Three-electron integral in a Gaussian basis set with linear terms

Explicitly correlated Gaussian functions allow an accurate determination of wave functions and energies of a few electron atoms. A recently introduced variant of these functions involving terms linear in the interparticle distances exhibits a faster convergence and correct behavior at coalescence points as shown in pilot calculations for the helium atom. The extension of this method to atoms with more than two electrons requires dealing with a different type of three-electron integral. In this work, we study in detail a generating three-electron integral ∫exp(−∑_i,j=1³ a_ijr_i·r_j)∕(r₁r₂r₃)dr₁dr₂dr₃, present a compact analytical form, and analyze its numerical evaluation.