Generating function density functional theory: free-energy functionals and direct correlation functions for hard-spheres

A recent density functional theory for the inhomogeneous hard-sphere fluid expresses the free-energy in terms of a set of derivatives, with respect to the particle radius, of a simple generating function. In the present paper this generating function approach (GFA) is reformulated in terms of general conditions on the coefficients of the differential operator involved in the theory. This gives more flexibility to the theory and allows for ready derivation of new functionals that are tested in the calculation of density profiles of hard spheres near a hard wall. Comparison with the results of the previous formulation of the GFA shows a better performance of the new functionals and also gives a much deeper understanding of the structure of the theory and its connections with the fundamental measures theory of Rosenfeld. A simple modification of the GFA pair direct correlation functions yields explicit analytic expressions with very good agreement with simulation results.