# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 5 Conclusions  
We have presented a detailed investigation of the accuracy of DFT+\(U\) calculations for 20 materials containing transition-metal or \(p\)-block (group III-IV) elements, including oxides, nitrides, sulfides, oxynitrides, and oxysulfides. Hubbard \(U\) parameters were computed from first principles using density-functional perturbation theory [38; 79], thus avoiding any fitting or tuning parameters. We have found that the accuracy of DFT+\(U\) band gaps depends strongly on the type of Hubbard projector functions used for applying the \(U\) corrections. More specifically, by comparing DFT+\(U\) results obtained using nonorthogonalized and orthogonalized atomic orbitals as Hubbard projectors, we found considerable deviations in the band gaps, with the orthogonalized projectors showing the highest accuracy, in overall close agreement with experimental data.  
Additionally, we have analyzed systematically which electronic states contribute predominantly to the valence band maximum and to the conduction band minimum in order to determine the states that might require Hubbard corrections. We have found that in the 14 TM-containing materials the \(d\) states of TM elements are mostly found at the bottom of conduction bands while the \(p\) states of light elements (O, N, S) appear at the top of the valence bands. We have not found any clear trend as to whether one should apply the \(U\) correction to both groups of states or only to the \(d\) states of TM elements--either of these two options perform equally (for the band gaps) or one of them is better than the other depending on the specific material. We note that all materials were considered to be nonmagnetic in our simulations, and hence future studies with more detailed investigation of various magnetic orderings would be useful. As for the remaining 6 materials containing the \(p\)-block (group III-IV) elements, we found that the \(p\) states of light elements (O, N, S) contribute mainly to the top of the valence bands and the \(p\) states of the group III-IV elements appear mostly at the bottom of the conduction bands. Application of the \(U\) corrections to both groups of states or only to the \(p\) states of light elements give essentially the same band gaps, which means that the application of \(U\) corrections to the \(p\) states of group III-IV elements leads to minor improvements (if any).  
Hence, this study conclusively shows that DFT+\(U\) with _ab initio_\(U\) may serve as a consistent and reliable electronic-structure method for improving band gaps beyond (semi)local functionals, provided that care is taken in choosing appropriate Hubbard projectors.  
Author contributions include conceptualization, I.D.; methodology, I.T.; software, Y.X., N.E.K.-H., W.Z., and I.T.; validation, N.E.K.-H. and W.Z.; formal analysis, N.E.K.-H.; investigation, N.E.K.-H. and W.Z.; resources, I.D.; data curation, N.E.K.-H. and W.Z.; writing-original draft preparation, N.E.K.-H.; writing-review and editing, I.D., I.T., W.Z., and Y.X.; visualization, N.E.K.-H.; supervision, I.D. and I.T.; project administration, I.D.; funding acquisition, I.D. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the DMREF and INFEWS programs of the National Science Foundation under Grant Agreement No. DMREF-1729338, and by the Swiss National Science Foundation (SNSF) through grant 200021-179138 and its National Centre of Competence in Research (NCCR) MARVEL.  
Not applicable.  
Data Availability Statement: The data used to produce the results of this work are available in the _Materials Cloud Archive_[137] and through the _HydroGEN DataHub_ website under the project NSF DMREF PSU PEC.  
The authors declare no conflict of interest.