Two-body integrals for hard sphere fluid in the first coordination layer

Hard sphere is a commonly used reference fluid, its two-body integrals are of great significance in perturbation theory. However, in order to model the highly orientated strong interactions, such as associating and ionic solvating with perturbation theory, the upper limit of the two-body integral should be restricted within the first coordination layer instead of infinite. In this paper, the two-body integrals confined to the first coordination shell for hard sphere fluid are calculated, which might be useful in representing semi-bonding interaction, such as hydrogen bonding and ionic solvating and pairing.