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Rüdenberg’s well-known letter of 1951 entitled “On the Three- and Four-Center Integrals in Molecular Quantum Mechanics” explicitly presents two approximation formulas for four-center repulsion integrals, only. When applied to some types of three-center repulsion integrals, however, these two recipes still imply considerable oversimplifications. Using both one-electron and two-electron routes of Rüdenberg’s truncated expansion, on the other hand, such shortcomings can be avoided strictly. Starting from four simple “Unrestricted and Combined” (U&amp;C) approximation schemes introduced elsewhere, an improved “Restricted and Combined” (R&amp;C) approximation picture for Fock-matrix elements now will be outlined, which does not tolerate any unnecessary oversimplifications. Although the simplicity of the U&amp;C scheme is lost in this case, R&amp;C-approximated Fock-matrix elements still can be constructed from one- and two-center integrals alone in an effective way. Moreover, due to their dependence on a single geometric parameter, all types of two-center integrals can be calculated in advance for about one hundred fixed interatomic distances at the desired level of sophistication and stored once and for all. A cubic spline algorithm may be taken to interpolate the actual integral value from each precomputed list.

On the Restricted and Combined Use of Rüdenberg’s Approximations in Molecular Orbital Theories of Hartree-Fock Type