Competing Ionization and Dissociation in the H$_2$ Gerade System

A numerically solvable two-dimensional (2D) model, employed by the authors to study the dissociative recombination of H$_2^+$ in the ungerade symmetry [Phys. Rev. A $\mathbf{98}$, 062706 (2018)], is extended to describe the collision process in the gerade symmetry of H$_2$. In this symmetry the ionization and dissociation processes are driven primarily by the direct, curve-crossing mechanism. The model is represented by a set of three coupled electronic channels in 2D, in the space of $s,p,d$ partial waves of the colliding electron. We demonstrate that the Born-Oppenheimer properties of the H$_2$ molecule in the relevant range of internuclear distances can be described by such a model. The molecular rotational degrees of freedom are accounted for by the rotational frame transformation. The numerical solution of the model is discussed and the resulting rovibrationally inelastic and dissociative recombination cross sections are compared with the available data.