Application to Density Functionals

As mentioned, in order to be able to apply the variational principle in DFT, it is necessary to extend the definition of the functionals beyond the domain of v-representable densities, and the standard procedure is here to apply the Levy constrained-search procedure [17]. This has led to the functionals known as the Levy-Lieb (FL[p ) and Lieb (FL[p ) functionals, respectively, and we shall now investigate the differentiability of these functionals. This will represent the main part of our paper. 

We consider a general electronic system with the Hamiltonian (1) 

we shall extend these definitions further by relaxing the normalization restriction of the wavefunctions and densities by using the definition of the density (3) also outside the normalization domain. 

Within the normalization domain the energy functional (39) has its minimum equal to the exact ground-state energy Ev of the system, when the density is equal to any of the exact ground-state densities pv, 

We shall now investigate the differentiability of the Levy-Lieb functional (38), and in doing so we shall largely follow the arguments of our recent Comment to the work of Nesbet [10], extended to the more general situation. 

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