Theory of Small-Energy-Transfer Collisions in Dominant Long-Ranged Forces: ${\mathrm{H}}^{+}$-${\mathrm{H}}_2$ and $e^{-}$-${\mathrm{H}}_2$ Vibrational Excitation

In the two-state approximation analytic formulas are derived for the inelastic cross section in the limit in which the particle velocity is large relative to the energy defect. These formulas involve a mixing parameter and two eigenphase shifts. For scattering in a $\frac{C\left(X\right)\mathrm{}}{R^n}$ potential, where $X$ is an internal coordinate of the target, the integral cross section is presented in closed form, with parameters given by the elastic-superelastic potential-difference strength, the coupling strength, the velocity, and a cutoff for the singular potential, taken to be the target radius. Results are given for ${\mathrm{H}}^{+}$-${\mathrm{H}}_2$, $e^{-}$-${\mathrm{H}}_2$, and $e^{-}$-${\mathrm{N}}_2$ vibrational excitation. The theory appears to be correct for the high-energy tail of the cross sections.