# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 4 Results and Discussion
### Band Gaps from DFT+\(U\) Calculations

Table 3: Comparison of the band gaps (in eV) as computed using DFT and DFT+\(U\) (using _ortho-atomic_ and _atomic_ projectors) and as measured in experiments for compounds containing TM elements, with the mean absolute error (MAE) given for each method. DFT+\(U\) results obtained with _ortho-atomic_ and _atomic_ projectors are split into two columns each: (1) band gaps predicted with \(U\) corrections applied only to the \(d\) states of TM elements (\(U\)-TM), and (2) band gaps predicted with \(U\) corrections applied to the \(d\) states of TM and \(p\) states of light elements (\(U\)-All). The computationally predicted band gaps with the closest values to the experimental band gaps are highlighted in bold. in some cases (e.g., SnO\({}_{2}\)). This finding suggests that DFT+\(U\) with _ortho-atomic_ projectors is a valuable tool for _improving_ band gaps over standard DFT and thus can be useful for high-throughput screening of thousands of materials, but whenever one is interested in very accurate predictions of band gaps then it is important to resort to more advances and more accurate methods (see Section 1). \begin{table}
\begin{tabular}{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-Light** & **U-All** & **U-Light** \\ \hline SnO\({}_{2}\) & 3.4 [131] & 0.65 & 4.69 & **4.65** & 15.81 \\ PbO\({}_{2}\) & 1.5 [132] & 0.09 & **1.10** & 1.06 & 10.27 \\ InN & 0.8 [133] & 0 & 0.85 & **0.80** & 8.87 \\ SnN\({}_{4}\) & 1.6 [134] & 0.21 & 2.09 & **2.08** & 9.63 \\ SnS\({}_{2}\) & 2.4 [135] & 1.53 & **2.00** & 1.98 & 3.00 \\ Ge\({}_{2}\)N\({}_{2}\)O & 3.78 [136] & **2.61** & 5.03 & 4.99 & 15.46 \\ \hline MAE & & 1.40 & 0.65 & 0.63 & 8.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the band gaps (in eV) as computed using DFT and DFT+\(U\) (using _ortho-atomic_ and _atomic_ projectors) and as measured in experiments for compounds containing group III-IV elements (Ge, In, Sn, Pb), with the mean absolute error (MAE) given for each method. DFT+\(U\) calculations using _ortho-atomic_ projectors are split into two columns: (1) band gaps predicted with \(U\) corrections applied only to the \(p\) states of light elements O, N, and S (\(U\)-Light), and (2) band gaps predicted with \(U\) corrections applied to the \(p\) states of group III-IV and light elements (\(U\)-All). For _atomic_ calculations, band gaps were predicted with \(U\) corrections applied only to the \(p\) states of light elements (\(U\)-Light). The computationally predicted band gaps with the closest values to the experimental band gaps are highlighted in bold. Figure 3: PDOS (in states/eV/cell) for all materials studied in this work obtained using DFT+\(U\) with _ortho-atomic_ projectors. The \(U\) correction was applied both to the \(d\) states of TM elements and to the \(p\) states of the light and \(p\)-block (group III-IV) elements (i.e., \(U\)-All). PDOS are color-coordinated such that \(s\) states are green, \(p\) states are red, and \(d\) states are blue. The zero of energy corresponds to the top of valence bands. Finally, for the sake of completeness, we present the PDOS for all 20 materials using DFT+\(U\) with _ortho-atomic_ projectors in Figure 3. The \(U\) correction was applied both to the \(d\) states of TM elements and to the \(p\) states of the light and \(p\)-block (group III-IV) elements (i.e., \(U\)-All). By comparing Figure 3 with Figure 1 we can see not only an increase in the value of the band gap thanks to the \(U\) correction but also \"sharpening\" of the PDOS due to the higher level of localization of states that are corrected. Although a comparative analysis of the PDOS on the basis of experimental spectroscopic data is beyond the scope of the present work, it would be an instructive topic for future studies.