The Hohenberg-Kohn Theorem for Degenerate Ground States

THE HOHENBERG-KOHN THEOREM FOR DEGENERATE GROUND STATES 

If we choose the complete set to be the set of eigenstates of Hamiltonian H then we find 

This defines the expectation value of an observable 0 in an ensemble described by density matrix D. For the particular case of the density operator we have 

We will denote densities n(r) of this type, which are obtained from an orthonormal set of ground states I i = 1- -q corresponding to a potential v, as ensemble v-representable densities, or for short E-V-densities. We further denote the set of all E-V-densities generated by potentials in L,/2 + L00 as B. A density will be called a pure state v-representable density or for short PS-V-density if it can be written as (r) = (1frlu(r)l1fr), where I P) is a ground state. Obviously PS-V-densities are special cases of E-V-densities and the set of PS-V-densities is therefore a subset of the set of E-V-densities. 

Suppose V generates the ground state multiplet Aj = I , ), i = l - -q and v2 generates the ground state multiplet A2 = I P,), i = 1 -q2. All the wavefunctions within these multiplets may, without loss of generality, be chosen orthonormal. Then none of the wavefunctions in the sets A1 and A2 are equal. This follows from 

---

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

---