Semiclassical Theory for Molecular Autoionization. II. Classical Treatment of the Rydberg Electron

The spontaneous ionization of molecular Rydberg states by vibrational-translational or rotational-translational conversion results in electrons ejected with close to zero energy. This condition means that, without serious initial-final trajectory discrepancies, the Rydberg electron can be treated to a good approximation as a classical object, moving in a closed Kepler orbit, calculated in the Coulomb potential located at the center of mass of the molecular ionic core. The interactions which produce ionization are the monopole contributions of the nuclei in the region between the inner turning point and the radius of the ionic core, $R_I\le R\le \frac{m_2}{(m_1+m_2)X}$, and the multipole moment and polarization potentials in the region between the core radius and the outer turning point, $\frac{m_2}{(m_1+m_2)X}\le R\le R_0$, plus additional interactions from the short-ranged part of the Hartree potential in both regions. The rate of ejection is given by the probability of ejection per period times the inverse period, $P(i\to j){\tau }^{-1\mathrm{}}$, for $i$, $j$ rotational-vibrational states. ${\tau }^{-1\mathrm{}}$ is proportioned to $n^{-3\mathrm{}}$, where $n$ is the principal quantum number of the Rydberg electron; thus the product $P{\tau }^{-1\mathrm{}}$ obeys the "propensity" rule for multi-quantum-mechanical vibrational transitions and the $n^{-3\mathrm{}}$ rule deduced from the normalization of Rydberg states in fully quantum-mechanical treatments. Rates for ${\mathrm{H}}_2$ are calculated in the Born and two-state approximations and compared with experimental and other theoretical results.