Quasi-static models of barrier suppression have played a major role in our
understanding of the ionization of atoms and molecules in strong laser fields.
Despite their success, in the case of diatomic molecules these studies have so
far been restricted to fields aligned with the molecular axis. In this paper we
investigate the locations and heights of the potential barriers in the hydrogen
molecular ion in an electric field of arbitrary orientation. We find that the
barriers undergo bifurcations as the external field strength and direction are
varied. This phenomenon represents an unexpected level of intricacy even on
this most elementary level of the dynamics. We describe the dynamics of
tunnelling ionization through the barriers semiclassically and use our results
to shed new light on the success of a recent theory of molecular tunnelling
ionization as well as earlier theories that restrict the electric field to be
aligned with the molecular axis