Theory of the vibrational structure of resonances in electron-molecule scattering

We derive a simple Hamiltonian representing the coupling of an electronic level of positive energy to the continuum of scattering states as well as to the molecular vibrations. By summing the perturbation series for the $S$ matrix to infinite order in the bound-state-continuum interaction, an effective non-Hermitian boson Hamiltonian describing the dynamics of vibrational motion in the resonance state is obtained. It is shown that the effective Hamiltonian can be diagonalized, yielding explicit expressions for the vibrational excitation cross sections. The theory is applied to two representative examples, the 3.8-eV shape resonance in C${\mathrm{O}}_2$ and the 2.4-eV shape resonance in ${\mathrm{N}}_2$. The results exhibit a clear improvement over those obtained with existing theories, which express the cross sections in terms of conventional Franck-Condon factors. The influence of the anharmonicity of the potential functions on the structure of the vibrational excitation functions is discussed.