Accurate exchange-correlation potentials and total-energy components for the helium isoelectronic series

Starting from very accurate many-body wave functions, we have constructed essentially exact densities, exchange-correlation potentials, and components of the total energy for helium and two-electron ions. These density-functional results are compared to the corresponding quantities obtained from a variety of commonly used approximate density functionals, namely, the local-density approximation and various generalized gradient approximations, in order to test the accuracy of the approximate functionals. Although the generalized gradient approximations yield improved energies compared to the local-density approximation, the exchange and correlation potentials (especially the latter) obtained from the generalized gradient approximations are in poor agreement with the corresponding exact potentials. The large-distance asymptotic behavior of the exact exchange-correlation potential to O(1/r⁴) is found to agree with theoretical predictions. The short-range behavior of the exchange-correlation potential is very close to quadratic. The prospects for improved generalized gradient approximations are discussed.