Simple and exact approach to the electronic polarization effect on the solvation free energy: Formulation for quantum-mechanical molecular-mechanical system and its applications to aqueous solutions

We present a simple and exact numerical approach to compute the free energy contribution δμ in solvation due to the electron density polarization and fluctuation of a quantum-mechanical solute in the quantum- mechanicalmolecular-mechanical (QMMM) simulation combined with the theory of the energy representation (QMMM-ER). Since the electron density fluctuation is responsible for the many-body QM-MM interactions, the standard version of the energy representation method cannot be applied directly. Instead of decomposing the QM-MM polarization energy into the pairwise additive and non-additive contributions, we take sum of the polarization energies in the QM-MM interaction and adopt it as a new energy coordinate for the method of energy representation. Then, it is demonstrated that the free energy δμ can be exactly formulated in terms of the energy distribution functions for the solution and reference systems with respect to this energy coordinate. The benchmark tests were performed to examine the numerical efficiency of the method with respect to the changes in the individual properties of the solvent and the solute. Explicitly, we computed the solvation free energy of a QM water molecule in ambient and supercritical water, and also the free-energy change associated with the isomerization reaction of glycine from neutral to zwitterionic structure in aqueous solution. In all the systems examined, it was demonstrated that the computed free energy δμ agrees with the experimental value, irrespective of the choice of the reference electron density of the QM solute. The present method was also applied to a prototype reaction of adenosine 5 ^′-triphosphate hydrolysis where the effect of the electron density fluctuation is substantial due to the excess charge. It was demonstrated that the experimental free energy of the reaction has been accurately reproduced with the present approach.