# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 4 Results and Discussion
### Band Gaps from DFT+\(U\) Calculations  
In this section we present our findings for the band gaps of all 20 materials computed using standard DFT and using DFT+\(U\) with _ab initio_ values of Hubbard \(U\) (see Section 4.2) together with _atomic_ and _ortho-atomic_ Hubbard projectors. The resulting predicted band gaps are plotted in Figure 2a, for the materials containing TM elements, and in Figure 2b, for the materials containing \(p\)-block (group III-IV) elements; this data is also summarized in Tables 3 and 4. We use the following short-hand notations: \((i)\) in the TM-containing materials we use \"\(U\)-TM\" for DFT+\(U\) calculations with the \(U\) correction applied only to the \(d\) states of TM elements, and \"\(U\)-All\" for DFT+\(U\) calculations with the \(U\) correction applied both to the \(d\) states of the TM elements and to the \(p\) states of the light elements (O, N, S); \((ii)\) in the materials containing the \(p\)-block (group III-IV) elements we use \"\(U\)-Light\" for DFT+\(U\) calculations with the \(U\) correction applied only to the \(p\) states of the light elements, and \"\(U\)-All\" for DFT+\(U\) calculations with the \(U\) correction applied both to the \(p\) states of the light and group III-IV elements (Ge, In, Sn, Pb). Upon inspection of the results for the TM-containing materials (Figure 2a) we can see large variations in the computed band gaps depending on the type of projector functions that are used (e.g., compare the slope of the lines obtained by fitting the data points). Overall, very good agreement with the experimental band gap values is obtained for DFT+\(U\) calculations using _ortho-atomic_ projectors with respective \(U\) values. In contrast, DFT+\(U\) calculations with the _atomic_ projectors result in large, unrealistic values (\(>\)8 eV) for the band gaps in some materials (Ta\({}_{2}\)O\({}_{5}\), Y\({}_{2}\)SO\({}_{2}\), ZrO\({}_{2}\), and HfO\({}_{2}\)) when Hubbard corrections are applied to TM and light elements (indicated by \(U\)-All) and underestimated band gaps when Hubbard corrections are applied to only TM (indicated by \(U\)-TM). In the case of _ortho-atomic_ projectors, both \(U\)-TM and \(U\)-All results are in good agreement (on average) with the experimental band gaps, with \(U\)-All fitting line (mean absolute error of 0.55) being closer to the exact trend than the \(U\)-TM one (mean absolute error of 0.66). This latter finding can be further clarified by looking at Table 3: we can see that when the _ortho-atomic_ projectors are used, both \(U\)-TM and \(U\)-All corrections perform equally well (which one is better depends on the material). It is worth highlighting the outliers from these trends: V\({}_{2}\)O\({}_{5}\) shows comparable accuracy between band gaps computed using  
Figure 2: Comparison of the computationally predicted band gaps (from DFT and DFT+\(U\)) and the experimental band gaps for materials containing (**a**) TM elements, and (**b**) \(p\)-block (group III-IV) elements of the periodic table. The dashed black line indicates where the computational band gap equals the experimental band gap. Data was fit to first order polynomials so that data points from DFT and different DFT+\(U\) calculations could be more easily compared using the resulting solid trendlines (note that all data points were included in these fits). DFT+\(U\) calculations were performed using _atomic_ and _ortho-atomic_ projectors. On panel (**a**), the legend also indicates whether \(U\) corrections are applied to only \(d\) states of the TM elements (\(U\)-TM), or to the \(d\) states of TM and \(p\) states of light elements (\(U\)-All). On panel (**b**), the legend also indicates whether \(U\) corrections are applied to \(p\) states of light elements (\(U\)-Light), or to \(p\) states of light and \(p\)-block (group III-IV) elements (\(U\)-All). Note that in panel (**b**), the “DFT+\(U\) Ortho-Atomic U-Light” trendline (blue) is dashed so that the “DFT+\(U\) Ortho-Atomic U-All” trendline (orange) can also be seen.