Relation between the free energy and the direct correlation function in the mean spherical approximation

It is shown that the direct correlation function of a mixture of hard ions in the mean spherical approximation (MSA) can be expressed in terms of overlap functions of charged spherical shells. In particular, if the system is a mixture of pairs of ions of equal size and opposite charge, then the MSA direct correlation function is given by the electrostatic energy of a pair of charged shells, of radius equal to the radius of the hard ion plus 1/(2). This direct correlation function can be derived from a free energy functional, and a simple extension to nonuniform systems is given.