The multicenter electron-repulsion integrals appearing in the Hartree-Fock-Roothaan (HFR) equations for molecules are expressed in terms of the overlap integrals and the basic two-center Coulomb or hybrid integrals by the use of expansion formulas for the charge-density over Slater-type orbitals (STOs) obtained by the one of authors (I.I. Guseinov, J. Mel. Struct. (Theochem) 417 (1997) 117), For the calculation of basic integrals the new auxiliary functions are introduced. The multicenter electron-repulsion integrals are calculated for extremely large quantum numbers using recurrence relations for the overlap integrals and auxiliary functions, (C) 2000 Elsevier Science B.V. All rights reserved.