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

Although every stationary-state density $n_i$(r→) 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 δ$E_v$/δn(r→)=λ] is a stationary-state density $n_i$(r→). Always, $E_v$[$n_i$]≤$E_i$, where $E_i$ is the lowest stationary-state energy for density $n_i$(r→); the equality holds if and only if $n_i$(r→) 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 δn(r→) about an extremum n(r→) do not lead to energy variations δ$E_v$ of order (δn${)}^2$ when δn(r→)/$n^{1/2}$(r→) fails to be square-integrable; in fact, variations δ$E_v$ of order ‖δn‖ 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 ${\delta }^2$ $E_v$/δn(r→)δn(r→’). 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.