New mean-energy formulae for free energy differences

We report a number of new exact expressions, within the realm of classical statistical mechanics, for the solvation contribution ΔA to the free energy change of a solute that undergoes a transformation between two states that differ in the way they interact with the solvent. Our basic result is a coupling-parameter expression that is equivalent to the usual cumulant expansion of ΔA. In our formula, which is derived from a linear coupling-parameter scheme, all the cumulant terms beyond the first one are collected into an integral over the coupling parameter. From this formula, with the help of the mean-value theorems of calculus, we derive exact one- and two-reference-point mean-energy expressions for the solvation free energy change, neither of which is limited to the linear response regime. Both the one- and the two-reference point formulae are given in terms of one distinguished value of the coupling parameter (ξ and ξ*, respectively) that is not known a priori (except in the linear response regime where we find it exactly). In the case of the two-reference-point formula, we relate ξ* with the cumulative changes of the solvation structure along the intermediate states sampled in the (ξ = 0) → (ξ = 1) transformation of the solute. Finally, and based on these developments, we present an analysis of the Linear Interaction Energy and of the Extended Linear Response semi-empirical computational methods (widely used for the determination of binding constants of ligand/receptor biological systems) in an effort to understand their statistical mechanical foundations as well as their domain of applicability.