Comparison of exact and approximate density functionals for an exactly soluble model

We consider a model, given by two interacting electrons in an external harmonic potential, that can be solved analytically for a discrete and infinite set of values of the spring constant. The knowledge of the exact electronic density allows us to construct the exact exchangtiorrelation potential and exchange-correlation energy by inverting the Kohn-Sham equation. The exact exchange-correlation potential and energy are compared with the corresponding quantities, obtained for the same densities, using approximate density functionals, namely the local density approximation and several generalized gradient approximations. We consider two values of the spring constant in order to study the system in the low correlation case (high value of the spring constant) and in the high correlation case (low value of the spring constant). In both cases, the exchang-orrelation potentials corresponding to approximate density functionals differ from the exact one over the entire spatial range. The approximate correlation potentials bear no resemblance to the exact ones. The exchange energy for generalized gradient approximation functionals is much improved compared to the result obtained within the local density approximation but the correlation energy is only a little improved.