Particle number and probability density functional theory and A -representability

Abstract

In Hohenberg-Kohn density functional theory, the energy E is expressed as a unique functional of the ground state density ρ (r): E=E [ρ ] with the internal energy component FHK [ρ ] being universal. Knowledge of the functional FHK [ρ ] by itself, however, is insufficient to obtain the energy: the particle number N is primary. By emphasizing this primacy, the energy E is written as a nonuniversal functional of N and probability density p (r): E=E [N,p]. The set of functions p (r) satisfies the constraints of normalization to unity and non-negativity, exists for each N;N=1,∞, and defines the probability density or p-space. A particle number N and probability density p (r) functional theory is constructed. Two examples for which the exact energy functionals E [N,p] are known are provided. The concept of A -representability is introduced, by which it is meant the set of functions ψp that leads to probability densities p (r) obtained as the quantum-mechanical expectation of the probability density operator, and which satisfies the above constraints. We show that the set of functions p (r) of p -space is equivalent to the A -representable probability density set. We also show via the Harriman and Gilbert constructions that the A -representable and N -representable probability density p (r) sets are equivalent. © 2010 American Institute of Physics.