Consistent integral equations for two- and three-body-force models: Application to a model of silicon

Abstract

Functional differentiation of systematic expansions for the entropy, in the grand ensemble [B. B. Laird and A. D. J. Haymet, Phys. Rev. A 45, 5680 (1992)], leads directly to consistent integral equations for classical systems interacting via two-body, three-body, and even higher-order forces. This method is both a concise method for organizing existing published results and for deriving previously unpublished, higher-order integral equations. The equations are automatically consistent in the sense that all thermodynamic quantities may be derived from a minimum on an approximate free-energy surface, without the need to introduce weighting functions or numerically determined crossover functions. A number of existing approximate theories are recovered by making additional approximations to the equations. For example, the Kirkwood superposition approximation is shown to arise from a particular approximation to the entropy. The lowest-order theory is then used to obtain integral-equation predictions for the well-known Stillinger-Weber model for silicon, with encouraging results. Further connections are made with increasingly popular density-functional methods in classical statistical mechanics.