Density functional theory of homogeneous states

Abstract

We consider the weighted density approximation of the density functional description of systems in thermal equilibrium. We show that knowledge of the intermolecular potential puts constraints on the theory which take the form of a small number of nonlinear integral equations of unusual type. We show that for homogeneous states of systems with purely repulsive potentials these equations are sufficient to determine the free energy functional completely, at least at densities where the virial expansion of the theory converges. We have not been able to find either analytic or numerical solutions of these equations at arbitrary densities. We have solved the equations in the density expansion to the lowest order in which it disagrees with the exact virial expansion of the system. This extended weighted density approximation (EWDA) gives the exact virial expansion of the pressure to third order and the pair distribution function to first order in the density, as do the other standard integral equation theories. In the next order the EWDA is not exact, but it gives very good numerical results for the pressure and pair distribution and for both hard and soft repulsive potentials. In addition, the difference between the pressure and the compressibility equations of state is numerically very small, indicating a high degree of thermodynamic consistency. Were these properties to persist at higher densities, the EWDA would be clearly preferred to the usual integral approach, at least for repulsive potentials. For potentials with an attractive part the EWDA becomes singular at low temperatures in a way that suggests there is a structural flaw in the assumed form of the free energy functional.