OnΛ-Type Doubling inΠ3States of Diatomic Molecules Intermediate Between Hund's Casesaandb

Abstract

Theoretical formulas based on the work of Van Vleck are developed for the $\Lambda$-type doubling in ${}_{}{}^3{}_{}{}^{}\Pi$ states with coupling intermediate between Hund's cases $a$ and $b$. Heretofore only the limiting cases $a$ and $b$ had been considered for triplet states. It is shown that, in addition to the usual splitting caused by the $\Pi -\Sigma$ interaction, there is a contribution to the doubling of the ${}_{}{}^3{}_{}{}^{}\Pi _0^{}{}_{}{}^{}$ component which arises from the magnetic spin-spin coupling between individual electrons. The expressions for the doubling necessitate the numerical solution of a cubic equation for each value of $J$ but are otherwise simple and easy of application. Comparison with experiment is made in the case of ${\mathrm{N}}_2$ where good numerical agreement is found. The fine structure of ${}_{}{}^3{}_{}{}^{}\Sigma$ states which is called $\rho$-type tripling is also considered. It is shown that this fine structure can arise from perturbation by $\Pi$ states—a sort of reflection of the $\Lambda$-type doubling in these states. Curiously, it is found that the formulas which are derived in this manner give separations depending on the rotational quantum number $K$ in exactly the same way as the formulas of Kramers which were obtained from quite a different origin, viz., spin-spin interaction and coupling of the spin to the rotational magnetic moment of the molecule.