Extrema of the density functional for the energy: Excited states from the ground-state theory

Although every stationary-state density ${n}_{i}$(r\ensuremath{\rightarrow}) of a many-particle system is not an extremum of the ground-state density functional ${E}_{v}$[n], every extremum of ${E}_{v}$[n] [i.e., every solution of the Euler equation \ensuremath{\delta}${E}_{v}$/\ensuremath{\delta}n(r\ensuremath{\rightarrow})=\ensuremath{\lambda}] is a stationary-state density ${n}_{i}$(r\ensuremath{\rightarrow}). Always, ${E}_{v}$[${n}_{i}$]\ensuremath{\le}${E}_{i}$, where ${E}_{i}$ is the lowest stationary-state energy for density ${n}_{i}$(r\ensuremath{\rightarrow}); the equality holds if and only if ${n}_{i}$(r\ensuremath{\rightarrow}) is an extremum of ${E}_{v}$[n]. The extrema lying above the absolute minimum are excited-state densities which fail to be pure-state v-representable. Surprisingly, infinitesimal number-conserving density variations \ensuremath{\delta}n(r\ensuremath{\rightarrow}) about an extremum n(r\ensuremath{\rightarrow}) do not lead to energy variations \ensuremath{\delta}${E}_{v}$ of order (\ensuremath{\delta}n${)}^{2}$ when \ensuremath{\delta}n(r\ensuremath{\rightarrow})/${n}^{1/2}$(r\ensuremath{\rightarrow}) fails to be square-integrable; in fact, variations \ensuremath{\delta}${E}_{v}$ of order \ensuremath{\Vert}\ensuremath{\delta}n\ensuremath{\Vert} about the ground state are exemplified by the recently discovered ``derivative discontinuities of the energy.'' This unconventional behavior of ${E}_{v}$[n] may be traced in part to an asymptotic divergence of ${\ensuremath{\delta}}^{2}$${E}_{v}$/\ensuremath{\delta}n(r\ensuremath{\rightarrow})\ensuremath{\delta}n(r\ensuremath{\rightarrow}'). Conditions are presented under which a self-consistent solution of the Kohn-Sham single-particle problem represents an extremum of ${E}_{v}$[n]. The multiplets of the ground-state orbital configuration of the carbon atom are examined. The local-density and Langreth-Mehl approximations are found to yield a remarkably accurate account of the degeneracy of the various ground-state densities for this system, but no estimate of the multiplet splitting is obtained. Finally, aspects of v-representability are discussed, with emphasis on the iron atom.