Existence of an energy functional minimized by po

Hohenberg and Kohn also proved an analogue of the variational principle (p. 196)  

FOr a given number ol electrons (tfie mtegral over p equals JV) and external potential v, there exists a functional of p, denoted by for which the 

It appears that the energy Eq represents the minimum value of a certain, unfortunately unknown, functional, and, this minimum value is obtained when inserting the density distribution p equal to the perfect ground-state density distribution po into the functional. 

We will prove this theorem using the variational principle in a way first given by Levy. The variational principle states that Eq = min( 7/ ), where we search among the wave functions normalized to I and describing N electrons. This minimization may be carried out in two steps  

It is easy to show that ( K ) may be expressed as an integral involving the density distribution p instead of Indeed, since 

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