New type of cluster theory for molecular fluids: Interaction site cluster expansion

Abstract

A new cluster series, called the interaction site cluster expansion, is derived for classical molecular fluids. For a general class of molecular models, the series provides an exact formula for the equilibrium pair correlations between interaction sites on different molecules. In the models considered, each molecule contains m of these interaction sites. The total intermolecular potential between two molecules is the sum of m² site–site potentials. A site–site potential depends on the separation between the sites only. However, because the sites are not necessarily located at the centers of molecules, the total molecular pair interaction can depend strongly on molecular orientations. Even though the interaction site cluster expansion is exact, the cluster integrals in the series involve the translational coordinates of the interaction sites only. The molecular orientational coordinates are removed by a transformation which introduces intramolecular correlation functions into the series. Thus, the new cluster series expresses the intermolecular site–site correlation functions in terms of diagrams involving site–site cluster bonds and intramolecular correlation bonds. The interaction site cluster expansions are used to derive several important results. It is shown that when the site–site interactions are hard core potentials, the pair correlation functions rigorously contain not only discontinuities but also cusps. The locations for these singularities are derived. A method is presented which shows how these singularities are smoothed when the hard core potentials are softened. The singularities as well as special limiting results derived herein are used to provide a justification for the reference interaction site model (RISM) equation.