Lack of Hohenberg-Kohn Theorem for Excited States

For a given excited state there exist densities that arise from more than one external potential. This is due to a qualitatively different energy-density relationship from that of the ground state and is related to positive eigenvalues in the nonlocal susceptibility for excited states. Resulting problems with the generalization of the density functional methodology to excited states are discussed.

Figure 1

Two potentials (upper panel) yielding the same first excited-state density (lower panel).

Figure 2

Lithium-like occupations of a $-1/r$ potential and similar alternatives yield the $1s^{1}2s^2$ density shown above.

Figure 3

Transition energies $E_1-E_0$ as a function of the first excited-state density parameterized by $\zeta$ (see text).

Figure 4

Densities at different values of $\zeta$ [see Fig. (3)].

Figure 5

$E_1$, $V_1=\int d^{3}rv(\mathbf{r}){\rho }_1(\mathbf{r})$ and $T_1$ as a function of $\zeta$ as in Fig. (3).