Bounds to some local electron-pair properties with application to two-electron ions.

Abstract

New bounds to the maximum characteristics ${\mathit{u}}_{\mathrm{max}}$ and ${\mathit{h}}_{\mathrm{max}}$=h(${\mathit{u}}_{\mathrm{max}}$) of the spherically averaged electron-pair density h(u) and to the electron-electron coalescence h(0)=〈\ensuremath{\delta}(u)〉 of a many-electron system are shown in a rigorous manner (i.e., no approximate wave functions were used). The resulting rigorous inequalities also allow one to bound a given interelectronic moment 〈${\mathit{u}}^{\mathrm{\ensuremath{\beta}}}$〉 from above and from below. In particular, an interesting inequality is obtained for the electron-electron repulsion energy ${\mathit{E}}_{\mathit{e}\mathit{e}}$ of an N-electron system: 2\ensuremath{\pi}h(0)${\mathit{u}}_{\mathrm{max}}^{2}$\ensuremath{\le}${\mathit{E}}_{\mathit{e}\mathit{e}}$\ensuremath{\le}3N(N-1)/4${\mathit{u}}_{\mathrm{max}}$. For completeness, just to have an idea of the worth of these results, some of the rigorous inequalities are numerically studied for two-electron ions with nuclear charge Z=1, 2, 3, 4, 5, and 10 using a highly accurate electron-pair density h(u) constructed from the 204-term Hylleraas wave functions. The accuracy is found to increase, generally, with increasing Z and decreasing order \ensuremath{\beta} of the involved moments.