Energy differences between Kohn-Sham and Hartree-Fock wave functions yielding the same electron density

Kohn-Sham wave functions yielding the Hartree-Fock ground-state densities of alkaline-earth and noble-gas atoms are calculated. From the Kohn-Sham wave functions the corresponding noninteracting kinetic energies and the exchange energies are calculated according to the density-functional definition. The difference between the density-functional and the Hartree-Fock exchange energies for a given electron density is found to be surprisingly small. This justifies, at least for the systems studied here, the common practice of using Hartree-Fock exchange energies as reference values to assess the quality of today’s approximate density functionals for the exchange energy. The calculated Kohn-Sham wave functions give energies with the atomic Hamiltonian operators which are de facto identical to the corresponding values obtained from the optimized potential method. The influence of a coupling parameter turning on the electron-electron interaction on the difference between Kohn-Sham and Hartree-Fock kinetic and exchange energies is investigated. Various hybrid schemes which combine the Kohn-Sham and the Hartee-Fock methods are compared. For the atomic systems investigated here, the different exchange and correlation energies occurring in these schemes are found in most cases to deviate very little from the standard Kohn-Sham exchange and correlation energies. Finally, the formulation and validity of the Hohenberg-Kohn theorem in basis set representations are discussed.