Is an interacting ground state (pure state) v-representable density also non-interacting ground state v-representable by a Slater determinant? In the absence of degeneracy, yes!
It is shown that in the absence of degeneracy a density,$n,(r)$, describing the probability of finding a particle in a small volume at position, r, that is known to be pure-state v-representable in terms of the ground state of an interacting system of particles evolving under an external potential,$v,(r)$, is also pure state v-representable in terms of a single Slater determinant describing the ground state of a system of non-interacting particles under a potential, $v_s(r)$ . This establishes the validity of the Kohn–Sham formalism of density functional theory. An explicit form of $v_s(r)$ is derived. We also derive the exact form of the correlation functional and the corresponding potential, $\mu_c(r)$, that lead to the exact density and energy of an interacting system's ground state. In addition, we demonstrate the existence of a one-to-one mapping between the ground-state densities of non-degenerate interacting and non-interacting systems. Finally, we show that practical implementations of the Kohn–Sham formalism can generate neither the exact density nor energy of an interacting system's ground state, a feature that is particularly true in the case of the so-called exact exchange, in which the correlation functional and potential are set equal to zero.