Constrained Hybrid Monte Carlo Sampling Made Simple for Chemical Reaction Simulations

Most electrochemical reactions should be studied under a grand canonical ensemble condition with a constant potential and/or a constant pH value. Free energy profiles provide key insights into understanding the reaction mechanisms. However, many molecular dynamics (MD)-based theoretical studies for electrochemical reactions did not employ an exact grand canonical ensemble sampling scheme for the free energy calculations, partially due to the issues of discontinuous trajectories induced by the particle-number variations during MD simulations. An alternative statistical sampling approach, the Monte Carlo (MC) method, is naturally appropriate for the open-system simulations if we focus on the thermodynamic properties. An advanced MC scheme, the hybrid Monte Carlo (HMC) method, which can efficiently sample the configurations of a system with large degrees of freedom, however, has limitations in the constrained-sampling applications. In this work, we propose an adjusted constrained HMC method to compute free energy profiles using the thermodynamic integration (TI) method. The key idea of the method for handling the constraint in TI is to integrate the reaction coordinate and sample the rest degrees of freedom by two types of MC schemes, the HMC scheme and the Metropolis algorithm with unbiased trials (M(RT)²-UB). We test the proposed method on three different systems involving two kinds of reaction coordinates, which are the distance between two particles and the difference of particles' distances, and compare the results to those generated by the constrained M(RT)²-UB method serving as benchmarks. We show that our proposed method has the advantages of high sampling efficiency and convenience of implementation, and the accuracy is justified as well. In addition, we show in the third test system that the proposed constrained HMC method can be combined with the path integral method to consider the nuclear quantum effects, indicating a broader application scenario of the sampling method reported in this work.