Compatibility of DFT+U with non-collinear magnetism and spin-orbit coupling within a framework of numerical atomic orbitals.

We report the extension of the density-functional theory plus Hubbard U (DFT+U) method to the case of non-collinear magnetism and spin-orbit coupling in a framework of numerical atomic orbitals. Both the Hubbard repulsion term U, and the exchange J parameters are explicitly included and treated separately. The occupation numbers of the localized orbitals belonging to the correlated shell are computed from the projections of the Kohn-Sham eigenfunctions onto a set of non-overlapping, orthogonal, localized projectors. We provide the detailed expressions for the total energy, forces and stresses including the Pulay corrections. Our implementation on the version 5.0 of the siesta package has been validated with simulations carried out in isolated atoms and bulk solids including atoms with a strong spin-orbit coupling.