Accelerating Instanton Theory for Intramolecular Proton Transfer Reactions Using the Line Integral String Method and Gaussian Process Regression

Quantum tunneling plays a fundamental role in many chemical reactions, in particular proton transfer reactions. Ring polymer instanton theory offers a practical framework for computing tunneling rates in complex molecular systems. However, applying the ring polymer instanton method to ab-initio potential energy surfaces is computationally demanding. Here, we present an efficient implementation of the ring polymer instanton method by combining the Line Integral String (LI-String) approach with the Gaussian Process Regression (GPR). We demonstrate that the number of ab-initio potential- and force-evaluations can be reduced by using the GPR uncertainty estimate and the selective Hessian training approach. In addition, we show that the computationally inefficient GPR training procedure can be accelerated using the GPU and Black Box Matrix Multiplication (BBMM) method. We apply the GPR enhanced LI-String method to computing the ground state tunneling splitting of two prototypical molecules: malonaldehyde and formic acid dimer. We show it achieves reasonable agreement with the experiment measurements and previous theoretical results. This work opens the opportunity for applying the instanton method to study proton transfer reaction in complex chemical systems.