Weighted-density-functional theory of inhomogeneous liquids and the freezing transition

Starting from an exact expression for the Helmholtz free-energy functional of an inhomogeneous classical liquid, an approximate functional is presented which depends on a weighted average of the physical density. It retains the nonlocal character of the exact expression, but requires only the properties of the homogeneous liquid. A physical choice of the weighting function used to construct the weighted density is made by appealing to the structure of the homogeneous liquid. The resulting weighted-density approximation (WDA) corresponds to an approximation for some of the third- and higher-order terms in a density-functional expansion of the true free energy, the first-, second-, and a subset of higher-order terms being retained exactly. In this respect the theory differs crucially from earlier functionals based on weighted-density ideas. As an application of the WDA, the freezing of simple liquids is considered within the framework of mean-field theory and is briefly compared to previous theories. In particular, the freezing transition of the hard-sphere system is studied using the WDA formalism, the resulting freezing parameters and the equation of state for the solid being in good agreement with computer-simulation studies.