Degenerate ground states Hohenberg-Kohn theorems

Theorem I (Mapping theorem) F(r) n(r). The ground state energy of a many-electron system is a universal functional of the ground state density, n(r), only for a non-degenerate ground state (Hohenberg and Kohn, (HK)). 

The electron density of a non-degenerate ground state system determines essentially all physical properties of the system. This statement of the Hohenberg-Kohn theorem of Density Functional Theory plays an exceptionally important role among all the fundamental relations of Molecular Physics. 

As a consequence of the Hohenberg-Kohn theorem [14], a non-degenerate ground state electron density p(r) determines the Hamiltonian H of the system within an additive constant, implying that the electron density p(r) also determines all ground state and all excited state properties of the system. 

The density functional methods were, in the views of many, legitimized by the introduction of the first Hohenberg-Kohn theorem [10]. The consequence of this celebrated theorem is that for a non-degenerate ground state and a given external potential, v(r), the electronic ground state energy can be expressed as... 

The Hohenberg-Kohn theorem for degenerate ground states 47... 

THE HOHENBERG-KOHN THEOREM FOR DEGENERATE GROUND STATES... 

Hohenberg and Kohn proved their theorems only for nondegenerate ground states. Subsequently, Levy proved the theorems for degenerate ground states (see Parr and Yang, Sec. 3.4). 

A quite different approach is based on the Hohenberg-Kohn theorem, which states that the ground state energy of a (non-degenerate) many particle system in an external applied potential is a unique functional [ni of the density n(j. Furthermore, they proved that this universal functional S[n] has as its minimum value the correct ground state energy... 

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