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 <i>ab initio</i> 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.