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DFT+$U$ within the framework of linear combination of numerical atomic orbitals

We present a formulation and implementation of the DFT+\textit{U} method within the framework of linear combination of numerical atomic orbitals (NAO). Our implementation not only enables single-point total energy and electronic-structure calculations but also provides access to atomic forces and stresses, hence allowing for full structure relaxations of periodic systems. Furthermore, our implementation allows one to deal with non-collinear spin texture, with the spin-orbit coupling (SOC) effect treated self-consistently. The key aspect behind our implementation is a suitable definition of the correlated subspace when multiple atomic orbitals with the same angular momentum are used, and this is addressed via the "Mulliken charge projector" constructed in terms of the first (most localized) atomic orbital within the $d/f$ angular momentum channel. The important Hubbard $U$ and Hund $J$ parameters can be estimated from a screened Coulomb potential of the Yukawa type, with the screening parameter either chosen semi-empirically or determined from the Thomas-Fermi screening model. Benchmark calculations are performed for four late transition metal monoxide bulk systems, i.e., MnO, FeO, CoO, and NiO, and for the 5$d$-electron compounds IrO$_2$. For the former type of systems, we check the performance of our DFT+$U$ implementation for calculating band gaps, magnetic moments, electronic band structures, as well as forces and stresses; for the latter, the efficacy of our DFT+$U$+SOC implementation is assessed. Systematic comparisons with available experimental results, and especially with the results from other implementation schemes are carried out, which demonstrate the validity of our NAO-based DFT+$U$ formalism and implementation.