Matching the Probability Distribution On-the-Fly: OPES-Based Dual-Level Free Energy Calculation

The free energy profile (FE Profile) is essential for understanding chemical processes in complex systems. However, achieving sufficient sampling with an accurate potential in the FE profile calculations for chemical reactions in complex environments is often computationally prohibitive. The reference potential method (RPM) addresses this challenge by sampling configurations on a low-level potential energy surface (PES) and then computing the FE profile at a high-level one using thermodynamic perturbation correction, significantly reducing computational costs. Nevertheless, when the low-level Hamiltonian exhibits poor similarity to the high-level one, the FE profile often converges slowly to the accurate result due to insufficient sampling of critical high-level Hamiltonian configurations. In this work, we propose an approach denoted as dual-level OPES, which combines the ideas of the on-the-fly probability enhanced sampling method (OPES) and RPM. In this method, an OPES bias potential V_opes along the reaction coordinate accelerates barrier crossing, while the V_corr probability distribution correction potential gradually improves the probability distribution overlap between the low-level and high-level Hamiltonians in a predefined orthogonal space. Our method has been validated on both a mathematical model system and a realistic case (intramolecular proton transfer of glycine in an aqueous solution). The results show that the dual-level OPES method improves the overlap between the sample distribution in the selected orthogonal space and the high-level Hamiltonian. In the model system, we observe accelerated FE profile convergence compared to the direct application of RPM. For the intramolecular proton transfer case, the dual-level OPES method not only significantly reduces the FE profile error but also achieves a several tens of times speedup over direct high-level simulations.