Variational calculation of quantum mechanical/molecular mechanical free energy with electronic polarization of solvent

Quantum mechanical/molecular mechanical (QM/MM) free energy calculation presents a significant challenge due to an excessive number of QM calculations. A useful approach for reducing the computational cost is that based on the mean field approximation to the QM subsystem. Here, we describe such a mean-field QM/MM theory for electronically polarizable systems by starting from the Hartree product ansatz for the total system and invoking a variational principle of free energy. The MM part is then recast to a classical polarizable model by introducing the charge response kernel. Numerical test shows that the potential of mean force (PMF) thus obtained agrees quantitatively with that obtained from a direct QM/MM calculation, indicating the utility of self-consistent mean-field approximation. Next, we apply the obtained method to prototypical reactions in several qualitatively different solvents and make a systematic comparison of polarization effects. The results show that in aqueous solution the PMF does not depend very much on the water models employed, while in nonaqueous solutions the PMF is significantly affected by explicit polarization. For example, the free energy barrier for a phosphoryl dissociation reaction in acetone and cyclohexane is found to increase by more than 10 kcal/mol when switching the solvent model from an empirical to explicitly polarizable one. The reason for this is discussed based on the parametrization of empirical nonpolarizable models.