# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## Appendix A Convergence Tests for the Self-Consistent Calculation of Hubbard Parameters  
In this Appendix we discuss how we performed convergence tests when computing Hubbard parameters using the DFPT approach (see Section 2.1). We did these tests for TiO\({}_{2}\), NbNO, and Y\({}_{2}\)SO\({}_{2}\) which represent the material families considered in this study. As explained in Ref. [79], Hubbard \(U\) must be converged with respect to the \(\mathbf{k}\)- and \(\mathbf{q}\)-points sampling of the BZ: the former is the Bloch wavevector that is used to describe Kohn-Sham wavefunctions and charge density, while the latter is used to describe modulations of monochromatic perturbations that are used in linear-response calculations when computing \(U\) (see Section 2.1). As the criterion to monitor the convergence of \(U\) we use the ratio \(N_{\mathbf{k}}/N_{\mathbf{q}}\), where \(N_{\mathbf{k}}\) and \(N_{\mathbf{q}}\) are the number of \(\mathbf{k}\)- and \(\mathbf{q}\)-points in the BZ, respectively. It is important to remark that we use this criterion with the condition that the \(\mathbf{k}\)-point mesh for each material remains unchanged (i.e., it is fixed such that the spacing between the \(\mathbf{k}\)-points is 0.04 A\({}^{-1}\)). The result is shown in Figure 11.  
In these plots, convergence occurs from right to left, i.e., the smaller the ratio \(N_{\mathbf{k}}/N_{\mathbf{q}}\) the denser the \(\mathbf{q}\)-point mesh at a fixed \(\mathbf{k}\)-point mesh. For example, \(N_{\mathbf{k}}/N_{\mathbf{q}}\)=1 is the case where the \(\mathbf{q}\)-points are equivalent to the \(\mathbf{k}\)-points for a given material and hence, the most computationally expensive option (in this case the spacing between the \(\mathbf{q}\)-points is also 0.04 A\({}^{-1}\)). Since we enforce a consistent \(\mathbf{k}\)-point sampling density in our calculations, \(\mathbf{k}\)-points are unique for each material in this study. Thus, when we choose the number of \(\mathbf{q}\)-points for the sampling of the BZ corresponding to the primitive cell, we look at the \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratio by which we divide the \(\mathbf{k}\)-points for each material and truncate down to the nearest integer value. For example, if the \(\mathbf{k}\)-point mesh is \(6\times 5\times 5\) and we have a \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratio of 2, then the resulting \(\mathbf{q}\)-point mesh is \(3\times 2\times 2\). The \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratios tested, as well as their subsequent \(\mathbf{q}\) meshes for each material are shown in Table 11. From the \(\mathbf{q}\)-point mesh testing, a \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratio of 2 was selected with a convergence threshold of \(\leq 0.1\) eV. By selecting a higher \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratio than 1, we are able to speed up Hubbard parameter calculations and reduce the computational cost. Once the convergence of \(U\) is reached with respect to the \(\mathbf{q}\)-point sampling of the BZ at a fixed \(\mathbf{k}\)-point sampling, the next step is to perform the self-consistency convergence test. This is explained in detail in Ref. [38]. In short, it is necessary to reach self-consistency between the computed value of Hubbard \(U\) and the crystal structure; to do so, we need to perform structural optimization at the DFT+\(U\) level of theory with the current value of \(U\), then recompute \(U\) using DFPT on top of the new geometry, then perform new structural optimization with the updated value of \(U\), and so on until convergence is reached [38]. The results for the selected systems are shown in Figure 12. It can be seen that the accuracy of \(\Delta U\leq 0.1\) eV is reached after 3 cycles for all systems. Figure 11: Dependence of the Hubbard \(U\) parameter on the \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratio showing the results of \(N_{\mathbf{k}}/N_{\mathbf{q}}\) testing for 3 representative materials: (**a**) TiO\({}_{2}\), (**b**) NbNO, and (**c**) Y\({}_{2}\)SO\({}_{2}\).