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

Abstract

An 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}}$${\mathit{r}}_{23}^{\mathit{l}}$${\mathit{r}}_{31}^{\mathit{m}}$${\mathit{r}}_{12}^{\mathit{n}}$ ${\mathit{e}}_{1}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}\mathit{r}}$-\ensuremath{\beta}${\mathit{r}}_{2}$-\ensuremath{\gamma}${\mathit{r}}_{3}$d${\mathbf{r}}_{1}$d${\mathbf{r}}_{2}$d${\mathbf{r}}_{3}$. General formulas are worked out for the two cases (i) l=-2, m\ensuremath{\ge}-1, n\ensuremath{\ge}-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.