Semiclassical instanton theory for reaction rates at any temperature: How a rigorous real-time derivation solves the crossover temperature problem

TL;DR

This paper develops a rigorous semiclassical instanton theory valid at all temperatures, bridging the gap at the crossover temperature and providing a smooth, accurate method for calculating reaction rates involving tunneling.

Contribution

It introduces a real-time derived, temperature-independent instanton theory that extends standard methods to high temperatures and incorporates multi-orbit corrections.

Findings

Accurately models reaction rates across all temperatures.

Shows excellent numerical agreement with model systems.

Provides a framework for future theory extensions.

Abstract

Instanton theory relates the rate constant for tunneling through a barrier to the periodic classical trajectory on the upturned potential energy surface whose period is $\tau=\hbar/(k_{\rm B}T)$. Unfortunately, the standard theory is only applicable below the "crossover temperature", where the periodic orbit first appears. This paper presents a rigorous semiclassical ($\hbar\to0$) theory for the rate that is valid at any temperature. The theory is derived by combining Bleistein's method for generating uniform asymptotic expansions with a real-time modification of Richardson's flux-correlation function derivation of instanton theory. The resulting theory smoothly connects the instanton result at low temperature to the parabolic correction to Eyring transition state theory at high-temperature. Although the derivation involves real time, the final theory only involves imaginary-time (thermal) properties, consistent with the standard version of instanton theory. Therefore, it is no more difficult to compute than the standard theory. The theory is illustrated with application to model systems, where it is shown to give excellent numerical results. Finally, the first-principles approach taken here results in a number of advantages over previous attempts to extend the imaginary free-energy formulation of instanton theory. In addition to producing a theory that is a smooth (continuously differentiable) function of temperature, the derivation also naturally incorporates hyperasymptotic (i.e. multi-orbit) terms, and provides a framework for further extensions of the theory.