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 so- lute in the quantum-mechanical/molecular-mechanical (QM/MM) simulation combined with the theory of the energy representation (QM/MM-ER). Since the electron density fluctuation is re- sponsible for the many-body QM-MM interactions, the standard version of the energy representa- tion 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 ener- gies in the QM-MM interaction and adopt it as a new energy coordinate for the method of en- ergy 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 so- lute. 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, irre- spective 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.
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