“Full numerical” diatomic matrix elements: Simplified shooting method

The problem of diatomic matrix elements Mnn′ = 〈Ψn|Q|Ψn′〉 related to the anharmonic oscillator is considered for standard operators Q of the form x = r – re (r is the radial variable), powers of x, or exponentials, or combinations of such operators; the quantum numbers (n, n′) may be equal or not. A “full numerical” method to determine Mnn′ is presented for any type of the potential U, analytic like that of Morse or numerical like the RKR potential. This numerical method is a simplified version of the standard Cooley shooting method (CSM). The present simplified shooting method (SSM): (1) shoots in one direction only (instead of two); (2) avoids starting problems and matching problems; (3) determines the “end” point automatically (without prior guesses); and (4) reduces thus the number of grid points effectively needed. Examples for analytic (Morse) and RKR potentials are presented. The numerical application to a standard example used by Delgado‐Barrio et al. [J. Comp. Chem., 7, 208 (1986)] using the CSM, and by Kobeissi et al. [J. Comp. Chem., 10, 358 (1989)] using the highly accurate “Canonical Functions” method, shows that when the SSM and CSM are used with the same integrator and the same mesh size the relative discrepancy ΔMnn′(between computed and exact M) is averaged for several (n, n′) to 5.4 × 10−4 for the CSM and to 8.5 × 10−6 for the present SSM. This improvement in accuracy is supplemented by a reduction in computer time consumption. © John Wiley & Sons, Inc.