Lieb functional convexity

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... 

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 ... 

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 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. 

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]... 

---