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Theory of Solutions in Energy Representation in NPT-ensemble: Derivation Details

Theory of solutions in energy representation (ER method) developed by Matubayasi and Nakahara provides with an approximate way of calculating solvation free energies (or, identically, the excess chemical potentials) from atomistic simulations. In this document we provide some derivation details of this, to our opinion, theoretically involved method, which will help a non-specialist to follow. There are three points which differ this document from a regular textbook on statistical mechanics or research articles: 1) Derivation is detailed and all approximations are explicitly stated; 2) Statistical mechanics derivations are performed in NPT-ensemble; 3) We perform the derivations for the case when a molecule is represented as a set of (atomic) sites interacting via spherically symmetric potentials (a classical Force Field representation). In ER method, a new collective coordinate is introduced - the interaction energy of a solute and a solvent molecule. The excess chemical potential is expressed as a functional of the solute-solvent density distribution defined over the collective variable. The functional can be approximated by the Percus's method of functional expansion, which leads to the end-point (not dependent on the $λ$-coupling path) free energy expression. As a side result, we prove that the solvation free energy is always equivalent to the excess (over ideal) chemical potential, and not only at infinite dilution or when internal molecule degrees of freedom are not affected by solvation as it is sometimes wrongly believed.