The Second Hohenberg-Kohn Theorem Variational Principle

2 The Second Hohenberg-Kohn Theorem Variational Principle 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p(r) 0. J p(r) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function T. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at 

Let us summarize what we have shown so far. First, all properties of a system defined by an external potential Vext are determined by the ground state density. In particular the ground state energy associated with a density p is available through the functional J P(f )VNedr + Fhk [p]. Second, this functional attains its minimum value with respect to all allowed densities if and only if the input density is the true ground state density, i. e., for p(r) = p0 (r). Of course, the applicability of this variational recipe is limited to the ground 

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