Calculation of some integrals for the atomic three-electron problem

An analysis is presented for the evaluation of integrals of the form F ${\mathit{r}}_1^{\mathit{i}}$ ${\mathit{r}}_2^{\mathit{j}}$ ${\mathit{r}}_3^{\mathit{k}}$ ${\mathit{r}}_{23}^{\mathrm{-}2}$ ${\mathit{r}}_{31}^{\mathit{m}}$ ${\mathit{r}}_{12}^{\mathit{n}}$ ${\mathit{e}}_1^{\mathrm{-}\mathrm{\alpha }\mathit{r}\mathrm{-}}$ ${\mathrm{\beta }}_{23}^{\mathit{r}\mathrm{-}\gamma \mathit{r}}$ d${\mathbf{r}}_1$ d${\mathbf{r}}_2$ d${\mathbf{r}}_3$ which arise in the determination of certain properties for atomic three-electron systems. All convergent integrals with i≥-2, j≥-2, k≥-2, m>=-1, and n≥-1 are discussed. These integrals are solved by reduction to one-dimensional quadratures of the form ${\mathcal{F}}_0^1$f(x)w(x) dx, where w(x) is one of the four functions ln[(1+x)/(1-x)], ${\mathit{x}}^{\mathrm{-}1}$ ln[(1+x)/(1-x)], ln[(1+x)/(1-x)] ln(${\mathit{x}}^{\mathrm{-}1}$), and ${\mathit{x}}^{\mathrm{-}1}$ ln[(1+x)/(1-x)] ln(${\mathit{x}}^{\mathrm{-}1}$), and f(x) is a well-behaved function on the interval [0,1]. The polynomials, which are mutually orthogonal over the interval [0,1] for each of the preceding four weight functions w(x), are determined. These polynomials allow specialized numerical quadrature calculations to be performed, which leads to an efficient algorithm for evaluation of the above integrals.