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The origin of non-collinear magnetic order in UO$_{2}$ is studied by an abinitio dynamical-mean-field-theory framework in conjunction with alinear-response approach for evaluating inter-site superexchange interactionsbetween U 5$f^{2}$ shells. The calculated quadrupole-quadruple superexchangeinteractions are found to unambiguously resolve the frustration offace-centered-cubic U sublattice toward stabilization of the experimentallyobserved non-collinear 3k-magnetic order. Therefore, the exotic 3kantiferromagnetic order in UO$_{2}$ can be accounted for by a purely electronicexchange mechanism acting in the undistorted cubic lattice structure. Thequadrupolar short-range order above magnetic ordering temperature $T_N$ isfound to qualitatively differ from the long-range order below $T_N$.

Quadrupolar Superexchange Interactions, Multipolar Order and Magnetic Phase Transition in UO$_2$