Ring-polymer instanton theory for tunneling between asymmetric wells

TL;DR

This paper introduces a new instanton theory formulation for asymmetric double wells, enabling accurate tunneling calculations where previous methods failed, and demonstrates its effectiveness on molecular models and biomolecules.

Contribution

A novel instanton theory based on projected flux correlation functions that accurately handles tunneling in asymmetric systems, extending the applicability of instanton methods.

Findings

High accuracy in 1D and 2D models compared to quantum results

Good agreement with experimental tunneling data in biomolecule

Clarification of the relationship between tunneling splittings and reaction rates

Abstract

Instanton theory has arisen as a practical tool for calculating tunneling splittings in molecular systems. Unfortunately, the original formulation of instanton theory fundamentally breaks down when trying to calculate the level splitting in asymmetric double wells, as there is no imaginary-time periodic orbit connecting the two non-degenerate minima. We have therefore developed a new formulation of instanton theory based on a projected flux correlation function that is applicable to these asymmetric systems. Comparison with exact quantum-mechanical results in one- and two-dimensional models demonstrates that it has a reasonably high accuracy, similar to that reported for instanton theory in the symmetric case. The theory is then applied to study tunneling between non-degenerate minima in the biomolecule $\alpha$-fenchol, for which we find good agreement with experiment. Finally, we use the connection to instanton rate theory, which is also derived from flux correlation functions, to discuss the often misunderstood relationship between tunneling splittings and reaction rate constants.