Analysis of probability of inserting a hard spherical particle with small diameter in hard-sphere fluid

The probability of inserting, without overlap, a hard spherical particle of diameter sigma in a hard-sphere fluid of diameter sigma(0) and packing fraction eta determines its excess chemical potential at infinite dilution, mu(ex)(sigma, eta). In our previous work [R. L. Davidchack and B. B. Laird, J. Chem. Phys. 157, 074701 (2022)], we used Widom's particle insertion method within molecular dynamics simulations to obtain high precision results for mu(ex)(sigma, eta) with sigma/sigma(0) <= 4 and eta <= 0.5. In the current work, we investigate the behavior of this quantity at small sigma. In particular, using the inclusion-exclusion principle, we relate the insertion probability to the hard-sphere fluid distribution functions and thus derive the higher-order terms in the Taylor expansion of mu(ex)(sigma, eta) at sigma = 0. We also use direct evaluation of the excluded volume for pairs and triplets of hard spheres to obtain simulation results for mu(ex)(sigma, eta) at sigma/sigma(0) <= 0.2247 that are of much higher precision than those obtained earlier with Widom's method. These results allow us to improve the quality of the small-sigma correction in the empirical expression for mu(ex)(sigma, eta) presented in our previous work.