# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 4 Results and Discussion  
### Determination of the States Requiring Hubbard Corrections  
One can assess the necessity of applying the Hubbard \(U\) correction to certain states of a given element by examining the PDOS of the material. In particular, the PDOS plots show which specific states have the highest density of states (DOS) near the valence band maximum (VBM) and conduction band minimum (CBM) in semiconductors and insulators. By applying \(U\) corrections to elements with states at the VBM and CBM, the value of the band gap is expected to change with respect to standard DFT predictions. In contrast, applying \(U\) values to elements with low DOS or lacking states near band edges is not expected to significantly affect the predicted band gap value. Using PDOS, we can also assess whether a given material is a band insulator, a Mott-Hubbard insulator, or a charge-transfer insulator [114]. In Mott-Hubbard insulators, the VBM and CBM are of the same kind (e.g., of \(d\) character), in charge-transfer insulators instead the VBM and CBM are of different kinds (e.g., the VBM is mainly of the \(p\) character and the CBM is of the \(d\) character), while in band insulators the band gap is not determined by strongly localized electrons of\(d\) or \(f\) character. Therefore, by determining the dominant character of the VBM and CBM in the PDOS, it is possible to identify the insulating type of a given material. Thus, prior to DFT+\(U\) calculations, PDOS calculations were performed in order to determine the dominant character of states at the VBM and CBM, and thus to know which states might require the Hubbard \(U\) correction. The PDOS for all materials studied here are shown in Figure 1. It can be seen that all 14 TM-containing materials can be classified as charge-transfer insulators, while all of the remaining materials that contain group III-IV elements (except InN that comes out to be metallic at the semilocal DFT level of theory) are band insulators. Also we note that in Figure 1 those cases that have multiple lines of the same color indicate the PDOS for non-equivalent atoms (e.g., multiple red lines for V\({}_{2}\)O\({}_{5}\) indicate 2\(p\) states of non-equivalent oxygen atoms). All the materials of this study are oxides, nitrides, and sulfides containing \(d^{0}/d^{10}\) cations. The PDOS of \(d^{0}\) TM-containing materials (TiO\({}_{2}\), ZrO\({}_{2}\), HfO\({}_{2}\), V\({}_{2}\)O\({}_{5}\), Ta\({}_{2}\)O\({}_{5}\), WO\({}_{3}\), TiS\({}_{2}\), ZrS\({}_{2}\), HfS\({}_{2}\), MoS\({}_{3}\), Sc\({}_{2}\)S\({}_{3}\), NbNO, TaNO, Y\({}_{2}\)SO\({}_{2}\)) in Figure 1 show dominant \(d\) states from the TM \(d^{0}\) cations at the CBM and dominant \(p\) states from the light elements (O, N, S) at the VBM. Hence, one might expect that the application of the Hubbard \(U\) correction to these states would allow us to obtain the most accurate prediction of the band gap at the DFT+\(U\) level of theory. The PDOS of \(d^{10}\)\(p\)-block (group III-IV) containing materials (SnO\({}_{2}\), PbO\({}_{2}\), InN, Sn\({}_{3}\)N\({}_{4}\), SnS\({}_{2}\), Ge\({}_{2}\)N\({}_{2}\)O) show dominant \(p\) states from the light elements (O, N, S) at the VBM and a mixture of \(p\) and \(s\) states from the group III-IV elements at the CBM, indicating that the \(U\) correction can be applied to the \(p\) states of the light and group III-IV elements to obtain the most accurate prediction of the band gap [115]. In addition, it might be useful to investigate in these materials whether the application of the \(U\) correction only to the \(d\) states of the TM elements in \(d^{0}\) TM-containing materials or only to the \(p\) states of  
Figure 1: PDOS (in states/eV/cell) for all materials studied in this work obtained from standard DFT calculations. PDOS are color-coordinated such that \(s\) states are green, \(p\) states are red, and \(d\) states are blue. Note that cases where there are multiple lines of the same color indicate non-equivalent atoms.