# Semiclassical analysis of the quantum instanton approximation

## TL;DR

This paper analyzes the relationship between quantum and semiclassical instanton approximations for reaction rates, revealing limitations in asymmetric systems and proposing a modified approach for improved accuracy.

## Contribution

It demonstrates the connection between quantum and semiclassical instanton methods and introduces a modification to improve predictions for asymmetric barriers.

## Key findings

- Two dominant minimum-action paths in quantum instanton expression
- Semiclassical instanton theory is accurate mainly for symmetric systems
- Proposed modification avoids unphysical path sampling, improving accuracy

## Abstract

We explore the relation between the quantum and semiclassical instanton approximations for the reaction rate constant. From the quantum instanton expression, we analyze the contributions to the rate constant in terms of minimum-action paths and find that two such paths dominate the expression. For symmetric barriers, these two paths join together to describe the semiclassical instanton periodic orbit. However, for asymmetric barriers, one of the two paths takes an unphysically low energy and dominates the expression, leading to order-of-magnitude errors in the rate predictions. Nevertheless, semiclassical instanton theory remains accurate. We conclude that semiclassical instanton theory can only be obtained directly from the semiclassical limit of the quantum instanton for symmetric systems. We suggest a modification of the quantum instanton approach which avoids sampling the spurious path and thus has a stronger connection to semiclassical instanton theory, giving numerically accurate predictions even for very asymmetric systems in the low temperature limit.