Representability Conditions

Abstract

In his paper titled “The Convex Structure of Electrons” John Coleman [22] described how convexity entered the discussion of N-representability: “At one of those memorable mid-night sessions at Sanibel when we were discussing N-representability, Per-Olov sprang forward, took the chalk from my hand and demanded — ‘Have you considered the more general problem of ensemble representability?’”A non-pure state being a convex combination of pure, the collection of all reduced density matrices — those for both pure and non-pure states — forms a convex set. Since this mid-night meeting many have resorted to convex structure as a means of interpreting their results on N-representability. Certainly one of the most striking results along these lines is Coleman’s proof [23] that an arbitrary antisymmetrized geminal power wave function “covers” an extreme point in the convex set of reduced density matrices. With this theorem the AGP wave function achieved a special status in the theory. It is interesting that the only large families of extreme points that are known correspond either to Slater determinant states, AGP states or correspond to wave functions generated by taking Grassman products of AGP functions with Slater determinant functions.