Evaluation of some integrals for the atomic three-electron problem using convergence accelerators

An analysis is presented for the evaluation of atomic integrals of the form Fr₁ⁱ{r}₂ʲr₃ᵏ{r}₂₃ˡr₃₁ᵐ{r}₁₂ⁿ$ ${e}₁^{{{-}}{{α}}r}-β{r}₂-γ{r}₃d{r}₁d{r}₂d{r}₃$. General formulas are worked out for the two cases (i) l=-2, m{≥}-1, n{≥}-1, and (ii) l=-2, m=-2. A series solution for both cases is obtained. The Levin u transformation and the Richardson extrapolation techniques are employed to obtain a reasonable number of digits of precision for the integral with minimum CPU requirements.