  First-principles calculations of the spin-orbit coupling in graphene with
hydrogen adatoms in dense and dilute limits are presented. The chemisorbed
hydrogen induces a giant local enhancement of spin-orbit coupling due to $sp^3$
hybridization which depends strongly on the local lattice distortion. Guided by
the reduced symmetry and the local structure of the induced dipole moments we
use group theory to propose realistic minimal Hamiltonians that reproduce the
relevant spin-orbit effects for both single-side semihydrogenated graphene
(graphone) and for a single hydrogen adatom in a large supercell. The principal
linear spin-orbit band splittings are driven by the breaking of the local
pseudospin inversion symmetry and the emergence of spin flips on the same
sublattice.
