Why Solvent Response Contributions to Solvation Free Energies Are Compatible with Ben-Naim's Theorem

We resolve a seeming paradox arising from a common misinterpretation of Ben-Naim's theorem, which rests on the decomposition of the Hamiltonian of a molecular solute/solvent system into solute-solvent and solvent-solvent interactions. According to this theorem, the solvation entropy can also be decomposed into a solute-solvent term and a remaining solvent-solvent term that is commonly referred to as the solvent reorganization term. Crucially, the latter equals the average solvent-solvent interaction energy such that these two solvent-solvent terms cancel and thus do not change the total solvation free energy. This analytical result implies that changes in the solvent-solvent interactions cannot contribute to any thermodynamic driving force. The solvent reorganization term is often identified with the contribution of many-body solvent correlations to the solvation entropy, which seems to imply that these correlations, too, cannot contribute to solvation. However, recent calculations based on atomistic simulations of a solvated globular protein and spatially resolved mutual information expansions revealed substantial contributions of many-body solvent correlations to the solvation free energy, which are not canceled by the enthalpy change of the solvent. Here, we resolved this seeming contradiction and illustrate by two examples- a simple Ising model and a solvated Lennard-Jones particle- that the solvent reorganization entropy and the actual entropy contribution arising from many-body solvent correlations differ both conceptually and numerically. Whereas the solvent reorganization entropy in fact arises from both solvent-solvent as well as solute-solvent interactions and thus has no straightforward intuitive interpretation, the mutual information expansion permits an interpretation in terms of the entropy contribution of solvent-solvent correlations to the solvation free energy.