Accompanying coordinate expansion formulas derived with the solid harmonic gradient

A series of accompanying coordinate expansion (ACE) formulas for calculating the electron repulsion integral (ERI) over both generally and segmentally contracted solid harmonic (SH) Gaussian-type orbitals (GTOs) can be rederived by the use of the modified operator (called solid harmonic gradient here) of the spherical tensor gradient of Bayman and the reducing solid harmonic gradient defined in this article. The final general formulas contain the reducing mixed solid harmonics defined in a previous article [Ishida, K. J Chem Phys 1999, 111, 4913] and the reducing triply mixed solid harmonics defined previously [Ishida, K. J Chem Phys 2000, 113, 7818]. Each general formula in the series is named ACEb1k1, ACEb2k3, or ACEb3k3. New general algorithm can be obtained inductively from the general formula named ACEb2k3, in addition to the previously developed ACEb1k1 and ACEb3k3. For calculating ERI practically, we select one of these ACE algorithms, as it gives the minimum floating-point operation (FLOP) count. Theoretical assessment by the use of the FLOP count is performed for the (LL/LL) class of ERIs over both generally and segmentally contracted SH-GTOs (L = 1-3). It is found that the present ACE is theoretically the fastest among all rigorous methods in the literature.