  We present first-principles calculations of the electronic band structure and
spin-orbit effects in graphene functionalized with methyl molecules in dense
and dilute limits. The dense limit is represented by a 2$\times$2 graphene
supercell functionalized with one methyl admolecule. The calculated spin-orbit
splittings are up to $0.6$ meV. The dilute limit is deduced by investigating a
large, 7$\times$7, supercell with one methyl admolecule. The electronic band
structure of this supercell is fitted to a symmetry-derived effective
Hamiltonian, allowing us to extract specific hopping parameters including
intrinsic, Rashba, and PIA (pseudospin inversion asymmetry) spin-orbit terms.
These proximity-induced spin-orbit parameters have magnitudes of about 1 meV,
giant compared to pristine graphene whose intrinsic spin-orbit coupling is
about 10 $\mu$eV. We find that the origin of this giant local enhancement is
the $sp^3$ corrugation and the breaking of local pseudospin inversion symmetry,
as in the case of hydrogen adatoms. Also similar to hydrogen, methyl acts as a
resonant scatterer, with a narrow resonance peak near the charge neutrality
point. We also calculate STM-like images showing the local charge densities at
different energies around methyl on graphene.
