Analytic evaluation of three-electron integrals

An analytic formula for the three-electron generating integral I(${\mathrm{\alpha }}_1$,${\mathrm{\alpha }}_2$,${\mathrm{\alpha }}_3$,${\mathrm{\alpha }}_{12}$,${\mathrm{\alpha }}_{23}$,${\mathrm{\alpha }}_{31}$) :=F ${(}_{123122331}$ ${)}^{\mathrm{-}1}$ exp(-${\mathrm{\alpha }}_1$ ${\mathrm{r}}_1$-${\mathrm{\alpha }}_2$ ${\mathrm{r}}_2$-${\mathrm{\alpha }}_{33}$-${\mathrm{\alpha }}_{12}$ ${\mathrm{r}}_{12}$-${\mathrm{\alpha }}_{23}$ ${\mathrm{r}}_{23}$ -${\mathrm{\alpha }}_{31\mathrm{r}31}$)${\mathrm{d}}^3$ ${\mathrm{r}}_1$ ${\mathrm{d}}^3$ ${\mathrm{r}}_{23}^3$ is given which is valid for all values of ${\mathrm{\alpha }}_1$,${\mathrm{\alpha }}_2$,${\mathrm{\alpha }}_3$,${\mathrm{\alpha }}_{12}$ ${\mathrm{\alpha }}_{23}$,${\mathrm{\alpha }}_{31}$ for which this integral converges. A large class of integrals can be evaluated analytically by taking derivatives of I with respect to the α’s. More general integrals whose integrands contain products of spherical harmonics can also be evaluated analytically. In particular, all of the matrix element integrals which arise in a variational calculation on the lithium atom with Hylleraas-type basis functions can be evaluated in closed form. Certain difficult two-electron integrals can be obtained as a limiting case. Two-center two-electron molecular integrals can be obtained via an inverse Laplace transform; this observation is used to discuss the computation and convergence of series expansions for these molecular integrals.