Optimized Cluster Expansions for Classical Fluids. I. General Theory and Variational Formulation of the Mean Spherical Model and Hard Sphere Percus-Yevick Equations

Abstract

Computationally convenient theoretical methods for calculating the thermodynamic properties and pair correlation functions of a classical multicomponent fluid are presented. The Mayer cluster series for the Helmholtz free energy and pair correlation function are transformed using topological reduction to more compact forms involving a renormalized potential. Then the convergence of the two series is improved by an optimal choice of the renormalized potential. The result is two rapidly convergent series which are useful both for ionic solutions and for simple liquids with short range intermolecular forces. When these series are truncated, very accurate and convenient approximations are obtained for both types of fluids. Another set of results is a variational formulation of the mean spherical model and hard sphere Percus-Yevick equations for the pair correlation functions of multicomponent fluids. The variational formulations greatly facilitate the process of solving the equations numerically. Each of these results can be extended to models for molecular fluids with only a moderate increase in computational effort.