# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 4 Results and Discussion
### Hubbard U Parameters from First Principles  
In this section we present the Hubbard parameters for all systems studied here, computed using the DFPT approach that was briefly discussed in Section 2.1. Based on the PDOS computed at the semilocal DFT level (see Section 4.1), we calculated \(U\) for those states that appear at the VBM and CBM, and the results are shown in Table 2. More specifically, for the \(d^{0}\) TM-containing materials we computed the Hubbard parameters for the partially occupied \(d\) states of TM elements (Sc, Ti, V, Y, Zr, Nb, Mo, Hf, Ta, W) that appear at the CBM and also for the \(p\) states of light elements (O, N, S) that appear at the VBM. In the case of \(d^{10}\)\(p\)-block (group III-IV) containing materials we computed the Hubbard \(U\) parameters for the \(p\) states of light elements at the VBM and for the \(p\) states of the group III-IV elements the CBM. First of all it is important to discuss the dependence of the computed values of the \(U\) parameters on the type of Hubbard projectors that are used. In this work we consider two types of projector functions, namely _atomic_ and _ortho-atomic_. The difference between the two is the inter-site orthogonalization of atomic orbitals that is present in the latter  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Formula** & \multicolumn{2}{c}{**Hubbard \(U\) (Ortho-Atomic) [eV]} & \multicolumn{2}{c}{**Hubbard \(U\) (Atomic) [eV]} \\ \hline TiO\({}_{2}\) & Ti(3\(d\)): 6.10, & O(2\(p\)): 8.23 & & Ti(3\(d\)): 3.81, & O(2\(p\)): 15.89 & \\ ZrO\({}_{2}\) & Zr(4\(d\)): 2.72, & O1(2\(p\)): 9.20, & O2(2\(p\)): 8.79  
and is absent in the former. As can be seen in Table 2, the values of Hubbard parameters depend strongly on the type of projector functions. In particular, we can see that \(U\) for the \(d\) states of TM elements is larger when _ortho-atomic_ projectors are used, while \(U\) for the \(p\) states of light elements are much smaller with respect to the case when _atomic_ projectors are employed. Notably, \(U\) for the \(p\) states of light elements in some materials are extremely large, and in fact this finding is not surprising and is well known in the literature [116]: linear-response theory for computing Hubbard parameters according to Ref. [34] is not suited for application to fully occupied states (closed-shell systems), which was observed when using _atomic_ Hubbard projectors. Indeed, it can be seen from Table 2 that the values of \(U\) for the \(p\) states of light elements (O, N, S) are unphysically large when computed using _atomic_ projectors. As will be seen in Section 4.3, this leads to dramatic worsening of the band gaps when computed using DFT+\(U\) with these values of \(U\) and _atomic_ projectors. However, interestingly we find that when _ortho-atomic_ Hubbard projectors are used, the values of Hubbard \(U\) for the \(p\) states of light elements are much smaller and they fall in a physically meaningful range (though still quite large). As will be seen in the following section, the band gaps computed using DFT+\(U\) and using these \(U\) values with the _ortho-atomic_ projectors are in good overall agreement with the experimental band gaps. Therefore, this observation highlights the crucial role played by the Hubbard projectors [90] when computing \(U\) for predicting various properties of materials, and in particular band gaps. Next, in the case of \(d^{0}\) TM-containing materials, the \(U\) values computed for the partially occupied \(d\) states were applied to either \(3d\), \(4d\), or \(5d\) states depending on the TM period in the periodic table. In general, the smaller the principal quantum number the larger the localization of the \(d\) orbitals (because they are closer to the nucleus); thus, the value of \(U\) is expected to be larger for \(3d\) states than for \(4d\) and \(5d\) states [117].