Levy-Lieb functional

The functionals FHK and FEHK have the unfortunate mathematical difficulty that their domains of definition A and B, although they are well defined, are difficult to characterize, i.e., it is difficult to know if a given density n belongs to A or B. It is therefore desirable to extend the domains of definition of FHK and FEHK to an easily characterizable (preferably convex) set of densities. This can be achieved using the constrained search procedure introduced by Levy [19]. We define the Levy-Lieb functional FLL as ... 

The differentiability of different functionals used in density-functional theory (DFT) is investigated, and it is shown that the so-called Levy Lieb functional FLL[p] and the Lieb functional FL[p] are Gateaux differentiable at pure-state v-representable and ensemble v-representable densities, respectively. The conditions for the Frechet differentiability of these functionals are also discussed. The Gateaux differentiability of the Lieb functional has been demonstrated by Englisch and Englisch (Phys. Stat. Solidi 123, 711 and 124, 373 (1984)), hut the differentiability of the Levy-Lieb functional has not been shown before. 

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. 

Since Ev is the ground-state energy, it follows that the expression (50) is nonnegative. This implies that the functional is locally convex in the neighborhood of the density pv. In the standard method for proving the differentiability the convexity of the functional is used [5,4,11], Since the Levy-Lieb functional is not necessarily convex, this procedure does not work. The reason that in spite of this fact it has been possible to demonstrate the differentiability here could be connected to the fact that the functional is locally convex in the neighborhood of the points of interest. 

We can then conclude that the Levy—Lieb functional FLL is Gateaux differentiable at any PS-v-representable density and that the derivative can be represented by a multiplicative local function. 

Using the constrained-search procedure, Lieb [5] has in analogy with the Levy-Lieb functional (38) extended the EHK functional (10) to... 

Using the constrained search, the correspondence of the Levy-Lieb functional (38) is for the noninteracting system the minimum of the kinetic energy,... 

Instead of basing the treatment on the Levy-Lieb functional (38), the corresponding result can be obtained by using the Lieb functional (64). 

We have shown that the Lieb functional (69) is Gateaux differentiable at all Zs-v-representable densities, which is consistent with the result of Englisch and Englisch [3,4], who demonstrated the differentiability by using the convexity of the functional. The same procedure is used by van Leeuwen [11], This procedure cannot be used for the Levy-Lieb functional, LLL[p], which is not manifestly convex. According to Englisch and Englisch, the differentiability of this functional is an open question. 

The procedure we have applied does not depend on the (global) convexity of the functional, and we have been able to demonstrate the Gateaux differentiability of the Levy-Lieb functional at all PS-v-representable densities, where this functional is locally convex. It seems plausible that both these functionals are also Frechet differentiable at the same densities, although we have not been able to find a rigorous proof. 

The decomposition of D2 in Eq. (25b) is sometimes called the Levy-Lieb partition of the 2-RDM [57,58]. Formulas essentially equivalent to Eqs. (25a)-(25e) were known long ago, in the context of time-dependent Green s functions [59-61], but this formalism was rediscovered in the present context by Mazziotti [33]. 

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. 

In the degenerate case we can have a situation, where a linear combination of ground-state densities is not necessarily itself a ground-state density. This has the consequence that the HK [equation (11)] and the Levy-Lieb [equation (38)] functionals are not necessarily convex, which for many applications is a disadvantage. A convex functional can be constructed by considering ensemble-v-representable (E-v-representable) densities [3,12,11]... 

We shall now show how the formalism examined above can be used to derive the standard Kohn-Sham scheme. We start by considering the Levy-Lieb energy functional (39), which is minimized under the normalization constraint... 

Above we have assumed that the minimization is carried out within the domain of normalized of densities. Alternatively, we can perform the minimization, using the Euler-Lagrange procedure. Then we use the extension of the functionals valid also outside the normalization domain and enforce the normalization constraint by a Lagrange multiplier.5 For the Levy-Lieb energy functional (70) this leads to... 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

It is illustrative to discuss the reformulation of the Hohenberg-Kohn theory originally carried out by Levy [54] (and later, also by Lieb [55-57]), where instead of the stronger v-representability condition, all that is asked for is compliance with the weaker TV-representability condition for the energy functionals. Our discussion is based on Eq. (18) plus the assumption that Av C M, where Av is the set of u-representable densities (namely, densities coming from ground-state wavefunctions for Hamiltonians // , with t/eV) and J f is the set of iV-representable densities. The latter is explicitly defined by... 

Among others, three conditions are of particular importance. The first constraint is related to the behavior in the small x region, where the GGA exchange functional should reduce to in order to recover the correct uniform gas limit. The second condition was defined by Levy, who showed that some scaling properties can be satisfied if the asymptotic form of the functional for large x is x , where a > Vi [68,69]. The last condition is the so called "Lieb-Oxford bound" [70], which states that ... 

The behavior of some of the most common functionals with respect to these three constraints is reported in table II. The B functional does not obey neither the Levy condition nor the Lieb-Oxford bound, but its numerical performances are better than those provided by the PW functional, which respects all the above mentioned constraints. 

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