Development of noncollinear-spin DFT$+$U method with spin-orbit interaction

We developed a DFT$+$U$+$SOI method by incorporating spin-orbit interaction (SOI) into a noncollinear-spin generalization of the density functional theory (DFT) plus Coulomb interaction among $d$ electrons, parameterized by U and J. The Coulomb interaction, which is based on the rotationally invariant form, is generalized for noncollinear-spin configuration, and the fully localized limit is adopted for the double-counting term. The spin-orbit interaction is treated in the $l$-dependent fully separable nonlocal form using additional Kleinman-Bylander projectors generated by relativistic calculations of atoms. We implemented our DFT$+$U$+$SOI method into the SIESTA code and performed test calculations for the 4$d$ or 5$d$ transition metal oxides, the all-in-all-out noncollinear magnetic insulator Cd$_{\mathrm{2}}$Os$_{\mathrm{2}}$O$_{\mathrm{7}}$, the canted antiferromagnetic order insulator Sr$_{\mathrm{2}}$IrO$_{\mathrm{4}}$, and the paramagnetic insulator Ca$_{\mathrm{2}}$RuO$_{\mathrm{4}}$. This work was supported by NRF of Korea (Grant No. 2011-0018306) and KISTI supercomputing center (Project No. KSC-2012-C3-046).