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 us to compute only the significant three- and four-electron integrals, avoiding the computation of the very large number of negligible integrals.
Note that, although Slater was the first to propose such a correlation factor, he suggested to set in order to ensure that the wave function fulfills Kato’s electron-electron cusp condition. However, Hartree and Ingman84 found this correlation factor physically unreasonable due to its behavior at large r12 and suggested that a correlation factor of the form (with ) would be more appropriate. We refer the interested reader to the review of Hattig et al.14 for a detailed historical overview.
Komornicki and King mentioned the crucial importance of an effective integral screening in Ref. 57.