Analysis of some integrals arising in the atomic three-electron problem

Abstract

A detailed analysis is presented for the evaluation of atomic integrals of the form F ${\mathit{r}}_1^{\mathit{i}}$ ${\mathit{r}}_2^{\mathit{j}}$ ${\mathit{r}}_3^{\mathit{k}}$ ${\mathit{r}}_{23}^{\mathrm{-}2}$ ${\mathit{r}}_{31}^{\mathit{m}}$ ${\mathit{r}}_{12}$ ${}_{}{}^{\mathit{n}}{}_{}{}^{}$ ${\mathit{e}}_1^{\mathrm{-}\mathrm{\alpha }\mathit{r}}$-β${\mathit{r}}_2$-γ${\mathit{r}}_3$d${\mathbf{r}}_1$ d${\mathbf{r}}_2$ d${\mathbf{r}}_3$, which arise in several contexts of the three-electron atomic problem. All convergent integrals with i≥-2, j≥-2, k≥-2, m≥-1, and n≥-1 are examined. These integrals are solved by two distinct procedures. A majority of the integrals can be evaluated by a reduction of the three-electron integrals to integrals arising in the atomic two-electron integral problem. A second approach allows all integrals with the aforementioned indices to be evaluated by the use of Sack’s expansion [J. Math. Phys. 5, 245 (1964)] of the interelectronic separation, which leads to a reduction of the above nine-dimensional integrals to a set of three-dimensional integrals. A discussion is given for the numerical evaluation of the three-dimensional integrals that arise.