Mutual neutralization of H$^-$ with H$^+$ and H$_2^+$*

We have previously studied the low energy mutual neutralization reaction in collisions of H$^+$ and H$^-$ using an ab initio molecular close-coupling approach. The reaction is driven by non-adiabatic couplings between the ion-pair state and the $n=2$ and $n=3$ covalent states at large internuclear distances. In present work we have calculated all states of $^1\Sigma_g^+$, $^1\Pi_g$, $^1\Sigma_u^+$ and $^1\Pi_u$ symmetries associated with the limits $n\leq4$ including the ion-pair states. We have computed adiabatic potential energy curves, non-adiabatic couplings, as well as the rotational couplings between the $\Sigma$ and $\Pi$ states. The goal is to perform a detailed study and investigate the importance of the higher lying states, rotational couplings as well as non-zero asymptotic non-adiabatic couplings. These effects have previously not been considered. To perform an ab initio study of mutual neutralization in collisions of H$^-$ with H$_2^+$ is a challenge. The multi-dimensional potential energy surfaces as well as non-adiabatic couplings for many excited electronic states have to accurately be computed. We perform full configuration interaction calculations of the states to identify the important avoided crossings/conical intersections driving the reaction.