  Hydrogen adatoms on graphene are investigated using DFT and analytical
approaches. We demonstrate that the level of lattice deformation due to the
hydrogen adsorption does not substantially change the coupling between the
graphene $p_{z}$ orbitals. The hybridization primarily takes place between the
adsorbate's s orbital and the graphene $p_{z}$ orbitals. We also show that the
impurity interaction with the graphene atoms is limited to only a few nearest
neighbors, allowing us to construct a compact TB model for the
impurity-graphene system with an arbitrary impurity distribution. The
complexity of our model scales with the number of impurities, not their
separation, making it especially useful in the study of low impurity
concentrations.
