  The origin of non-collinear magnetic order in UO$_{2}$ is studied by an ab
initio dynamical-mean-field-theory framework in conjunction with a
linear-response approach for evaluating inter-site superexchange interactions
between U 5$f^{2}$ shells. The calculated quadrupole-quadruple superexchange
interactions are found to unambiguously resolve the frustration of
face-centered-cubic U sublattice toward stabilization of the experimentally
observed non-collinear 3k-magnetic order. Therefore, the exotic 3k
antiferromagnetic order in UO$_{2}$ can be accounted for by a purely electronic
exchange mechanism acting in the undistorted cubic lattice structure. The
quadrupolar short-range order above magnetic ordering temperature $T_N$ is
found to qualitatively differ from the long-range order below $T_N$.
