Energy expressions for Kohn–Sham potentials and their relation to the Slater–Janak theorem

Direct approximation of exchange-correlation potentials is a promising approach to accurate prediction of molecular response properties. However, little is known about ways of obtaining total energies from model potentials other than by using the Levy-Perdew virial relation. We introduce and explore several alternative formulas which arise as line integrals of potentials taken along density scaling and aufbau-filling paths, and which are not limited to the exchange term. The relaxed-orbital variant of the aufbau-path energy expression is shown to be closely related to the Slater-Janak theorem. Although the Levy-Perdew relation generally yields reasonable energies for all model exchange potentials, the relaxed-orbital aufbau path gives better results for those potentials that predict accurate highest-occupied orbital eigenvalues, such as the potential of Räsänen, Pittalis, and Proetto [J. Chem. Phys. 132, 044112 (2010)]. The ideas presented in this work may guide the development of new types of density-functional approximations for exchange and correlation.