The many to one correspondence between wavefunctions and densities

It is illustrative to discuss the reformulation of the Hohenberg-Kohn theory originally carried out by Levy [54] (and later, also by Lieb [55-57]), where instead of the stronger v-representability condition, all that is asked for is compliance with the weaker TV-representability condition for the energy functionals. Our discussion is based on Eq. (18) plus the assumption that Av C M, where Av is the set of u-representable densities (namely, densities coming from ground-state wavefunctions for Hamiltonians // , with t/eV) and J f is the set of iV-representable densities. The latter is explicitly defined by 

Bearing in mind that the same density p(r) can come from anyone of the wave-functions W for i = 1,. ..,oo, the Levy variational principle [54] can be stated as  

Notice that because p r) = p jixed(r), the last term is just a constant  

In Fig. 1, we have rearranged the wavefunctions in Hilbert space such that in each row we find all wavefunctions which yield the same density. Since pv0 is the exact one-particle density, it follows that the exact wavefunction must lie in the row containing all wavefunctions which yield pv0. In Fig. 1, we identify the exact wavefunction with h In fact, we characterize the vertical line denoted by 

In order to highlight the particular way in which the minimization process is carried out in the Levy variational principle, let us consider Fig. 1. In the inner variation of the Levy principle one searches for the optimal wavefunction in the sense that it yields the lowest energy, corresponding to a fixed density p(r) = Pfixed(r) Af. This means, that one moves along the row whose wavefunctions yield p(r) = p/ixedix) Thus, the inner variation becomes 

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