On the Quantum Mechanics of the Rotational Distortion of Multiplets in Molecular Spectra

Abstract

Most of the paper is on the effect of molecular rotation on spin multiplets, but the last section 5 considers the rather different subject of "$\ensuremath{\sigma}$-type doubling." 1.Hund's limiting cases (a) and (b), those commonly considered, are realized when the coupling between the electronic spin and the molecular axis of figure is very strong or very weak. Preparatory to the new analysis for the intermediate case, amplitude matrices are given both for cases (a) and (b), including phase factors in (b).2.The general intermediate case can be handled (except for purely algebraic difficulties) both for frequencies and intensities by our mathematical method. The procedure is to start with case (b) and to introduce a coupling energy which is proportional to the cosine of the angle between the axis of electronic spin $s$ and the molecular axis of figure, and which when increased adaibatically converts the system over into case (a). The formula for the energy $W$ is the solution of an algebraic equation of order $2s+1$. Approximate solutions are given for nearly case (b) with any $s$ and for nearly case (a) in triplet spectra ($s=1$).3.The doublet case ($s=\frac{1}{2}$) is particularly satisfactory as (unlike the old quantum theory) the analysis yields a simple closed formula for the energy, viz., $W=[{(j+\frac{1}{2})}^{2}\ensuremath{-}{{\ensuremath{\sigma}}_{k}}^{2}\ifmmode\pm\else\textpm\fi{}\frac{1}{2}{{4{(j+\frac{1}{2})}^{2}+\ensuremath{\lambda}(\ensuremath{\lambda}\ensuremath{-}4){{\ensuremath{\sigma}}_{k}}^{2}}}^{\frac{1}{2}}](\frac{{h}^{2}}{8{\ensuremath{\pi}}^{2}I})$ which holds throughout the interval from (a) to (b) both for regular and inverted multiplets. Here $\ensuremath{\lambda}$ is an abbreviation for $\frac{8{\ensuremath{\pi}}^{2}\mathrm{AI}}{{h}^{2}}$, where $A$ is the proportionality factor in the magnetic coupling energy $\mathrm{As}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\sigma}}_{k}$. This formula yields an adiabatic correlation of energy levels in case (a) with those in case (b) which is precisely that predicted by Hund and Kemble including the anomalous behavior of the component $j={\ensuremath{\sigma}}_{k}\ensuremath{-}\frac{1}{2}$ in "regular" multiplets ($A>0$). The agreement with the experimental doublet widths in the OH band 2811 is slightly better than in the old quantum theory. Intensity formulas are given which apply throughout the range from (a) to (b). Here account is taken of the fact that the moment of inertia I and coupling constant A are different in the initial and final states if there are changes in "electronic" quantum numbers.4.Simple special cases of the doublet intensity formulas arise when there is (a) type coupling in the initial states and (b) in the final or vice versa. The $^{2}P\ensuremath{\rightarrow}^{2}S$ bands usually meet this condition and formulas for them are developed; the main new result is that for a given initial state the transitions ending on ${j}_{k}=j\ensuremath{-}\frac{1}{2}$ and ${j}_{k}=j+\frac{1}{2}$ are of equal intensity. As another illustration intensity formulas are given for $^{2}D_{a}\ensuremath{\rightarrow}^{2}P_{b}$.5.An elementary theory of $\ensuremath{\sigma}$-type doubling is developed by using mathematics very similar to that in the preceding but introducing adiabatically a coupling proportional to the square rather than first power of the cosine of the angle between an angular momentum vector $k$ and a "core" consisting of the non-gyroscopic "dumb-bell" molecular model. In a stationary molecule the sign of $\ensuremath{\sigma}$ is arbitrary and if $\ensuremath{\sigma}\ensuremath{\ne}0$ there are two states of identical energies. It is shown that actually the rotation removes this degeneracy and creates a small splitting into two levels for a given value of ${\ensuremath{\sigma}}^{2}$ which Mulliken calls "$\ensuremath{\sigma}$-type doubling." Kronig's result is obtained that the doubling is smaller for large ${\ensuremath{\sigma}}^{2}$. The combination relations predicted by Hulth\'en and treated mathematically by Kronig are shown to apply exactly even when the perturbing effect of the angular momentum perpendicular to the figure axis in case (a) is considered, and the coupling is no longer rigorously of type (a). Interaction of $\ensuremath{\sigma}$-type degeneracy with the spin is reserved for a later paper, so section 5 applies primarily to singlets.