Issues with J-dependence in the LSDA+U method for non-collinear magnets

We re-examine the commonly used density functional theory plus Hubbard $\mathit{\text{U}}$ (DFT$+U$) method for the case of non-collinear magnets. While many studies neglect to explicitly include the exchange correction parameter $\mathit{\text{J}}$, or consider its exact value to be unimportant, here we show that in the case of non-collinear magnetism calculations the $\mathit{\text{J}}$ parameter can strongly affect the magnetic ground state. We illustrate the strong $\mathit{\text{J}}$-dependence of magnetic canting and magnetocrystalline anisotropy by calculating trends in the magnetic lithium orthophosphate family LiMPO${}_4$ (M = Fe and Ni) and difluorite family MF${}_2$ (M = Mn, Fe, Co and Ni). Our results can be readily understood by expanding the usual DFT$+U$ equations within the spinor scheme, in which the $\mathit{\text{J}}$ parameter acts directly on the off-diagonal components which determine the spin canting.