General formula evaluation of the electron-repulsion integral (ERI) and its derivatives over Gaussian-type orbitals. II. ERI evaluation improved

The general recurrence relation (RR) of paper I [J. Chem. Phys. 95, 5198 (1991)] can be improved. This improved RR, with the use of the Head–Gordon–Pople (HGP), is able to take the contraction of Gaussian-type orbitals (GTO’s) efficiently into account for the electron-repulsion-integral (ERI) evaluation. With the use of the RR by Hamilton and Schaefer and by Lindh, Ryu, and Liu (HSLRL), this improved RR makes the evaluation more rapid. Furthermore, the calculation of the roots and weights of the Rys polynomial can be improved. As a result, an extremely rapid code can be obtained for a vector computer. For example, the measured time is 135 ns per one ERI for [HH‖HH] shell block of the uncontracted GTO’s, which is 16 times faster than that of paper I. When the degree of contraction K=2 for (HH‖HH) shell block of the contracted GTO’s, the measured time is 40 ns per one primitive ERI. It is confirmed that the present method is most efficient for the ERI evaluation of GTO’s higher than or equal to f. It is found that both the HGP and the HSLRL RR’s are numerically unstable in their bad cases. Several procedures to avoid their instabilities are discussed.