Toward understanding the exchange-correlation energy and total-energy density functionals

If an accurate ground-state electron density ${\mathrm{\rho }}_0$ for a system is known, it is shown from calculations on atoms that a strikingly good estimate for the total electronic energy of atoms is provided by the formula E[${\mathrm{\rho }}_0$]=${\mathit{tsum}}_{\mathit{i}}$ ${\mathrm{\varepsilon }}_{\mathit{i}}$-(1-1/N)J[${\mathrm{\rho }}_0$], where N is the number of electrons, J[${\mathrm{\rho }}_0$] is the classical Coulomb repulsion energy for ${\mathrm{\rho }}_0$, and the ${\mathrm{\varepsilon }}_{\mathit{i}}$ are the Kohn-Sham orbital energies determined by the Zhao-Morrison-Parr procedure [Phys. Rev. A 50, 2138 (1994)] for implementation of the Levy-constrained search determination of the Kohn-Sham kinetic energy. The surprising accuracy of this formula is attributed to the fact that the exchange-correlation functional is equal to -J/N plus a functional that behaves as if it were approximately homogeneous, of degree 1 in the electron density. A corresponding exact formula is given, and various approximate models are constructed.