Coincidence theorem for the direct correlation function of hard-particle fluids

The Mayerf-function for purely hard particles of arbitrary shape satisfiesf²(1, 2)=–f(1, 2). This relation can be introduced into the graphical expansion of the direct correlation functionc(1, 2) to obtain a graphical expression for the case of exact coincidence, in position and orientation, of two identical hard cores. The resulting expression forc(1, 1)+1 contains only graphsG fromc(1), the sum of irreducible graphs with one labeled point. Relative to its coefficient inc(1),G occurs inc(1, 1) with an additional factorR_cwhich is 1 for the leading graph in the expansion and of the form 2–2L(G) for all other graphs. HereL(G)=0, 1, 2,..., is a nonnegative integer. Topological analysis is used to derive an expression forL(G) in terms of the connectivity properties ofG.