Bounds to the central electron-pair density with applications to two-electron atoms.

Abstract

The local electron-electron quantity 〈\ensuremath{\delta}(${\mathbf{r}}_{12}$)〉 of atomic systems, which plays an important role in the realization of the so-called correlation cusp condition, as well as in the description of the relativistic and radiative corrections to the ground-state energy, is investigated by means of the interelectronic radial expectation values 〈${\mathit{r}}_{12}^{\mathrm{\ensuremath{\alpha}}}$〉, \ensuremath{\alpha}g-3. Starting from the unimodal character of the spherically averaged electron-pair density h(${\mathit{r}}_{12}$) and the recently found inequality h(${\mathit{r}}_{12}$)gh'(${\mathit{r}}_{12}$), upper ${\mathit{U}}_{\mathit{k}}$ and lower ${\mathit{L}}_{\mathit{k}}$ bounds, k=0,1,2,..., to the electron-pair density at the origin or central electron-pair density h(0), which is equal to the quantity 〈\ensuremath{\delta}(${\mathbf{r}}_{12}$)〉, are found analytically. For the two-electron atoms with Z=1, 2, 3, 5, and 10, the quality of these bounds is analyzed by means of the optimum 20-term Hylleraas-type wave functions recently obtained by two of us [T. Koga and K. Matsui, Z. Phys. D (to be published)]. It is shown that the accuracy of both types of bounds increases with k for fixed Z, being greater for the lower bounds than for the upper bounds when k is small. Both bounds are of similar quality for big values of k. Moreover, any desired accuracy may be reached by increasing the k value.