# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 3 Technical Details  
All calculations were performed using the plane-wave (PW) pseudopotential method as implemented in the Quantum ESPRESSO distribution [104; 105; 106]. We used GGA for the xc functional constructed with the PBE prescription [107], and we selected the ultrasoft pseudopotentials from the GBRV library [108; 109]. KS wavefunctions and charge density were expanded in plane waves up to a kinetic-energy cutoff of 60 and 480 Ry, respectively. Electronic ground states were computed by sampling the BZ with uniform \(\Gamma\)-centered \(\mathbf{k}\)-point meshes with a spacing of 0.04 A\({}^{-1}\).  
DFT+\(U\) calculations were performed using the simplified rotationally invariant formulation [33]. As projectors for the Hubbard manifold, we used _atomic_ and _ortho-atomic_ orbitals [84]; for the latter we employed the Lowdin orthogonalization method [95]. Hubbard \(U\) parameters were computed using the linear-response approach [34] as implemented using DFPT [79] (see Section 2.1). We have used the self-consistent procedure for the calculation of \(U\) as described in detail in Ref. [38], which consists of cyclic calculations containing structural optimizations and recalculations of Hubbard parameters for each new geometry. We have performed convergence tests with respect to the \(\mathbf{q}\)-point sampling for 3 materials (TiO\({}_{2}\), NbNO, Y\({}_{2}\)SO\({}_{2}\)); the convergence criteria was \(\Delta U<0.1\) eV. The details can be found in the Appendix A. After selecting the \(\mathbf{q}\)-point sampling, hybrid functional (HSE06) calculations were performed for two representative materials (TiO\({}_{2}\) and NbNO) to compare the computational costs of HSE06 and \(U\) calculations. The results of this analysis are detailed in Appendix B.  
All materials were tested for possible finite magnetization by performing collinear spin-polarized calculations within standard DFT. In order to allow for fractional occupations we have used the Marzari-Vanderbilt smearing [110] with a broadening parameter of \(5\times 10^{-3}\) Ry. In this case we also sampled the BZ using the uniform \(\Gamma\)-centered \(\mathbf{k}\)-point meshes with a spacing of 0.04 A\({}^{-1}\). Materials were only tested for ferromagnetic ordering; thus, this study is not exhaustive and more testing would be needed to check for other types of possible magnetic orderings. Based on our computational testing for ferromagnetic ordering, all of the materials in this study showed zero magnetization. Therefore, all materials were modeled as nonmagnetic, which could lead to errors in predicted band gap values for materials that do show magnetic ordering other than ferromagnetic. In fact, some of the materials are known to be antiferromagnetic (V\({}_{2}\)O\({}_{5}\)) or ferromagnetic (SnO\({}_{2}\), SnS\({}_{2}\)) in their defective forms [111; 112; 113], but the primary focus of this work is to investigate the pristine bulk phase.  
We have computed the projected density of states (PDOS) using both DFT and DFT+\(U\) with the Gaussian smearing and a broadening parameter of \(5\times 10^{-3}\) Ry. The PDOS was calculated by summing up states from all equivalent atoms in the unit cell.