A unified framework for semiclassical reaction rate theory

TL;DR

The paper introduces a unified semiclassical framework that connects instanton theory and SCTST through a conjectured exact relation between the cumulative reaction probability and a generalized instanton action, allowing all-orders in $\hbar$ treatment. By expressing the effective action $\tilde{W}_{\mathbf{n}}(E;\hbar)$ in terms of instanton data and exact-WKB theory, it recovers standard instanton theory, yields VPT2-SCTST connections, and provides systematic pathways to higher-order corrections and multidimensional anharmonic effects. The approach yields explicit expressions for thermal rates both below and above the crossover temperature (instanton and sphaleron regimes) and introduces anharmonic Wigner-like corrections, all within a transparent semiclassical language. These results pave the way for rigorously improvable semiclassical rate methods and highlight connections to ring-polymer instanton and microcanonical approaches, with potential extensions to rotations, nonadiabaticity, and full-dimensional simulations.

Abstract

A general semiclassical theory for the calculation of reaction rate constants is developed. The theory can be understood as a formal framework that encompasses existing semiclassical methods: instanton theory and semiclassical transition state theory (SCTST). Unlike SCTST, the present formalism does not start from the concept of "good" action-angle variables. Instead, it is based on a conjectured connection between the cumulative reaction probability and the instanton contribution to the formally exact generalisation of Gutzwiller's formula for the trace of the Green's function. The formalism effectively generalises the "imaginary free-energy" formulation of instanton theory to microcanonical scattering rates and all orders in $\hbar$. In one dimension, explicit expressions are derived for the generalised reduced action up to $O(\hbar^4)$ using exact WKB/quantum Hamilton-Jacobi theory. The connection between the present formalism and the standard second order vibrational perturbation theory (VPT2) version of SCTST is explored. It is also shown that the standard thermal instanton rate theory, as well as higher order (dividing surface independent) "perturbative" corrections can be straightforwardly derived from the framework. Above the crossover temperature, first-order corrections in $\hbar$ to the parabolic barrier ("sphaleron") rate are also derived. A simple anharmonic transition state theory and anharmonic version of the Wigner tunneling correction are presented. Finally, the potential for the development of new and improved semiclassical methods for modelling reaction kinetics is discussed.

Figures (1)

• Figure 1: Graphical confirmation that $\lim_{\hbar\to0}[\tilde{W}(E;\hbar)-W_0(E)]/\hbar^2=W_2(E)$ for the Gaussian barrier defined by Eq. (\ref{['eq:Gaussian_potential']}). The plotted ratios were calculated on a grid between $\hbar=0.1$ and $\hbar=1.0$, and cubic splines are used to guide the eye.