Accelerating Instanton Theory with the Line Integral String Method, Gaussian Process Regression, and Selective Hessian Modeling

TL;DR

This paper introduces a Gaussian process regression enhanced line integral string method to efficiently compute tunneling rates and splittings in molecular reactions, significantly reducing computational costs while maintaining accuracy.

Contribution

The authors develop a novel surrogate modeling approach that accelerates instanton calculations and reduces force and Hessian evaluations using selective Hessian training and GPU acceleration.

Findings

Achieves order of magnitude speedup in force evaluations

Predicts tunneling rates within 20% of exact values

Accurately computes tunneling splittings in complex molecules

Abstract

We develop a Gaussian process regression enhanced line integral string method to accelerate ring polymer instanton calculations of tunneling rates and tunneling splittings in molecular proton transfer reactions. By exploiting uncertainty estimates from the surrogate representation, we show that the number of force evaluations required to converge an instanton path becomes effectively independent of the number of beads used to discretize the pathway. To reduce the computational overhead associated with training, particularly when Hessian information is included, we implement graphics processing unit accelerated black box matrix matrix multiplication, achieving an order of magnitude speedups relative to standard implementations. For rate calculations, we introduce a selective Hessian training strategy that distinguishes flexible modes strongly coupled to the transferring proton from more rigid modes weakly coupled to the reaction coordinate. This enables the construction of accurate surrogate potential energy surfaces with substantially fewer Hessian evaluations. Applications to malonaldehyde and Z-3-aminopropenal demonstrate that tunneling rates can be predicted within 20% of exact values while reducing force and Hessian evaluations. The approach is further extended to tunneling splitting calculations for the formic acid dimer and malonaldehyde, yielding splittings in reasonable agreement with experiment and high level theoretical results.