Effective potentials in density-functional theory

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Abstract
The problem of deducing the effective single-particle potential of density-functional theory is addressed. A systematic approach is introduced based on the reduction of the system of N one-body Schrödinger equations to a system of N-1 nonlinear differential equations which involve the given density directly. The approach is useful for systems consisting of a small number of particles and applications are made to Be and Ne atoms for which exchange and exchange-correlation potentials are found, and to one-dimensional systems. Densities corresponding to two and three spinless fermions in one dimension are considered and all the examples treated are found to be ground-state v-representable. It is shown that any density for two spinless fermions in one dimension is v-representable and we speculate that the same is likely to be true for any number of particles in one dimension. In contrast densities in three dimensions are given which are not v-representable. However, all these are of the class of ensemble ground-state densities introduced by Levy and Lieb. It remains to be seen if all densities are either ground-state or ensemble–ground-state v-representable.