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To explore the formation of noncollinear magnetic configurations in materialswith strongly correlated electrons, we derive a noncollinear LSDA+$U$ modelinvolving only one parameter $U$, as opposed to the difference between theHubbard and Stoner parameters $U-J$. Computing $U$ in the constrained randomphase approximation, we investigate noncollinear magnetism of uranium dioxideUO$_2$ and find that the spin-orbit coupling (SOC) stabilizes the 3$\textbf{k}$ordered magnetic ground state. The estimated SOC strength in UO$_2$ is as largeas 0.73 eV per uranium atom, making spin and orbital degrees of freedomvirtually inseparable. Using a multipolar pseudospin Hamiltonian, we show howoctupolar and dipole-dipole exchange coupling help establish the 3$\textbf{k}$magnetic ground state with canted ordering of uranium $f$-orbitals. Thecooperative Jahn-Teller effect does not appear to play a significant part instabilizing the noncollinear 3$\textbf{k}$ state, which has the lowest energyeven in an undistorted lattice. The choice of parameter $U$ in the LSDA+$U$model has a notable quantitative effect on the predicted properties of UO$_2$,in particular on the magnetic exchange interaction and, perhaps trivially, onthe band gap: The value of $U=3.46$ eV computed fully $ab$ $initio$ deliversthe band gap of 2.11~eV in good agreement with experiment, and a balancedaccount of other pertinent energy scales.

Parametrization of LSDA+$U$ for noncollinear magnetic configurations: Multipolar magnetism in UO$_2$