Extended Use of the Coulomb Approximation: Mean Powers of $r$, a Sum Rule, and Improved Transition Integrals

Expectation values of $r$ and of $r^2$ have been computed for a range of effective quantum numbers up to 8.5 by use of Coulomb-approximation wave functions. The results show that the hydrogenic formulas for these expectation values are accurate for noninteger quantum numbers $\nu$ to better than 1% for all $\nu -l>\mathrm{}1.0\mathrm{}$, and much greater accuracy is attained for larger $\nu$. The sum rule wherein the squares of the dipole radial transition integrals sum to $〈r^2〉$ can therefore be used with the hydrogenic formula for the latter quantity as an excellent approximation for all but small $\nu$. A number of radial dipole integrals for calculating transition probabilities are obtained for Oi, Oii, Oiii, and Oiv by use of the Hartree-Hartree-Swirles wave functions for the lower state and Coulomb-approximation functions for the excited states. The results are compared with analytic Hartree-Fock and Hartree-Fock-Slater values, and with the results of Stewart and Rotenberg. Some effects of configuration interaction are noted in the case of Oii. Particular attention is paid to the contribution to the dipole integrals from large distances which present calculations omit or evaluate inaccurately. To demonstrate this problem, a few dipole integrals for Oi, Oii, and Ni are computed more accurately.