Variational and perturbative formulations of QM/MM free energy with mean-field embedding and its analytical gradients

TL;DR

This paper develops variational and perturbative QM/MM free energy methods with mean-field embedding, providing analytical gradients and insights into non-mean-field effects, demonstrated on a water SN2 reaction.

Contribution

It introduces a variational and perturbative framework for QM/MM free energy calculations with analytical gradients and discusses non-mean-field effects.

Findings

Analytical free energy gradient derived as the gradient of effective QM energy in averaged MM potential.

Perturbative approach shown to be equivalent to first-order expansion of the QM energy.

Non-mean-field effects estimated to be less than 0.5 kcal/mol for the studied SN2 reaction.

Abstract

Conventional quantum chemical solvation theories are based on the mean-field embedding approximation. That is, the electronic wavefunction is calculated in the presence of the mean field of the environment. In this paper a direct quantum mechanical/molecular mechanical (QM/MM) analog of such a mean-field theory is formulated based on variational and perturbative frameworks. In the variational framework, an appropriate QM/MM free energy functional is defined and is minimized in terms of the trial wavefunction that best approximates the true QM wavefunction in a statistically averaged sense. Analytical free energy gradient is obtained, which takes the form of the gradient of effective QM energy calculated in the averaged MM potential. In the perturbative framework, the above variational procedure is shown to be equivalent with the first-order expansion of the QM energy (in the exact free energy expression) about the self-consistent reference field. This helps understand the relation between the variational procedure and the exact QM/MM free energy as well as existing QM/MM theories. Based on this, several ways are discussed for evaluating non-mean-field effects (i.e., statistical fluctuations of the QM wavefunction) that are neglected in the mean-field calculation. As an illustration, the method is applied to an SN2 Menshutkin reaction in water, NH3 + CH3CL -> NH3CH3^{+} + CL^{-}, for which free energy profiles are obtained at the HF, MP2, B3LYP, and BH&HLYP levels by integrating the free energy gradient. Non-mean-field effects are evaluated to be < 0.5 kcal/mol using a Gaussian fluctuation model for the environment, which suggests that those effects are rather small for the present reaction in water.