Degeneracy in Density Functional Theory: Topology in the $v$ and $n$ Spaces

This paper clarifies the topology of the mapping between the $v$ and $n$ spaces in fermionic systems. Density manifolds corresponding to degeneracies $g=1$ and $g>1$ are shown to have the same mathematical measure: Every density near a $g$-ensemble-$v$-representable ($g$-VR) $n(\mathbf{r})$ is also $g$-VR (except “boundary densities” of lower measure). The role of symmetry and the connection between $T=0$ and $T=0^{+}$ are discussed. A lattice model and the Be series are used as illustrations.

Figure 1

Two interior regions in the $M^{\prime }$-D $\overrightarrow{n}$ space, $S^g$ and $S^{g+1}$, separated by a $(M^{\prime }-1)$-D boundary ${\Sigma }^g$.

Figure 2

13-point fcc lattice with $D_{4h}$ symmetry: $v_l=V_1$ ($V_2$) on the sites $l=\mathrm{1}\mathrm{–}\mathrm{8}$ (9–12), and $v_{13}=0$.

Figure 3

Top: ground states of two noninteracting spinless fermions on the lattice of Fig. 2, with potentials of $D_{4h}$ symmetry. Bottom: associated region of allowed densities.

Figure 4

Energies and ensemble weights of the next two levels above $1A_{1g}$, along paths in the $(V_1,V_2)$ and $(N_1,N_2)$ planes (dotted lines in Fig. 3).