  The electronic structure of selected rare-earth atoms adsorbed on a
free-standing graphene was investigated using methods beyond the conventional
density functional theory (DFT+U, DFT+HIA and DFT+ED). The influence of the
electron correlations and the spin-orbit coupling on the magnetic properties
has been examined. The DFT+U method predicts both atoms to carry local magnetic
moments (spin and orbital) contrary to a nonmagnetic $f^6$ ($J=0$) ground-state
configuration of Sm in the gas phase. Application of DFT${}+{}$Hubbard-I (HIA)
and DFT${}+{}$exact diagonalization (ED) methods cures this problem, and yields
a nonmagnetic ground state with six $f$ electrons and $J=0$ for the Sm adatom.
Our calculations show that Nd adatom remains magnetic, with four localized $f$
electrons and $J=4.0$. These conclusions could be verified by STM and XAS
experiments.
