Toward Improving Multiple Time Step QM/MM Simulations with Δ-Machine Learning

Semiempirical methods are popular for QM/MM simulations of chemical reactions in the condensed phase due to their immense speedup compared to higher-level ab initio or density functional theory (DFT) methods but can be much less accurate depending on the system of interest. Multiple time step (MTS) methods can improve the accuracy of semiempirical QM/MM simulations by using higher-level calculations at less frequent outer time steps, yet accurate results and efficient sampling require the low- and high-level methods to be sufficiently similar. In this work, we show the limitations of standard semiempirical methods (e.g., AM1) for MTS applications. In a condensed-phase reaction of proton transfer from water to methyl phosphate, for which DFT (B3LYP) is used as the high-level method, we can only reach an outer time step of 4, even with the stochastic isokinetic thermostat. We then explore the value of employing Δ-machine learning to enhance the efficiency of MTS QM/MM simulations. As an initial step, we train neural network potentials and Δ-learning corrections to the AM1 method for the corresponding gas-phase reaction. We show that Δ-corrections outperform ML potentials and that the amount of training data has a major impact on the accuracy and transferability of the learned correction. In a gas-phase MTS simulation with appropriate machine-learned Δ-corrections, we can reach an outer integration frequency of 25 for a nearly exact result compared to B3LYP and 30 if we accept an error of 0.3 kcal·mol^-1 for the free energy profile. The study validates and provides guidance to Δ-learning based MTS simulations, setting the stage for future development and realistic applications in the condensed phase.