The N‐particle wave function as a homogeneous functional of the density

Abstract

ABSTRACT:It is shown that requiring consistency with the structure of the equation that determines the wave function associated to a density ͑r ៝͒ by density-functional theory, yields the N-particle wave function as a degree-half homogeneous functional of the density, and leads to a separation A͓N, ͔ of N dependence (with N ϭ ͵ ͑r ៝͒ dr ៝) of density functionals A͓͔ ϭ A͓ ͵ , ͔ for which A͓ ͵ , ͔ ϭ A͓ ͵ , ͔; as a consequence of the linearity of quantum mechanical operators. This implies that the ground-state value of any quantum mechanical observable arises naturally as a degree-one homogeneous N-particle density functional. This general scheme for the structure of density functionals can be considered as the conceptual generalization of the Weizsäcker functional, which is the exact degree-one homogeneous one-particle kinetic-energy density functional.

Treatments of the Exchange Energy in Density-Functional Theory

“…(54) It is worth noting that requiring N-particle density functionals to be homogeneous of some degree eliminates the ambiguity present in their definition [15], meaning that the only possible N-particle noninteracting kinetic-energy and exchangeenergy density functionals homogeneous of degree one and two, respectively, are those constructed in [14] and here (see also the Appendix). Finally, the question of separability [41] for N-particle density functionals may be worth taking a look at, referring also to a recent work [42], which investigates the separability problem of (explicitly) N-dependent density functionals.…”

“…which, for a ρ( ) v r , gives the minimum of the sum T (15) over the set of normalized Slater determinants (ψ D ) that yield ρ( ) v r , that is,…”

“…It is worth noting that requiring N-particle density functionals to be homogeneous of some degree eliminates the ambiguity present in their definition [15],…”

“…Beside the HFKS scheme, another mixture of DFT and the Hartree-Fock theory, the Kohn-Sham formulation of the Hartree-Fock scheme, will also be considered. In the second part of the paper, the concept of N-particle exchange-energy density functionals will be introduced, following also [14] (see also [15]), and it will be shown that they emerge naturally as second-degree homogeneous functionals, for every particle number N. For simplicity, spin is not considered throughout the paper.…”

Density functional for the ground-state energy of artificial two-electron atoms with harmonic confinement

“…for the CWB ground-state density. It can be observed that rewriting equation (11) in the form h2 16m ρ (r) + h2 8m…”