## Abstract

By using the impact parameter formulation $\mathrm{H}^+_2(v'_0)-\mathrm{H}_2(v''_0)$ collisions leading to H$^+_2$(v')-H$_2$(v'') or to H$_2$($v''$)-H$^+_2$($v'$) are treated. It is assumed that the ion and the molecule are in the ground electronic state so that only two electronic states of the complex are involved. The electronic interaction is evaluated by a simple extension of the method of Firsov (1951). Taking it to be spherically symmetrical, the problem effectively reduces to solving a set of coupled first order differential equations, the coefficients of which depend on the H$^+_2$ - H$_2$ vibrational overlap integrals. Cross-section curves for vibrational excitation with (X) and without (D) charge transfer are presented. If the impact velocity V is low, corresponding D and X cross-sections are about equal, but in the region of high V the former tends to fall off more rapidly than the latter. The total charge transfer cross-section in H$^+_2$(0) - H$_2$(0) collisions, Q$^X$(00 $\rightarrow \Sigma|$V) is calculated. As V is increased from zero Q$^X$(0 0 $\rightarrow \Sigma\mid$V) first shows the slow fall off characteristic of symmetrical resonance charge transfer; but it reaches a minimum when V is near 1 x 10$^{-1}$ a.u., then rises and passes through a maximum. This behaviour arises from the increase in the relative contribution to charge transfer from the inelastic collisions. To facilitate comparison with experiment a study is also made of charge transfer when the H$^+_2$ ions are distributed among the vibrational levels as they would be if they were produced by the impact of fast electrons on H$_2$ molecules. The calculations on the weighted mean total charge transfer cross-section $\overline{Q}^X(\Sigma\mid V)$ are confined to the low and high velocity limits. However, they are sufficient to indicate that Q$^X$(00 $\rightarrow \Sigma\mid$V) and $\overline{Q}^X(\Sigma\mid V)$ are not markedly different. The predicted form of the cross-section curve is consistent with the laboratory data available.