Factorization of the direct correlation function. III. Integral equations and thermodynamics

Integral equations for pair correlations are one of the standard methods for calculating the thermodynamics of fluid systems. One of the routes to the thermodynamic quantities involves the evaluation of a certain functional of the direct correlation function(s), namely the sum of ring graphs. We show that this functional can be evaluated directly in terms of the factorizing functions ξij and their first two derivatives at their lower cutoff. In numerical computations, this is much simpler and more accurate than the previous method of evaluation using Fourier transform techniques. For certain equations of Percus–Yevick type, where analytic solution is feasible, it provides a much faster route to the compressibility pressure. We give a very general derivation for molecular fluids, using the closed form of the factorizing functions ξij. The range of validity includes recently derived integral equations for fluids of molecules with singly bondable bonding sites. The resulting subdivision of molecular species into subspecies representing different bonding states is absorbed into highly condensed notation. The concept of a support plane of the hard core of a molecule was identified in previous work as fundamental in defining the lower cutoff of the factorizing functions. The same idea is found to be the key to arriving at the evaluation of the ring functional obtained here.