Inelastic Scattering of Electrons by the Hydrogen Molecule Ion

The inelastic scattering of electrons by ${\mathrm{H}}_2^{+}$ is investigated by means of the first Born approximation. Using the exact electronic wave functions, the integral that must be evaluated to find the differential cross section for fixed internuclear separation is studied for an electronic excitation from the ground state to any discrete excited state. Values related to this integral are tabulated for the processes $1s{\sigma }_g-2p{\sigma }_u$, $1s{\sigma }_g-2p{\pi }_u$, $1s{\sigma }_g-2s{\sigma }_g$ at the equilibrium internuclear distance, while the first case is studied at two additional internuclear separations. The corresponding total cross sections are calculated for incident energies up to 400 eV. Assuming that rotational and vibrational levels of the final electronic state cannot be resolved, it is known that an observed cross section necessarily depends on the initial vibrational state of the molecular ion. The effect of two different initial vibrational states ($\nu =0, 3$) is investigated for the $1s{\sigma }_g-2p{\sigma }_u$ case and it is observed that the vibrational state has a marked influence on the total cross section.