Interaction-energy statistics from structural correlations

We introduce a unified statistical framework for quantifying system-environment coupling by treating the interaction energy VSE as a stochastic variable. Using our proposed reference-particle decomposition, we derive exact, closed-form expressions for the mean and variance of VSE in terms of the single-particle density and correlation functions up to fourth order. These expressions are valid throughout all coupling regimes, from strongly coupled finite systems to the weak-coupling thermodynamic limit. They connect interaction-energy statistics directly to equilibrium structural information and provide a systematic route for evaluating coupling across open system-environment boundaries. We further show how the moment framework enters free energy constructions, including the Hamiltonian of mean force and the quasichemical-theory outer-shell contribution. When the relevant interaction-energy distribution is approximately Gaussian, the first two moments determine the corresponding second-cumulant free energy estimate. To validate our framework, we performed explicit Monte Carlo simulations of a Lennard-Jones system across a range of system sizes, generating reference distributions of the system-environment interaction energy VSE. We also performed independent empty-cavity Monte Carlo calculations to validate the quasichemical outer-shell specialization. We then applied our derived analytical formulas to predict these distributions and found excellent agreement in both the weak- and strong-coupling regimes. The results demonstrate that interaction-energy distributions and the associated thermodynamic contributions can be reconstructed from microscopic structural correlations.