Density functional theory-based calculations of the (10,0) zigzag single-walled carbon nanotube with hydrogen chemisorbed exohedrally show that electron pairing and strain minimization lead hydrogen atoms to cluster and preferentially sit in axial configurations. This tendency to confine in highly ordered configurations contrasts with the results we obtain when we employ the widely used force field AIREBO that predicts a preference for a sparse hydrogen distribution. The nature of the frontier orbitals is significantly dependent on the specific configuration of the adsorbate, being either unperturbed delocalized states of the bare nanotube or localized "impurity" states. The infrared absorption spectrum calculated for a model with hydrogen bound both on the surface and at the edges of the nanotube allows an unambiguous assignment of the characteristic features observed for hydrogenated single-walled carbon nanotubes.