Equations for Nonionic and Ionic Systems

Sfl (l. 2) + fa (l, 2) = the sum of all distinct simple connected graphs consisting of two white 1 circles labeled and 2/, one or more black p circles, f bonds of all possible orders greater than two, and /z bonds the graphs are free of articulation circles, and all black circles are interconnected by at least one route that does not include any white circles there is no pairwise bond between the white circles 

tab is the subsum consisting of those graphs which contain one or more cutting circles, and Sab is the remainder. From the discussion above, it is also clear that tab is given by 

All of the graphs in Cat can therefore be obtained by summing the following sets of graphs  

We note that the combination in (b) is given by exp(tab +Sab)— tab Sai in view of the rule for combinatorial factors given in Eq. (75). Adding all of the contributions from (a) to (d), we find 

If we neglect Sab, then Eq. (148) together with Eq. (142) can be solved for the two unknown functions tab and hab—gab 1. In this way we can achieve a summation over an infinite number of the graphs in hab, even though we also neglect an infinite number. This is the hypemetted chain (HNC) approximation. 

---