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

A detailed analysis is presented for the evaluation of atomic integrals of the form Fr₁ⁱ{r}₂ʲr₃ᵏ{r}₂₃^{{{-}}2}r₃₁ᵐ{r}₁₂$ $ⁿe₁^-αr-{β}r₂-{γ}r₃dr₁ dr₂ dr₃, 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.