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.”
I. INTRODUCTION
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 ℏ2.16 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‡ 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 Tn ≃ n · 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 so that Tn ≃ n · 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‡ 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 . We end with a discussion of the advantages, drawbacks, and future avenues of research of the multidimensional uniform semiclassical rate theory.
II. MULTIDIMENSIONAL UNIFORM INSTANTON SEMICLASSICAL RATE THEORY EXPRESSION
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 does not vanish even when the second derivative of the action does vanish. As a result, the thermal transmission probability , as given in Eq. (2.19), is well defined and does not lead to any divergences. The action 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.
III. DISSIPATIVE SYSTEMS
A. Numerical solution in the general case
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.
B. Extension of the high-temperature rate expression
1. The dissipative parabolic barrier limit
2. Overcoming the divergence
IV. DISCUSSION
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.
ACKNOWLEDGMENTS
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.
AUTHOR DECLARATIONS
Conflict of Interest
The author has no conflicts to disclose.
Author Contributions
Eli Pollak: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Writing – original draft (lead); Writing – review & editing (lead).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.