Three-electron integral evaluation in the transcorrelated method using a frozen Gaussian geminal

Formulas are developed for evaluating the three-electron integrals, which appear in the transcorrelated method using a frozen Gaussian geminal. First, we derive expressions with intermediates, in which the angular momentum indices of the inter-electronic operators are temporarily increased. Following the inspection on the factorization property, we further derive more straightforward recurrence relations corresponding to the Obara–Saika one for electron repulsion integrals. It is shown that the relations can be applied outside the contraction loops over geminal primitives by introducing three-indexed auxiliary integrals.