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

To explore the formation of noncollinear magnetic configurations in materials with strongly correlated electrons, we derive a noncollinear LSDA+$U$ model involving only one parameter $U$, as opposed to the difference between the Hubbard and Stoner parameters $U-J$. Computing $U$ in the constrained random phase approximation, we investigate noncollinear magnetism of uranium dioxide UO$_2$ 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 large as 0.73 eV per uranium atom, making spin and orbital degrees of freedom virtually inseparable. Using a multipolar pseudospin Hamiltonian, we show how octupolar and dipole-dipole exchange coupling help establish the 3$\textbf{k}$ magnetic ground state with canted ordering of uranium $f$-orbitals. The cooperative Jahn-Teller effect does not appear to play a significant part in stabilizing the noncollinear 3$\textbf{k}$ state, which has the lowest energy even 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, on the band gap: The value of $U=3.46$ eV computed fully $ab$ $initio$ delivers the band gap of 2.11~eV in good agreement with experiment, and a balanced account of other pertinent energy scales.