Levy variational principle

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 ... 

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... 

The outer variation spans over all possible TV-representable densities, that is, densities which belong to Af. Notice, that since the set of u-representable densities Av is a subset of Af, in the Levy variational principle one considers densities such as p(r) and p r) (see Fig. 1) which belong to Af but not to A, namely, densities which are TV-representable but not v-representable. In this sense, the Levy variational principle is more lax than the original Hohenberg-Kohn one it leads, however, to the same extremum. 

The following comments are in order with regard to the Levy variational principle. 

An actual realization of the Levy variational principle for some concrete system has not taken place until now. 

If the kinetic balance condition (5) is fulfilled then the spectrum of the Levy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

P. W. Ayers, S. Golden, and M. Levy, Generalizations of the Hohenberg—Kohn theorem I. Legendre transform constructions of variational principles for density matrices and electron distribution functions. J. Chem. Phys. 124, 054101 (2006). 

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 will prove this theorem using the variational principle in a way shown first by Levy. The variational principle states that... 

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 ... 

Although the use of electronic density to describe the properties of a system started with the Drude s model of homogeneous electron gas in the beginning of XX century, it was only in 1964 that DFT received solid bases with the publication of two theorems by Hohenberg and Kohn." The first theorem states that DFT is an exact theory, i.e., the external potential of an electronic system is uniquely determined by the electron density and, consequently, the total energy and the other observables. The second one establishes the variational principle for DFT. One year later, Kohn and Sham proposed a scheme, which is mainly used to perform DFT calculations. According to Levy, the Kohn-Sham method uses a reference system, in which non-interacting electrons are under the effects of a local effective potential, which provides the same electron density than the real system. 

The variational Dirac-Coulomb and the corresponding Levy-Leblond problems, in which the large and the small components are treated independently, are analyzed. Close similarities between these two variational problems are emphasized. Several examples in which the so called strong minimax principle is violated are discussed. 

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