.
Talkner
, and
M.
Borkovec
,
Rev. Mod. Phys.
62
,
251
341
(
1990
).
42.
A. O.
Caldeira
and
A. J.
Leggett
,
Ann. Phys.
149
,
374
(
1983
).
43.
A. I.
Larkin
and
Yu. N.
Ovchinnikov
,
Sov. Phys. JETP
59
,
420
(
1984
).
44.
H.
Grabert
and
U.
Weiss
,
Phys. Rev. Lett.
53
,
1787
(
1984
).
45.
P. G.
Wolynes
,
Phys. Rev. Lett.
47
,
968
(
1981
).
47.
R. F.
Grote
and
J. T.
Hynes
,
J. Chem. Phys.
73
,
2715
(
1980
).
48.
M. C.
Gutzwiller
,
J. Math. Phys.
12
,
343
(
1971
).
49.
50.
H.
Wang
,
M.
Thoss
, and
W. H.
Miller
,
J. Chem. Phys.
115
,
2979
(
2001
).
51.
S. R.
McConnell
,
A.
Löhle
, and
J.
Kästner
,
J. Chem. Phys.
146
,
074105
(
2017
).
53.
E.
Pollak
,
J. Chem. Phys.
85
,
865
(
1986
).
54.
J. L.
Liao
and
E.
Pollak
,
Chem. Phys.
268
,
295
313
(
2001
).
55.
R.
Ianconescu
and
E.
Pollak
,
J. Chem. Phys.
151
,
024703
(
2019
).
56.
J. E.
Lawrence
and
J. O.
Richardson
,
Faraday Discuss.
238
,
204
235
(
2022
).
57.
R. F.
Dashen
,
B.
Hasslacher
, and
A.
Neveu
,
Phys. Rev. D
10
,
4114
(
1974
).