On the Born-Green-Yvon equation and triplet distributions for hard spheres

The Born–Green–Yvon integral equation for hard spheres is studied using two closures which provide improvements to the traditional Kirkwood superposition approximation (KSA). These rigorous corrections to the KSA arise from a diagrammatic expansion of the triplet potential of mean force which can be carried out in terms of either the Mayer f-function or the total correlation function h. While the short-ranged f-bond corrections improve the calculated pair distribution function at contact, they otherwise distort this function and thus give very poor compressibility results. The long-ranged h-bond corrections are found to give overall improvement to the pair distribution function and, in particular, give nearly the correct phase of this function. Furthermore, the triplet distribution function computed with the second-order h-bond correction is found to be reasonably close to Monte Carlo results.