Rigorous lower bounds to average electron radial and momentum densities for atomic systems.

Abstract

Rigorous lower bounds to the average electron radial density 〈\ensuremath{\rho}〉 in terms of any two moments of the charge density have been found for N-electron systems. Similar bounds to the average electron momentum density 〈\ensuremath{\gamma}〉 are also given by means of any two momentum expectation values. These bounds allow us to establish rigorous inequalities to 〈\ensuremath{\rho}〉 and 〈\ensuremath{\gamma}〉 (which have been recently found to be experimentally measurable) and other fundamental and/or measurable quantities such as the electronic energy, the diamagnetic susceptibility, and the Compton profile peak height J(0). The quality of the bounds is analytically discussed and numerically studied within the Hartree-Fock framework. Best bounds turn out to be (1) 〈\ensuremath{\rho}〉\ensuremath{\ge}〈${r}^{\mathrm{\ensuremath{-}}1}$${〉}^{3}$/9\ensuremath{\pi}N, (2) 〈\ensuremath{\gamma}〉\ensuremath{\ge}8[J${(0)]}^{3}$/9\ensuremath{\pi}N. .AE