Density functional formulation of the random-phase approximation for inhomogeneous fluids: Application to the Gaussian core and Coulomb particles

Using the adiabatic connection, we formulate the free energy in terms of the correlation function of a fictitious system, h(lambda)(r, r'), in which interactions lambda u(r, r') are gradually switched on as lambda changes from 0 to 1. The function h(lambda)(r, r') is then obtained from the inhomogeneous Ornstein-Zernike equation and the two equations constitute a general liquid-state framework for treating inhomogeneous fluids. The two equations do not yet constitute a closed set. In the present work we use the closure c(lambda)(r, r') approximate to -lambda beta u(r, r'), known as the random-phase approximation (RPA). We demonstrate that the RPA is identical with the variational Gaussian approximation derived within the field-theoretical framework, originally derived and used for charged particles. We apply our generalized RPA approximation to the Gaussian core model and Coulomb charges.