Radial distribution function of penetrable sphere fluids to the second order in density

The simplest bounded potential is that of penetrable spheres, which takes a positive finite value epsilon if the two spheres are overlapped, being zero otherwise. In this paper we derive the cavity function to second order in density and the fourth virial coefficient as functions of T-*equivalent to k(B)T/epsilon (where k(B) is the Boltzmann constant and T is the temperature) for penetrable sphere fluids. The expressions are exact, except for the function represented by an elementary diagram inside the core, which is approximated by a polynomial form in excellent agreement with accurate results obtained by Monte Carlo integration. Comparison with the hypernetted-chain (HNC) and Percus-Yevick (PY) theories shows that the latter is better than the former for T-*less than or similar to 1 only. However, even at zero temperature (hard sphere limit), the PY solution is not accurate inside the overlapping region, where no practical cancellation of the neglected diagrams takes place. The exact fourth virial coefficient is positive for T-*less than or similar to 0.73, reaches a minimum negative value at T-*approximate to 1.1, and then goes to zero from below as 1/T-*4 for high temperatures. These features are captured qualitatively, but not quantitatively, by the HNC and PY predictions. In addition, in both theories the compressibility route is the best one for T-*less than or similar to 0.7, while the virial route is preferable if T-*greater than or similar to 0.7.