We study in three dimensions a diatomic homo-nuclear molecule with many bound states incident upon a potential barrier. We consider different initial states for the molecule and take into account transitions between rotational states of the molecule during the process in which the entire molecule tunnels past or is reflected by the external potential barrier. We show the manner in which the transmission resonances are affected by having many bound states. We find that the transmission probability profile in three dimensions is markedly different from that for two bound states in three dimensions, and for many bound states in one dimension. We find important new results. One result is that resonant or near resonant tunnelling and resonant or near resonant reflection can both occur within a small energy range of the molecule. We also find that tunnelling with transmission probability close to unity can occur over a wide energy range.

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