# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 1 Introduction
It is thus expected that DFT+\(U\) would be more accurate at predicting optical band gaps for inorganic semiconductors than for organic semiconductors [53; 54]. To improve the precision of the computed electronic band gaps, DFT+\(U\) incorporates a corrective \(U\) term designed to restore piecewise linearity of the total energy with respect to the occupations of the orbitals within the Hubbard manifold, which acts to approximate derivative discontinuities [28; 35]. Often the \(U\) parameter is chosen semiempirically by fitting it to reproduce experimental band gaps [55; 12; 56], experimental oxidation enthalpies [57; 58], or other properties. However, fitting \(U\) to reproduce a subset of experimental data is not a predictive approach, especially in cases when no prior data is available. If instead the Hubbard \(U\) parameter is determined using _ab initio_ methods, the predicted band gap might not exactly match the experimental gap, but this approach typically improves band gap predictions drastically (over semilocal DFT) and provides a more accurate, overall description of the material [12]. In practical terms, the Hubbard \(U\) parameter can be calculated using _ab initio_ methods, such as constrained DFT (cDFT) [59; 60; 61; 62; 63; 64; 65; 66; 67], constrained random phase approximation (cRPA) [68; 69; 70; 71; 72; 73; 74], and Hartree-Fock-based methods [75; 76; 77; 78]. In particular, a linear-response formulation of cDFT was introduced in Ref. [34]: it is mainly based on the fact that the Hubbard corrections are meant to remove SIE from approximate energy functionals that manifests itself through a curvature of the total energy as a function of atomic occupations. Recently, this formulation was recast in the framework of density-functional perturbation theory (DFPT) [38, 79], allowing the replacement of computationally expensive supercells by primitive cells at the cost of having multiple monochromatic perturbations instead of just one (localized). In all these _ab initio_ methods for computing \(U\), key is the choice of the Hubbard projector functions. There are different types of Hubbard projectors [80], including nonorthogonalized atomic orbitals [34, 81, 82, 83], orthogonalized atomic orbitals [36, 37, 39, 84], nonorthogonalized Wannier functions [85], orthogonalized Wannier functions [86], linearized augmented plane-wave approaches [87], and projector-augmented-wave projector functions [88, 89]. It is well known that the values of the computed Hubbard \(U\) parameters strongly depend on the type of Hubbard projectors used [79, 84]. Moreover, the final properties of interest (e.g., band gaps, oxidation reaction energy, among others) obtained using DFT+\(U\) and different projectors are not always consistent [90]. Therefore, specific attention must be given to the choice of Hubbard projectors when computing band gaps using DFT+\(U\). Historically, DFT+\(U\) was introduced to accurately describe the \(d\)- and \(f\)-type electrons of transition-metal (TM) and rare-earth compounds, as it was thought that the \(U\) parameter would correct these strongly correlated, localized states (i.e., \(d\)- and \(f\)-type states). It was later discovered that the Hubbard \(U\) parameter corrects for SIE, which is present in not only \(d\) and \(f\), but also \(s\) and \(p\) states [91, 92]. While light elements (e.g., O, N, S) contain only \(s\) and \(p\) states and thus do not have as large SIE as TM, the \(U\) correction has been increasingly applied to light elements in recent years to improve the computational prediction of properties, such as the band gap [78, 93, 94]. Additionally, in the literature, \(U\) has been applied to \(d^{10}\) elements from the \(p\)-block (specifically group III-IV) of the periodic table, such as Ga\({}^{2+}\)[35]. While elements, like Ga\({}^{2+}\), are not TM, they have electronic configurations that match that of a \(d^{10}\) TM under certain ionizations.