Accelerated Computation of Free Energy Profile at Ab Initio QM/MM Accuracy via a Semi-Empirical Reference-Potential. III. Gaussian Smoothing on Density-of-States

Calculations of free energy profile, aka potential of mean force (PMF), along a chosen collective variable (CV) are now routinely applied to the studies of chemical processes, such as enzymatic reactions and chemical reactions in condensed phases. However, if the ab initio QM/MM level of accuracy is required for the PMF, it can be formidably expensive even with the most advanced enhanced sampling methods, such as umbrella sampling. To ameliorate this difficulty, we developed a novel method for the computation of free energy profile based on the reference-potential method recently, in which a low-level reference Hamiltonian is employed for phase space sampling and the free energy profile can be corrected to the level of interest (the target Hamiltonian) by energy reweighting in a nonparametric way. However, when the reference Hamiltonian is very different from the target Hamiltonian, the calculated ensemble averages, including the PMF, often suffer from numerical instability, which mainly comes from the overestimation of the density-of-states (DoS) in the low-energy region. Stochastic samplings of these low-energy configurations are rare events. If a low-energy configuration has been sampled with a small sample size N, the probability of visiting this energy region is ~ 1/N (shall be exactly 1/N for a single ensemble), which can be orders-of-magnitude larger than the actual DoS. In this work, an assumption of Gaussian distribution is applied to the DoS in each CV bin, and the weight of each configuration is rescaled according to the accumulated DoS. The results show that this smoothing process can remarkably reduce the ruggedness of the PMF and increase the reliability of the reference-potential method.