  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.
