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Conventional quantum chemical solvation theories are based on the mean-fieldembedding approximation. That is, the electronic wavefunction is calculated inthe presence of the mean field of the environment. In this paper a directquantum mechanical/molecular mechanical (QM/MM) analog of such a mean-fieldtheory is formulated based on variational and perturbative frameworks. In thevariational framework, an appropriate QM/MM free energy functional is definedand is minimized in terms of the trial wavefunction that best approximates thetrue QM wavefunction in a statistically averaged sense. Analytical free energygradient is obtained, which takes the form of the gradient of effective QMenergy calculated in the averaged MM potential. In the perturbative framework,the above variational procedure is shown to be equivalent with the first-orderexpansion of the QM energy (in the exact free energy expression) about theself-consistent reference field. This helps understand the relation between thevariational procedure and the exact QM/MM free energy as well as existing QM/MMtheories. Based on this, several ways are discussed for evaluatingnon-mean-field effects (i.e., statistical fluctuations of the QM wavefunction)that are neglected in the mean-field calculation. As an illustration, themethod 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 Gaussianfluctuation model for the environment, which suggests that those effects arerather small for the present reaction in water.

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