By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\rho }_{\alpha }, {\rho }_{\beta },\dots ,{\rho }_{\omega }$ are ground-level densities for interacting or noninteracting Hamiltonians $H_1, H_2,\dots ,H_M$ ($M$ arbitrarily large) with local potentials $v_1$,$v_2$,$\dots$,$v_M$, but given that we do not know which $\rho$ belongs with which $H$, the correct mapping is possible and is obtained by minimizing $\int d\overrightarrow{\mathrm{r}} \left[v_1\right(\overrightarrow{\mathrm{r}}\left){\rho }_{\alpha }\right(\overrightarrow{\mathrm{r}})+v_2(\overrightarrow{\mathrm{r}}\left){\rho }_{\beta }\right(\overrightarrow{\mathrm{r}})+\cdots v_M(\overrightarrow{\mathrm{r}}\left){\rho }_{\omega }\right(\overrightarrow{\mathrm{r}}\left)\right]$ with respect to optimum permutations of the $\rho$'s among the $v$'s. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, $a$ density will not generally be associated with any ground-state wave function (is not wave function $v$ representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-$v$-representable as well as for $v$-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-$v$ representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-$v$ representability for some Kohn-Sham noninteracting systems.

• Received 20 October 1981

DOI:https://doi.org/10.1103/PhysRevA.26.1200