Analytic continuation of Wolynes theory into the Marcus inverted regime

TL;DR

This paper extends Wolynes theory to the Marcus inverted regime by analytically continuing the imaginary time correlation function, enabling accurate rate calculations where previous methods diverged.

Contribution

We demonstrate how to analytically continue the imaginary time correlation function to evaluate rates in the Marcus inverted regime, overcoming divergence issues in Wolynes theory.

Findings

Accurately computes rates in the Marcus inverted regime.

Matches exact golden rule results for model systems.

Shows robustness of the extended Wolynes theory in complex models.

Abstract

The Wolynes theory of electronically nonadiabatic reaction rates [P. G. Wolynes, J. Chem. Phys. 87, 6559 (1987)] is based on a saddle point approximation to the time integral of a reactive flux autocorrelation function in the nonadiabatic (golden rule) limit. The dominant saddle point is on the imaginary time axis at $t_{\rm sp}=i\lambda_{\rm sp}\hbar$, and provided $\lambda_{\rm sp}$ lies in the range $-\beta/2\le\lambda_{\rm sp}\le\beta/2$, it is straightforward to evaluate the rate constant using information obtained from an imaginary time path integral calculation. However, if $\lambda_{\rm sp}$ lies outside this range, as it does in the Marcus inverted regime, the path integral diverges. This has led to claims in the literature that Wolynes theory cannot describe the correct behaviour in the inverted regime. Here we show how the imaginary time correlation function obtained from a path integral calculation can be analytically continued to $\lambda_{\rm sp}<-\beta/2$, and the continuation used to evaluate the rate in the inverted regime. Comparisons with exact golden rule results for a spin-boson model and a more demanding (asymmetric and anharmonic) model of electronic predissociation show that the theory it is just as accurate in the inverted regime as it is in the normal regime.