# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 4 Results and Discussion
### Band Gaps from DFT+\(U\) Calculations
_ortho-atomic_ projectors with Hubbard corrections applied to only TM and standard DFT, and TiS\({}_{2}\) shows higher accuracy for the band gap computed with DFT+\(U\) calculations using _atomic_ projectors. For V\({}_{2}\)O\({}_{5}\), it is important to remark that the antiferromagnetic ordering seen experimentally in V\({}_{2}\)O\({}_{5}\) nanostructures may lead to inaccurate predictions of the band gap using DFT and DFT+\(U\) since all materials in this study were modeled as nonmagnetic (see Section 3). Materials containing \(p\)-block (group III-IV) elements show even more striking differences between the band gaps computed using _atomic_ and _ortho-atomic_ projectors with their corresponding values of \(U\) parameters (see Figure 2b). Notably, it can be seen that the DFT+\(U\) band gaps computed using _atomic_ projectors are much larger than the experimental gaps, and, surprisingly, they are even worse than DFT band gaps. However, DFT+\(U\) band gaps computed using _ortho-atomic_ projectors with respective \(U\) values are in good overall agreement with the experimental band gaps, and importantly these are better than the DFT band gaps. In addition we find that in the majority of cases \(U\)-All corrections give slightly better band gaps than \(U\)-Light, though these differences are very small. This latter finding suggests that the application of the \(U\) correction to the \(p\) states of the \(p\)-block (group III-IV) elements is not very important. Thus, \(U\) values were not applied to the \(p\) states of group III-IV elements for the case of DFT+\(U\) band gaps computed using _atomic_ projectors. Therefore, from our analysis of the band gaps computed using DFT+\(U\) for all of the 20 materials considered here, we can conclude that the Hubbard projectors play a fundamental role in the accuracy of band gap predictions. More specifically, depending on the type of projector functions, the Hubbard \(U\) parameters vary dramatically (see Section 4.2) and as a consequence the DFT+\(U\) band gaps also vary largely. Hence, the projector functions must be chosen with care prior to DFT+\(U\) calculations. In particular, in our study we find that the _ortho-atomic_ Hubbard projectors give band gaps that are in much better agreement with the experimental values, compared to DFT+\(U\) with the _atomic_ projectors. This result should be taken into account in future DFT+\(U\) studies. In addition, it is important to mention that from Tables 3 and 4 we can see that still our best DFT+\(U\) band gaps are not perfect and differ from the experimental ones by as much as almost \(\sim\)40% \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Formula**} & \multirow{2}{*}{**Expt.**} & \multirow{2}{*}{**DFT**} & \multicolumn{2}{c}{**DFT+\(U\) (Ortho-Atomic)**} & \multicolumn{2}{c}{**DFT+\(U\) (Atomic)**} \\  & & & _U-TM_ & _U-All_ & _U-TM_ & _U-All_ \\ \hline TiO\({}_{2}\) & 3 [118] & 1.85 & **2.60** & 3.95 & 2.28 & 4.96 \\ ZrO\({}_{2}\) & 5.8 [119] & 3.67 & 4.16 & **5.81** & 4.04 & 11.75 \\ HfO\({}_{2}\) & 6 [120] & **4.11** & 4.68 & **6.55** & 4.59 & 15.44 \\ V\({}_{2}\)O\({}_{5}\) & 2.35 [121] & **2.01** & **2.69** & 3.50 & 1.69 & 2.82 \\ Ta\({}_{2}\)O\({}_{5}\) & 3.9 [122] & 3.12 & **3.41** & 4.85 & 3.22 & 8.08 \\ WO\({}_{3}\) & 2.6 [123] & 0.69 & 0.99 & **2.20** & 0.81 & 3.62 \\ TiS\({}_{2}\) & 1 [124] & 0.02 & 1.53 & 1.86 & **1.03** & 1.56 \\ ZrS\({}_{2}\) & 1.7 [125] & 1.13 & **1.92** & 2.27 & 1.45 & 2.81 \\ HfS\({}_{2}\) & 2.85 [126] & 1.30 & 2.16 & **2.51** & 1.78 & 3.73 \\ MoS\({}_{3}\) & 1.35 [127] & 0.62 & 0.98 & **1.52** & 0.19 & 0.58 \\ Sc\({}_{2}\)S\({}_{3}\) & 2 [128] & 1.08 & **2.04** & 2.55 & 1.61 & 3.36 \\ NbNO & 2.1 [129] & 1.65 & **2.05** & 2.58 & 1.67 & 3.04 \\ TaNO & 2.5 [129] & 1.95 & **2.36** & 3.03 & 2.07 & 4.05 \\ Y\({}_{2}\)SO\({}_{2}\) & 4.6 [130] & 3.11 & 3.23 & **4.38** & 3.14 & 11.88 \\ \hline MAE & & 1.10 & 0.66 & 0.55 & 0.87 & 2.68 \\ \hline \hline \end{tabular}
\end{table}