Levy-Lieb functional differentiability

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 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 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. 

---