Noncollinear relativistic Hubbard parameters and DFT+U with ultrasoft pseudopotentials

Density-functional theory (DFT) has proven to be an extremely successful theory to understand and predict ground state properties of real materials. However, it systematically fails when applied to systems in which the low-energy physics is characterized by localized electrons, for example with d or f character. To remedy this deficiency, Hubbard-augmented DFT functionals (e.g. DFT+U) have been developed in the last 30 years, and successfully applied to the study of weakly correlated oxides. In this work, we extend an implementation of the DFT+U functional in a plane-wave electronic structure code, from a scalar-relativistic to a fully-relativistic pseudopotential (FR-PP) framework. Our formulation is able to deal with FR-PP within the ultrasoft formalism, which allow an efficient plane-wave expansion of the pseudowavefunctions, especially when the electronic charge is heavily localized around the nuclei. We enhance our theory not only with total-energy, forces and stresses calculations, but we also develop a noncollinear linear-response approach based on density-functional perturbation theory which allow us to evaluate the interaction parameter U from first-principles, in an entirely parameter free scheme.