  Recent experiments on isolated Co clusters have shown huge orbital magnetic
moments in comparison with their bulk and surface counterparts. These clusters
hence provide the unique possibility to study the evolution of the orbital
magnetic moment with respect to the cluster size and how competing interactions
contribute to the quenching of orbital magnetism. We investigate here different
theoretical methods to calculate the spin and orbital moments of Co clusters,
and assess the performances of the methods in comparison with experiments. It
is shown that density functional theory in conventional local density or
generalized gradient approximations, or even with a hybrid functional, severely
underestimates the orbital moment. As natural extensions/corrections we
considered the orbital polarization correction, the LDA+U approximation as well
as the LDA+DMFT method. Our theory shows that of the considered methods, only
the LDA+DMFT method provides orbital moments in agreement with experiment, thus
emphasizing the importance of dynamic correlations effects for determining
fundamental magnetic properties of magnets in the nano-size regime.
