Highly accurate evaluation of atomic three-electron integrals of lowest orders

Calculations of three-electron atomic systems in Hylleraas coordinates require integrals involving all the interparticle distances $rij,$ which have usually been evaluated by introducing series expansions. For integrals with the smallest powers of $rij$ these expansions do not converge at a satisfactory rate, leading some investigators to introduce convergence-acceleration procedures. This paper recommends the alternative of evaluating these integrals in closed form and presents stable explicit formulas for so doing. Some of the formulas are more compact versions of those in the literature; others have not been previously reported. It is also shown that finite-difference methods can be used with advantage to obtain additional low-order integrals. Sample integral values have been provided for test purposes.