Consequences of extending 1‐matrix energy functionals from pure–state representable to all ensemble representable 1 matrices

TLDR: In this article, the authors extended Levy's energy functional to include all ensemble-representable l-matrices in its domain, which constitutes both a generalization and a simplification of earlier observations by Gilbert.

Abstract: Levy’s 1‐matrix energy functional (Ref. 4) is modified and extended to include all ensemble–representable l‐matrices in its domain. This constitutes both a generalization and a simplification of earlier observations by Gilbert. The generalization negates some criticisms of the Donnelly–Parr analysis of a 1‐matrix energy functional (Ref. 3) since it was assumed to be defined only for pure‐state representable 1‐matrices. Further study of this analysis suggests that the taking of arbitrary variations in the 1‐matrix at a certain point in their study may result in an invalid Euler equation for the 1‐matrix energy functional, although this does not alter the main results of that work. The redefinition of Levy’s functional to accommodate the larger domain allows direct application of Harriman’s analysis of the geometry of density matrices in finite dimensional situations. A decomposition of the equivalent of the Vee expectation value into two terms, one with explicit and another with implicit occupation number dependence, illustrates the role of N‐representability. The Vee expectation value is approximated by means of the Schwartz inequality.