Theory of electron-$\mathrm{H}_2^{}{}_{}{}^{+}$ dissociative collisions

Two terms of the large-collision-energy expansion of the first Born cross section, the Bethe-Born (BB) cross section, are presented for ionization and for the total inelastic scattering in electron-$\mathrm{H}_2^{}{}_{}{}^{+}$ collisions. An approximation made in an earlier calculation of the total inelastic BB cross section, which is known to produce an error in the second BB coefficient, is removed. Knowledge of these two theoretical cross sections makes possible a comparison of theory with the experimentally measured proton production, total dissociation, dissociative-excitation, and ionization cross sections for large collision energies. The theoretical dissociative-excitation cross section is calculated in the first Born approximation. Comparison between theory and experiment for all collision energies is accomplished by use of the Born dissociative-excitation cross section appropriately combined with the experimental ionization cross section. Experiment and theory agree within the expectations for first Born and BB theories. The measurements of the various cross sections from the different laboratories are shown to be mutually consistent.