Density functional theory for hard-sphere fluids: A generating function approach

A new density functional for the inhomogeneous hard-sphere fluid is proposed which expresses the free-energy density in terms of a set of derivatives, with respect to the particle radius, of a simple generating function. The three-dimensional version of the theory is used to calculate density profiles for hard spheres near walls and to investigate the bulk fluid g(r), via the test particle procedure. While the performance of the theory is generally poorer than that of a related theory, the fundamental-measure approach of Rosenfeld, it is better than that of approaches based on a single, density-independent weight function. Unlike earlier approaches, the theory is remarkably successful at describing situations where the effective dimensionality D is reduced below three. More specifically the three-dimensional functional yields rather accurate equations of state in the D = 1 and D = 2 limits and is exact for the D = 0 limit (a cavity that cannot hold more than one particle). The strict one-dimensional version of the theory yields the exact free-energy functional for hard rods whilst the free-energy functional for D = 2 is equivalent to that obtained from the fundamental-measure approach. The extension of the theory to hard-sphere mixtures is also described.