Integral Equation Methods

Abstract

In this chapter we describe some of the integral equation methods which have been devised for calculating the angular pair correlation function g(rω1ω2) and the site-site pair correlation function gαβ( r ) for molecular liquids. These methods are in the main natural extensions of methods devised for calculating the pair correlation function g(r) for atomic liquids. They can be derived from infinite-order perturbation theory (an example is given in § 5.4.8), whereby one partially sums the perturbation series of Chapter 4 to infinite order usually with the help of diagrams, or graphs, but alternative methods of derivation are also available, e.g. functional expansions. The original integral equation theories are in a certain sense more complete than perturbation theories, in that the full correlation function g (or gαβ) is calculated, whereas in perturbation theory one calculates the correction g — g0 to the reference fluid value g0. On the other hand the perturbation theory approximations are controlled; one can estimate the error by calculating the next term. I t is extremely difficult to estimate a priori the error in integral equation approximations, since certain terms are neglected almost ad hoc. Their validity must therefore be a posteriori, according to agreement with computer simulation results (or, less satisfactorily, with experiment). Of particular interest are theories which are a combination of perturbation theory and an integral equation, which tend to have some of the advantages of both approaches (see also §5.3.1). An example is the GMF/SSC theory of §5.4.7. The structure of the integral equation approach for calculating g(r ω1 ω2) is as follows. One starts with the Ornstein-Zernike (OZ) integral equation (3.117) between the total correlation function h = g — 1 and the direct correlation function c, which we write here schematically as . . . h = c + pch (5.1) . . . or, even more schematically, as . . . h = h[c], (OZ) (5.2) . . . where h[c] denotes a functional of c.