A new horizontal recurrence relation (HRR) is derived which may be used in the generation of two-electron repulsion integrals (ERIs). This work focuses primarily on the application of the new HRR to the 1988 Head-Gordon and Pople method and will be complementary to the more recent Gill, Head-Gordon and Pople procedures applicable to highly contracted basis sets. Special Cartesian coordinate axes may be used in reducing the number of nonzero intermediates dramatically. In direct SCF it may not be desirable to back-transform integrals, but evaluate the contributions to the Fock matrix for many classes of ERIs in the same special axis system, and then back-transform the Fock matrix contribution. Integral derivative evaluation is also discussed. Both the new HRR and the Head-Gordon and Pople HRR may be used in relating ERI derivatives. © 1991.