Journal of Statistical Physics, Vol. 37, Nos. l/2, 1984. p. 215-236.
The thermodynamic properties of the mean spherical (MSA), Percus-Yevick (PY), and hypernetted-chain (HNC) approximations are derived by a simple and unified approach by considering the RPA free-energy functional and employing an Ewald-type identity. It is demonstrated that with decreasing relative contribution of the hard-core insertion to the thermodynamic functions, the MSA changes its nature from PY-like to HNC-like, with ~: changing its role from excess pressure to excess free energy, respectively. It is found that the condition of continuity of the MSA pair functions is equivalent to a stationarity condition for ~: and leads to thermodynamic consistency between the "virial" and "energy" equations of state for the (thus defined) "soft"-MSA (SMSA),
with ~: playing the role of the excess free energy. It is shown that the PY-"compressibility" and "virial" equations of state for D-dimensional hard spheres may be simply obtained one from the other without knowing any details of the solution of the model. Using this relation we find an indication that the PY approximation for hard spheres becomes less accurate with increasing dimensionality. A general variational formulation is presented for the application of the MSA for soft potentials, and results for the one-component plasma are discussed and extended.