Density-functional theory for two noninteracting spinless fermions

A method is presented for relating a ground-state density distribution to the effective single-particle potential which has this density for its ground state for a noninteracting system of particles. The method is useful for systems consisting of a small number of spinless fermions and in the case of two particles a second-order, nonlinear differential equation relates the density and the effective potential. The question of $v$ representability is addressed through examples of densities for one- and three-dimensional systems. It is found that densities of the form $e^{-ar}$ for two-particle systems are not ground-state $v$ representable. The added flexibility of three dimensions over one allows an effective potential which does not necessarily have the chosen density as its ground state, but as its lowest nondegenerate state.