Reaction Path-Force Matching in Collective Variables: Determining Ab Initio QM/MM Free Energy Profiles by Fitting Mean Force

Abstract

First-principles determination of free energy profiles for condensed-phase chemical reactions is hampered by the daunting costs associated with configurational sampling on ab initio quantum mechanical/molecular mechanical (AI/MM) potential energy surfaces. Here, we report a new method that enables efficient AI/MM free energy simulations through mean force fitting. In this method, a free energy path in collective variables (CVs) is first determined on an efficient reactive aiding potential. Based on the configurations sampled along the free energy path, correcting forces to reproduce the AI/MM forces on the CVs are determined through force matching. The AI/MM free energy profile is then predicted from simulations on the aiding potential in conjunction with the correcting forces. Such cycles of correction−prediction are repeated until convergence is established. As the instantaneous forces on the CVs sampled in equilibrium ensembles along the free energy path are fitted, this procedure faithfully restores the target free energy profile by reproducing the free energy mean forces. Due to its close connection with the reaction path-force matching (RP-FM) framework recently introduced by us, we designate the new method as RP-FM in collective variables (RP-FM-CV). We demonstrate the effectiveness of this method on a type-II solution-phase SN 2 reaction, NH 3 + CH 3 Cl (the Menshutkin reaction), simulated with an explicit water solvent. To obtain the AI/MM free energy profiles, we employed the semiempirical AM1/MM Hamiltonian as the base level for determining the string minimum free energy pathway, along which the free energy mean forces are fitted to various target AI/MM levels using the Hartree−Fock (HF) theory, density functional theory (DFT), and the second-order Møller−Plesset perturbation (MP2) theory as the AI method. The forces on the bond-breaking and bond-forming CVs at both the base and target levels are obtained by force transformation from Cartesian to redundant internal coordinates under the Wilson B-matrix formalism, where the linearized FM is facilitated by the use of spline functions. For the Menshutkin reaction tested, our FM treatment greatly reduces the deviations on the CV forces, originally in the range of 12−33 to ∼2 kcal/mol/Å. Comparisons with the experimental and benchmark AI/MM results, tests of the new method under a variety of simulation protocols, and analyses of the solute−solvent radial distribution functions suggest that RP-FM-CV can be used as an efficient, accurate, and robust method for simulating solution-phase chemical reactions.

Machine-Learning-Assisted Free Energy Simulation of Solution-Phase and Enzyme Reactions

“…Needless to say, the accuracy of both indirect and multiple-time-step simulations is controlled by the quality of the se -QM/MM potential in use. In many cases, it is beneficial to reoptimize the se -QM/MM parameters , or directly modify the internal forces to ensure a proper thermodynamic perturbation or interpolation correction or to maintain a stable multiple-time-step trajectory.…”

“…Needless to say, the accuracy of both indirect and multipletime-step simulations is controlled by the quality of the se-QM/MM potential in use. In many cases, it is beneficial to reoptimize the se-QM/MM parameters 23,38 or directly modify the internal forces 39 to ensure a proper thermodynamic perturbation or interpolation correction or to maintain a stable multiple-time-step trajectory. Recently, Yang, 40−42 Gastegger, 43 York, 44 and Riniker 45 have proposed machine learning (ML) as a new strategy to address the computational cost of direct ai-QM/MM free energy simulations.…”

Accelerating ab initio QM/MM Molecular Dynamics Simulations with Multiple Time Step Integration and a Recalibrated Semi-empirical QM/MM Hamiltonian

“…Instead of removing the fast components of the correction force, one can in principle apply the MTS correction only on key slow degrees of freedom associated with the reaction coordinate, in the spirit of the reaction-path force-matching in the collective variables (RF-FM-CV) method. 100 The effectiveness of this strategy will be examined in future studies. We tested the stability of the MTS ai-QM/MM MD simulations with different combinations of inner step models and outer step sizes using microcanonical (NVE) ensemble MD simulations.…”

Development of Range-Corrected Deep Learning Potentials for Fast, Accurate Quantum Mechanical/Molecular Mechanical Simulations of Chemical Reactions in Solution

“…Other recent works have applied force matching to collective variables within QM/MM simulations to calculate free-energy surfaces with an efficient semiempirical method. 55 In this approach, the interactions between QM and MM atoms are not explictly corrected, but the reproduction of the net mean forces in the space of the reaction coordinate collective variables describing the reaction pathway implicitly accounts for their effect within the scope of the parameterization. The ultimate goal is to develop new QM/MM and QMFF models based on high-level ab initio QM data in environments that mimic the condensed phase.…”