High-density properties of liquid-state theories: Physically intuitive meaning for the direct correlation functions

Abstract

We prove that the hypernetted-chain and soft-mean-spherical integral-equation theories, together with the variational perturbation theory (based on the Percus-Yevick approximation for the hard-sphere reference system) are all identical in the asymptotic high-density limit: They share the same Madelung energy that constitutes an exact lower bound to the true potential energy of the system. The Percus-Yevick equation for hard spheres is shown to diverge at total packing fraction equal to unity, with a direct correlation function c(r) given by the overlap volume of two spheres with separation r. We present the exact analytic solution of the mean-spherical approximation for any repulsive potential φ(r) satisfying φ̃(k)≥0 which is a Green’s function for an operator of the Sturm-Liouville-type. The corresponding direct correlation function is given by the interaction between the particles when they are ‘‘uniformly smeared’’ inside a sphere. Our results are general, applicable to any number of dimensions and to mixtures. The physically intuitive meaning for the direct correlation functions is offered as a starting point for the statistical thermodynamic analysis of nonspherical hard, and/or charged, objects.