It is shown that there exists a variational Kohn-Sham density-functional theory, with a minimum principle, for the self-consistent determination of an individual excited-state energy and density. Exact properties of the required functional are ascertained, including a Koopmans theorem. This knowledge allows the employment of an effective potential that gives encouraging numerical results, and also helps to explain the success of a recent perturbation theory and its time-dependent counterpart.

Ensemble non-interacting kinetic energy functional is constructed for spherically symmetric systems. The differential virial theorem is derived for the ensemble. A first-order differential equation for the functional derivative of the ensemble non-interacting kinetic energy functional and the ensemble Pauli potential is presented. This equation can be solved and a special case of the solution provides the original non-interacting kinetic energy of the density functional theory.