Three- and four-electron integrals involving Gaussian geminals: fundamental integrals, upper bounds and recurrence relations

We report the three main ingredients to calculate three- and four-electron integrals over Gaussian basis functions involving Gaussian geminal operators: fundamental integrals, upper bounds, and recurrence relations. In particular, we consider the three- and four-electron integrals that may arise in explicitly-correlated F12 methods. A straightforward method to obtain the fundamental integrals is given. We derive vertical, transfer and horizontal recurrence relations to build up angular momentum over the centers. Strong, simple and scaling-consistent upper bounds are also reported. This latest ingredient allows to compute only the $\order{N^2}$ significant three- and four-electron integrals, avoiding the computation of the very large number of negligible integrals.