On $\sigma$-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules

1. The distortion due to molecular rotation causes the width of a spin multiplet to depend on $j$, and tends to uncouple the spin axis from quantization relative to the axis of figure, thus bringing about a gradual passage from Hund's case (a) to (b). Another rotational effect is the "sigma-type doubling" of spectral lines due to removal of the degeneracy associated with the equality in energy of left- and right-handed axial rotations in stationary molecules. The present paper treats these two effects, especially their interrelation.

2. As a needed mathematical preliminary we calculate the perturbing matrix elements due to the components of angular momentum perpendicular to the figure axis, which are neglected in the usual treatment of the rotating molecule as a symmetrical top. This calculation would be similar to Kronig's and Wigner and Witmer's were it not for inclusion of the spin. This is handled by Pauli's scheme of two wave functions per electron, especially his method of transforming them from one Cartesian system to another by the Cayley-Klein parameters. The results also hold with Dirac's "quantum theory of the electron," as Dirac's quartet of wave functions transform under a rotation like two independent Pauli pairs. Although the orbital and spin angular momentum operators look superficially different, it is shown that their gyroscopic effects enter additively as commonly supposed, and that in the first approximation the effect of the spin is to make the rotational energy (except for an additive constant) that of the symmetrical top with ${\sigma }_l+{\sigma }_s$ in place of ${\sigma }_l$.

3. Neglect of the relatively small sigma-doubling yields identically the formulas for the rotational distortion of spin multiplets which Hill and Van Vleck obtained by a different, alternative method that used Hund's case (b) rather than (a) as the unperturbed system.

4.Singlet $P$ states should exhibit a sigma doubling proportional to $j(j+1\mathrm{})\mathrm{}$ and $D$ states ordinarily a negligible doubling. Our technique of calculation differs slightly from that of Kronig, who obtained a similar result for singlet spectra (as did Hill and V. V. with a simple model), in that the degeneracy is removed in the final rather than in the initial approximation.

5. In ${}_{}{}^2{}_{}{}^{}P$ states the spin profoundly modifies the sigma doubling. In case (b) both spin components should exhibit equal doublings proportional to $j_k(j_k+1)$, but in case (a) the $P_{\frac{3}{2}}$ sigma doubling should be negligible, but the $P_{\frac{1}{2}}$ fairly large and proportional to $j+\frac{1}{2}$. Formulas are developed for the sigma-doublet width which apply throughout the range between (a) and (b). The pronounced doubling of the $P_{\frac{1}{2}}$ component in case (a) is due to a rather complicated superposition of the rotational distortion on the magnetic coupling between the components of spin and orbital angular momentum which are perpendicular to the axis of figure.

6.${}_{}{}^2{}_{}{}^{}S$ states. A similar superposition explains the so-called "rho-type doubling" in ${}_{}{}^2{}_{}{}^{}S$ states whereby levels of like $j_k$ but unlike $j$ differ by small amounts proportional to $j_k+\frac{1}{2}$. This explanation differs from the usual interpretation of this doubling as due to magnetic moment developed by orbital motions, and seems to give a slightly larger effect.

7.${}_{}{}^3{}_{}{}^{}P$ states. In case (b) all three components should have equal doublings proportional to $j_k(j_k+1)$ but in (a) the $P_0$ and $P_1$ doublings should be respectively independent of $j$ and proportional to $j(j+1\mathrm{})\mathrm{}$, while the $P_2$ is negligible. The large splitting of the $P_0$ component, whose exceptional behavior is due to its having $\sigma =0\mathrm{}$, is due largely to magnetic action effective even in a stationary molecule, and so has a different origin than the usual sigma doubling due to rotational distortion.

8.A summary and comparison with experimental data taken from Mulliken's following paper are finally given. There is a striking agreement between theory and experiment on the type of variation with $j$ in the various cases, especially the asymmetrical behavior of the various multiplet components in case (a). The theoretical orders of absolute magnitude are also confirmed, as evidenced by the reasonable values of the frequencies which must be assumed to give the observed separations.