Theory of the Fine Structure of the Molecular Oxygen Ground State

Abstract

A rather complete solution for the fine-structure problem in the oxygen molecule is given in the framework of the Born-Oppenheimer approximation. The reduction of the effect of the electronic state on the fine structure to an effective Hamiltonian, involving only the resultant electronic spin in addition to rotational and vibrational quantum numbers, is demonstrated. In this Hamiltonian the parameters $\ensuremath{\lambda}$ and $\ensuremath{\mu}$ measure the effective coupling of the spin to the figure axis and the rotational angular momentum, respectively. The contributions to these parameters which are diagonal in electronic quantum numbers, namely ${\ensuremath{\lambda}}^{\ensuremath{'}}$ and ${\ensuremath{\mu}}^{\ensuremath{'}}$, are evaluated by using an expression for the electronic wave function as a superposition of configurations. It turns out that ${\ensuremath{\lambda}}^{\ensuremath{'}}$ gives almost all of $\ensuremath{\lambda}$, whereas ${\ensuremath{\mu}}^{\ensuremath{'}}$ gives only 4 percent of $\ensuremath{\mu}$. The second-order contributions of spin-orbit coupling and rotation-induced electronic angular momentum to $\ensuremath{\lambda}$ and $\ensuremath{\mu}$, and the electronic contribution to the effective moment of inertia are related to each other and to certain magnetic effects to be given later. This interrelation enables them all to be essentially evaluated experimentally.The effective Hamiltonian is diagonalized through terms in ${(\frac{B}{\ensuremath{\hbar}\ensuremath{\omega}})}^{2}$ and the eigenvalues compared with the experimental spectra. The fitting establishes the constants: $\ensuremath{\mu}=252.67\ifmmode\pm\else\textpm\fi{}0.05$ Mc/sec; ${\ensuremath{\lambda}}_{e}=59386\ifmmode\pm\else\textpm\fi{}20$ Mc/sec; ${\ensuremath{\lambda}}_{1}={[\frac{\mathrm{Rd}\ensuremath{\lambda}}{\mathrm{dR}}]}_{e}=16896\ifmmode\pm\else\textpm\fi{}150$ Mc/sec; ${\ensuremath{\lambda}}_{2}={[(\frac{{R}^{2}}{2})(\frac{{d}^{2}\ensuremath{\lambda}}{d{R}^{2}})]}_{e}=(5\ifmmode\pm\else\textpm\fi{}2)\ifmmode\times\else\texttimes\fi{}{10}^{4}$ Mc/sec; ${\ensuremath{\lambda}}_{\mathrm{eff}}(v=0)=19501.57\ifmmode\pm\else\textpm\fi{}0.15$ Mc/sec. The transformation that diagonalizes the Hamiltonian is given with respect to both Hund case (a) and case (b) bases. These transformations are applied to matrix elements of ${S}_{Z}$. The results are tabulated and applied to calculate the exact intensity factors for spectral lines. This calculation shows slight deviations from the usual case (b) results for allowed lines and predicts quite sizeable intensities for the "forbidden" $\ensuremath{\Delta}K=2$ lines.