  We introduce a $p_{z}-d$ coupling model Hamiltonian for the $\pi$-graphene/Au
bands that predicts a rather large intrinsic spin-orbit (SO) coupling as are
being reported in recent experiments and DFT studies. Working within the
analytical Slater-Koster tight-binding approach we were able to identify the
overlapping orbitals of relevance in the enhancement of the SO coupling for
both, the sublattice symmetric (BC), and the ATOP (AC) stacking configurations.
Our model effective Hamiltonian reproduces quite well the experimental spectrum
for the two registries, and in addition, its shows that the hollow site
configuration (BC), in which the A/B sites remain symmetric, yields the larger
increase of the SO coupling. We also explore the Au-diluted case keeping the BC
configuration and showed that it renders the preservation of the SO-gap with a
similar SO interaction enhancement as the undiluted case but with a smaller
graphene-gold distance.

Materials Science;Mesoscale and Nanoscale Physics