  Density functional calculations are employed to study the molecular
dissociation of hydrogen on graphene, the diffusion of chemisorbed atomic
species, and the electronic properties of the resulting hydrogen on graphene
system. Our results show that applying stress to the graphene substrate can
lower the barrier to dissociation of molecular hydrogen by a factor of six, and
change the process from endothermic to exothermic. These values for the barrier
and the heat of reaction, unlike the zero stress values, are compatible with
the time scales observed in experiments. Diffusion, on the other hand, is not
greatly modified by stress. We analyse the electronic structure for
configurations relevant to molecular dissociation and adsorption of atomic
hydrogen on a graphene single layer. An absolute band gap of 0.5 eV is found
for the equilibrium optimum configuration for a narrow range of coverages
($\theta \approx 0.25$). This value is in good agreement with experiment [Elias
et al., Science {\bf 323}, 610 (2009)].
