Kohn-Sham Theory by Legendre Transforms

The method described in this section goes back to the work of De Dominicis and Martin [6]. This work discusses the relations between /V-body potentials and iV-particle density matrices, of which the density-potential relation to be discussed 

Our goal is now to go from the potential as our basic variable, to a new variable, which will be the electron density. The deeper reason that this is possible is that the density and the potential are conjugate variables. With this we mean that the contribution of the external potential to the total energy is simply an integral of the potential times the density. We make use of this relation if we take the functional derivative of the energy functional [v] with respect to the potential v  

This is our first basic relation. In order to derive the Kohn-Sham equations we define the following energy functional for a system of noninteracting particles with external potential vs and with ground state wavefunction I f fvJ)  

We see that Fs[n] in equation (48) is the kinetic energy of a noninteracting system with potential vs and density n. For this reason the functional Fs is usually denoted by Ts. In the following we will adopt this notation. Finally we define the exchange-correlation functional Exc[n by the equation 

This equation assumes that the functionals F[n] and Ts[n] are defined on the same domain of densities. We thus assume that for a given ground state density of an interacting system there is a noninteracting system with the same density. In other words, we assume that the interacting density is noninteracting-v-representable. If we differentiate equation (51) with respect to the density n we obtain 

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