Analytic evaluation of three-electron atomic integrals with Slater wave functions

The formulas of Fromm and Hill and of Remiddi, for the three-electron correlated atomic integral using Slater-type atomic orbitals, are shown to be equivalent in their common region of applicability. The demonstration required the derivation of an identity connecting dilogarithm functions which is either new or not widely known. it is then shown how to modify the more general of these formular (that of Fromm and Hill) to eliminate the necessity of branch tracking in the complex plane and thereby to achieve a straightforwardly computable formalism. By obtaining a formal cancellation of the individual-term singularities in the Fromm-Hill formula, problems of numerical instability are also avoided. The work is validated with a computer program and sample calculations.