Bridging the gap between correlation entropy functionals in the mean spherical and the hypernetted chain approximations: a field theoretic description

Abstract: The correlation entropy as a functional of radial distribution function $g(r)$ (or the total correlation function $h(r)=g(r)-1$) in classical fluids has been obtained from the second Legendre transform of the grand potential. We focus on the correlation entropy difference between the two typical functionals in the mean spherical approximation (MSA) and the hypernetted chain (HNC) approximation. While the entropy functional difference between these approximations is of a simple form, the diagrammatic approaches in the liquid state theory are quite different from each other. Here we clarified the gap between the MSA and HNC functionals by developing a field theoretic description of the correlation functional theory that combines the variational principle of lower bound free energy, the conventional saddle-point approximation of a reference system to be optimized based on the variational principle, and the hybrid treatment of the saddle-point approximation and the fugacity expansion for modifying the primary optimization. Our formulation demonstrates that the MSA functional is reproduced by the first maximization of the variational functional in the saddle-point approximation, and that the HNC functional is obtained from the improved maximization of the virial term due to the fugacity expansion around the MSA functional. The virial term leads to the modification of a reference system interacting via the direct correlation function, thereby creating the correlation entropy difference.