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 ... 

In this section we will prove that the Lieb functional is differentiable on the set of E-V-densities and nowhere else. The functional derivative at a given E-V-density is equal to — v where v is the external potential that generates the E-V-density at which we take the derivative. To prove existence of the derivative we use the geometric idea that if a derivative of a functional G[n] in a point n0 exists, then there is a unique tangent line that touches the graph of G in a point (n0, G[ 0 ). To discuss this in more detail we have to define what we mean with a tangent. The discussion is simplified by the fact that we are dealing with convex functionals. If G B — TZ is a differentiable and convex functional from a normed linear space B to the real numbers then from the convexity property it follows that for n0,nj 5 and 0 < A < 1 that... 

For the Lieb functional FL we will now prove the following statement ... 

We have seen that the Lieb functional Fh is differentiable at the set of E-V-densities in S and nowhere else. For this reason it is desirable to know a bit more about these densities. The question therefore is which densities are ensemble v-representable In this section we will prove a useful result which will enable us to put the Kohn-Sham approach on a rigorous basis. 

We are now ready to apply the Bishop-Phelps theorem to the Lieb functional Fh. We take e = 1, n0 G S and let —vk correspond to the F-bounded functional L0 of the theorem. According to the theorem we can then find a tangent functional — wk (the Le of the theorem) such that... 

We see that this is simply the Lieb functional with the two-particle interaction omitted. All the properties of the functional Fh carry directly over to Th. The reason is that all these properties were derived on the basis of the variational principle in which we only required that 7 + IV is an operator that is bounded from below. This is, however, still true if we omit the Coulomb repulsion W. We therefore conclude that Th is a convex lower semicontinuous functional which is differentiable for any density n that is ensemble v-representable for the noninteracting system and nowhere else. We refer to such densities as noninteracting E-V-densities and denote the set of all noninteracting E-V-densities by >0. Let us collect all the results for 7) in a single theorem ... 

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... 

We can now demonstrate the Gateaux differentiability of the Lieb functional (64) for all E-v densities using the same procedure as in the previous section. We know that the energy functional (65) has its lowest value when all functions belong to the ground state. It then follows that the Lieb functional for an E-v density becomes... 

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. 

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