  The magnetic state of UO$_2$ was determined experimentally to be
anti-ferromagnetic. Starting from this experimental fact, researchers have
calculated other properties within the Hubbard-corrected density-functional
theory, DFT+U. Up to now, the Hubbard parameters for UO$_2$ were usually so
chosen that the calculations give good results for some experimental data.
Also, to our knowledge there exists no valid theoretical research report on the
energetically stable magnetic state of this system. In present work, employing
the new method which is based on density-functional perturbation theory, we
have determined self-consistently the Hubbard parameters and ground-state
energies for UO$_2$ crystal in both ferromagnetic and anti-ferromagnetic
configurations, and the calculated results show that UO$_2$ crystal
energetically favors an anti-ferromagnetic state with a small energy
difference. In all the calculations the PBE-sol approximation was used for the
exchange-correlation energy functional.
