Degenerate ground states and nonunique potentials: Breakdown and restoration of density functionals

The Hohenberg-Kohn (HK) theorem is one of the most fundamental theorems of quantum mechanics, and constitutes the basis for the very successful density-functional approach to inhomogeneous interacting many-particle systems. Here we show that in formulations of density-functional theory (DFT) that employ more than one density variable, applied to systems with a degenerate ground state, there is a subtle loophole in the HK theorem, as all mappings between densities, wave functions, and potentials can break down. Two weaker theorems which we prove here, the joint-degeneracy theorem and the internal-energy theorem, restore the internal, total, and exchange-correlation energy functionals to the extent needed in applications of DFT to atoms, molecules, and solids. The joint-degeneracy theorem constrains the nature of possible degeneracies in general many-body systems.

Figure 1

(Color online) Schematic illustration of the breakdown of mappings in the presence of degeneracy (full lines) and the additional complication posed by nonuniqueness (red/dashed lines). Large ovals are sets of functions, medium-size ovals collect degenerate wave functions, and small ovals enclose degenerate wave functions that give rise to the same density. ($v$ and $n$ are used generically to represent sets of conjugate potentials and densities.)

Figure 2

(Color online) The joint-degeneracy theorem states that two degenerate ground states with equal density in a potential $V_1$ can also be ground states of a second potential $V_2$, but only as a degenerate pair. Case II, in which one of the two states is a ground state of $V_2$ but the other is not, is excluded by the theorem.