  We study orbital and spin-orbit proximity effects in graphene adsorbed to the
Cu(111) surface by means of density functional theory (DFT). The proximity
effects are caused mainly by the hybridization of graphene $\pi$ and copper d
orbitals. Our electronic structure calculations agree well with the
experimentally observed features. We carry out a graphene-Cu(111) distance
dependent study to obtain proximity orbital and spin-orbit coupling parameters,
by fitting the DFT results to a robust low energy model Hamiltonian. We find a
strong distance dependence of the Rashba and intrinsic proximity induced
spin-orbit coupling parameters, which are in the meV and hundreds of $\mu$eV
range, respectively, for experimentally relevant distances. The Dirac spectrum
of graphene also exhibits a proximity orbital gap, of about 20 meV.
Furthermore, we find a band inversion within the graphene states accompanied by
a reordering of spin and pseudospin states, when graphene is pressed towards
copper.
