Lieb functional differentiability

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... 

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 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. 

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... 

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. 

As a next step we will show that this implies that the functional FL is Gateaux differentiable at every E-V-density and nowhere else. For the application of the next theorem it is desirable to extend the domain of FL to all of L1 DL3. We follow Lieb 1 and define... 

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. 

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