Bound excited states in density-functional theory

Using recent modifications of the original Hohenberg-Kohn theorem due to Levy [Proc. Natl. Acad. Sci. USA 76, 6062 (1979)] and Valone [J. Chem. Phys. 73, 1344 (1980)] and the modified Ritz variational principles [J.K.L. MacDonald, Phys. Rev. 46, 828 (1934)] alternative density functionals are exhibited which respect the bounds of the modified principles. Excited-state energies and electron densities may be calculated by direct minimization of the new functionals. The density functional $R_2[\rho ,U]$ obeys the bound $R_2[\rho ,U]\ge {(E_b-U)}^2$, where $U$ is a fixed constant energy and $E_b$ is the bound-state energy of the system closest to $U$. At present, it appears that the functional depends nontrivially on the external potential. Some properties of reduced density-matrix functionals are presented. The nature of the 2 matrix functional, $R_2[\rho ,U]$, provides clues to the nature of the density functional. The evident dependence of $R_2[\rho ,U]$ on the external potential indicates that the presence of excited states in the Levy density functional is very unlikely. The point of view in this paper is found to complement the recently proposed Theophilou density functional for excited states.