# Accelerating equilibrium isotope effect calculations: I. Stochastic thermodynamic integration with respect to mass

## TL;DR

This paper introduces a stochastic thermodynamic integration method that accelerates isotope effect calculations by eliminating integration errors through mass variation and biasing potentials, without increasing statistical errors.

## Contribution

It presents a novel approach to remove thermodynamic integration errors in isotope effect calculations by stochastic mass changes and umbrella biasing, improving efficiency.

## Key findings

- Effective removal of integration errors demonstrated on model systems.
- No increase in statistical errors with the new method.
- Significant acceleration in isotope effect calculations achieved.

## Abstract

Accurate path integral Monte Carlo or molecular dynamics calculations of isotope effects have until recently been expensive because of the necessity to reduce three types of errors present in such calculations: statistical errors due to sampling, path integral discretization errors, and thermodynamic integration errors. While the statistical errors can be reduced with virial estimators and path integral discretization errors with high-order factorization of the Boltzmann operator, here we propose a method for accelerating isotope effect calculations by eliminating the integration error. We show that the integration error can be removed entirely by changing particle masses stochastically during the calculation and by using a piecewise linear umbrella biasing potential. Moreover, we demonstrate numerically that this approach does not increase the statistical error. The resulting acceleration of isotope effect calculations is demonstrated on a model harmonic system and on deuterated species of methane.