The observation of intrinsic magnetic order in graphene and graphene-based materials relies on the formation of magnetic moments and a sufficiently strong mutual interaction. Vacancies are arguably considered the primary source of magnetic moments. Here we present an in-depth density functional theory study of the spin-resolved electronic structure of (monoatomic) vacancies in graphene and bilayer graphene. We use two different methodologies: supercell calculations with the SIESTA code and cluster-embedded calculations with the ALACANT package. Our results are conclusive: The vacancy-induced extended $pi$ magnetic moments, which present long-range interactions and are capable of magnetic ordering, vanish at any experimentally relevant vacancy concentration. This holds for $sigma$-bond passivated and un-passivated reconstructed vacancies, although, for the un-passivated ones, the disappearance of the $pi$ magnetic moments is accompanied by a very large magnetic susceptibility. Only for the unlikely case of a full $sigma$-bond passivation, preventing the reconstruction of the vacancy, a full value of 1$mu_B$ for the $pi$ extended magnetic moment is recovered for both mono and bilayer cases. Our results put on hold claims of vacancy-induced ferromagnetic or antiferromagnetic order in graphene-based systems, while still leaving the door open to $sigma$-type paramagnetism.
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