Andrew and Meissner have shown that a first-approximation formula of the theory of Bacher and Goudsmit leads to better results than a second-approximation formula in several spectra with s and p electrons. The first-approximation formulas are not uniquely defined, and the theory as a whole does not account for the effects of strong configuration-interaction, even when this interaction can be estimated with second-order perturbation theory. In O iii experimental data are adequate to demonstrate roughly that errors from these two sources cancel, and thus explain the good agreement.
In spectra with s and d electrons, where configuration interaction is less a factor, two first-approximation formulas are shown to yield better results than higher approximations if Racah’s modification of the theory of Bacher and Goudsmit is used. The superiority is most marked in spectra of the normal atoms, while in spectra of ions the first three approximations are more or less equivalent and all are very accurate. The Slater theory as recently expanded to include polarization effects (i.e., the linear theory) implies the possibility of getting the best significant agreement in a first approximation, but this is not expected in Racah’s modification. It is probably obtained because the parameters increase twice as much with change of a d to an s electron as they do with addition of a d electron and a unit increase in the atomic number. It is not likely that this sort of relationship would be valid in entirely different types of configurations. In the latter it is expected that a second-approximation formula would give best results, but a suitable first approximation could be found by trial and error.
It is shown that the normal spectra follow the same pattern as the spectra of ions if configuration interaction or an irregular variation of the polarization parameters is assumed to cause the d2 configuration to be 3155 K too high above the ds configuration in Ca i, and 6542 K too high in Sr i. The equivalence of the linear theory and the theory of Bacher and Goudsmit theory is therefore, strongly indicated, insofar as effects of third- or higher order perturbations seem not to be observable, even in the normal spectra where they are expected to be strongest.
Third- and higher-order approximations may be required when the Bacher and Goudsmit theory is applied in its original formulation, if the variation of parameters of the linear theory is complicated with respect to change in the degree of ionization. In all our formulas for spectra with s and d electrons, the original theory improves, the higher the order of the approximation. However, it is noted that the agreement obtained with the third-approximation formulas is not as good as expected on the basis of the linear theory if a simple power series variation of the parameters is assumed to apply.
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