The tunneling splitting and the Kramers theory of activated processes

TL;DR

This paper introduces a new method for analyzing ground-state tunneling splitting in one-dimensional systems, leveraging an isomorphism with quantum Hamiltonians and localization functions, resulting in improved accuracy over traditional semiclassical estimates.

Contribution

It presents a novel approach based on the localization function and the Fokker-Planck-Smoluchowski operator to accurately estimate tunneling splittings, outperforming WKB methods.

Findings

The new method provides more accurate tunneling splitting estimates than WKB.

The approach is explicitly applied to one-dimensional systems.

Comparison with exact values confirms improved accuracy.

Abstract

The study of tunneling splitting is fundamental to get insight into the dynamics of a multitude of molecular systems. In this paper, a novel approach to the analysis of the ground-state tunneling splitting is presented and explicitly applied to one-dimensional systems. The isomorphism between the Fokker-Planck-Smoluchowski operator and the Born-Oppenheimer quantum Hamiltonian is the key element of this method. The localization function approach, used in the field of stochastic processes to study the Kramers problem, leads to a simple, yet asymptotically justified, integral approximation for the tunneling splitting. The comparison with exact values of the tunneling splittings shows a much better accuracy than WKB semiclassical estimates.