# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## Appendix B Computational Cost Comparison of HSE06 and Hubbard Parameter Calculations  
In this Appendix we present a comparison of the computational costs of hybrid functional (HSE06 [96; 97]) calculations and linear-response calculations of Hubbard \(U\) parameters [138]. This comparison is made for two representative materials, TiO\({}_{2}\) and NbNO. These materials were selected as examples of relatively small unit cells (rutile TiO\({}_{2}\) has 6 atoms per unit cell) and relatively large unit cells (NbNO has 12 atoms per unit cell)--in comparison to the diverse materials studied in this work. HSE06 self-consistent-field calculations were performed using the optimized geometry of the materials at the DFT+\(U\) level of theory. The computational setup is the same as described in Section 3. We recall that hybrid functional calculations are computationally very expensive because the calculation of the non-local exact-exchange potential contains double sums over electronic bands and double sums over \(\mathbf{k}\)-points (one of these \(\mathbf{k}\)-point meshes is replaced by a scarcer \(\mathbf{q}\)-point mesh to speed-up the calculations). As said above, it is common to lower the computational cost of HSE06 calculations (with some losses in the accuracy) by reducing the number of \(\mathbf{q}\)-points relative to the \(\mathbf{k}\)-points of a given material by setting a ratio \(N_{\mathbf{k}}/N_{\mathbf{q}}\) higher than 1 [139]. For the current HSE06 calculations, \(N_{\mathbf{k}}/N_{\mathbf{q}}\) was set to 3 for each material [140]. The resulting \(\mathbf{k}\)- and \(\mathbf{q}\)-point meshes for the \(U\) linear-response and HSE06 calculations are listed in Table 11 (we stress that the \(\mathbf{q}\)-point sampling has different physical meaning in these two types of calculations). The Hubbard parameters calculations reported in Table 11 were performed using a \(N_{\mathbf{k}}/N_{\mathbf{q}}\) ratio of 2, as discussed in Appendix A. It is clear from Table 11 that the HSE06 calculations are computationally more expensive than Hubbard \(U\) calculations, even in this particular case where a lower sampling density of \(\mathbf{q}\)-points was used for HSE06 (\(N_{\mathbf{k}}/N_{\mathbf{q}}=3\)) than for the Hubbard \(U\) calculations (\(N_{\mathbf{k}}/N_{\mathbf{q}}=2\)) (even though these two \(\mathbf{q}\)-points samplings are not really directly comparable). Additionally, it is seen from Table 11 that the relative computational cost increases rapidly with the system size, with the HSE06 calculations being 2.7 times more costly than Hubbard \(U\) calculations for TiO\({}_{2}\) and 8.2 times more computationally costly than Hubbard \(U\) calculations for NbNO. However, it is important to remark that in Table 11 we report the timing of only the one-shot calculation of the \(U\) parameters, while in practice we need to use the self-consistency procedure that requires typically 2-3 such calculations (see Appendix A). Thus, by considering the factor of 3 that was used in this work for the self-consistency loop, we obtain the ratios \(t_{\text{HSE06}}/t_{U}\) of 0.9 for TiO\({}_{2}\) and 2.7 for NbNO. Hence, we can still see that DFT+\(U\) with self-consistent \(U\) values is more advantageous for systems with more than \(\sim\)10 atoms in the unit cell. In fact, in Ref. [36] it was shown that DFT+\(U\) with \(U\) computedusing linear response theory can be used for systems as large as 320 atoms in the supercell, while this is beyond the reach for hybrid functionals with the current high-performance computing hardware. However, we note that the comparison shown in Table 10 is very approximate as each of these calculations can be optimized and speed-up further (e.g., by lowering the kinetic-energy cutoff for the density and potentials for the exact-exchange term in the HSE06 calculations, by lowering the convergence threshold for the response matrices in the \(U\) calculations (see Equation (6)), by using more efficient parallelization strategies, for instance).