# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 2 Methods and Materials  
### DFT+\(U\)  
In this section, we briefly review DFT+\(U\) in the simplified, rotationally invariant formulation [33] which is used in this work. For the sake of simplicity, the formalism is presented in the framework of norm-conserving pseudopotentials in the collinear spin-polarized case. The generalization to the ultrasoft pseudopotentials and the projector augmented wave method are described in Ref. [38]. In this section we use Hartree atomic units. DFT+\(U\) is based on an additive correction to the approximate DFT energy functional [33]:  
\[E_{\text{DFT+}U}=E_{\text{DFT}}+E_{U}, \tag{1}\]  
where \(E_{\text{DFT}}\) is the approximate DFT energy (constructed, e.g., within the spin-polarized LDA or GGA), while \(E_{U}\) contains the additional Hubbard term:  
\[E_{U}=\frac{1}{2}\sum_{l\
u m_{1}m_{2}}U^{I}\Big{(}\delta_{m_{1}m_{2}}-n_{m_{1} m_{2}}^{Ie}\Big{)}n_{m_{2}m_{1}}^{Ie}\,, \tag{2}\]  
where \(I\) is the atomic site index, \(m_{1}\) and \(m_{2}\) are the magnetic quantum numbers associated with a specific angular momentum, and \(U^{I}\) are the effective on-site Hubbard parameters. The atomic occupation matrices \(n_{m_{1}m_{2}}^{Ie}\) are based on a projection of the lattice-periodic parts of KS wavefunctions, \(u_{v,\mathbf{k}}^{\sigma}(\mathbf{r})\), on the Hubbard manifold:  
\[n_{m_{1}m_{2}}^{Ie}=\frac{1}{N_{\mathbf{k}}}\sum_{\mathbf{k}}^{N_{\mathbf{k}}} \sum_{v}f_{v,\mathbf{k}}^{\sigma}(u_{v,\mathbf{k}}^{\sigma}|\hat{P}_{m_{2}m_{ 1}\mathbf{k}}^{I}|u_{v,\mathbf{k}}^{\sigma})\, \tag{3}\]  
where \(v\) and \(\sigma\) represent, respectively, the band and spin labels of the KS wavefunctions, \(\mathbf{k}\) indicate points in the first Brillouin zone (BZ) and \(N_{\mathbf{k}}\) is the number of \(\mathbf{k}\)-points, \(f_{v,\mathbf{k}}^{\sigma}\) are the occupations of the KS states, and \(\hat{P}_{m_{2}m_{1}\mathbf{k}}^{I}\) is the projector on the Hubbard manifold:  
\[\hat{P}_{m_{2}m_{1}\mathbf{k}}^{I}=|\varphi_{m_{2}\mathbf{k}}^{I}\rangle \langle\varphi_{m_{1}\mathbf{k}}^{I}|\, \tag{4}\]  
where \(\varphi_{m_{1}\mathbf{k}}^{I}(\mathbf{r})\) is the Bloch sum computed using \(\varphi_{m_{1}}^{I}(\mathbf{r}-\mathbf{R}_{I})\) which are the localized orbitals centered on the \(I\)th atom at the position \(\mathbf{R}_{I}\). The Hubbard manifold can be constructed using different types of projector functions, as was discussed in Section 1; here we consider two types of localized functions, _atomic_ and _ortho-atomic_. In DFT+\(U\), the Hubbard parameters are not known _a priori_, and they are often adjusted semiempirically, by matching the value of properties of interest, which is fairly arbitrary. In this work we instead determine Hubbard parameters from a piecewise linearity condition implemented through linear-response theory [34], based on DFPT [38; 79]. Within this framework the Hubbard parameters are the elements of an effective interaction matrix computed as the difference between bare and screened inverse susceptibilities [34]:  
\[U^{I}=\left(\chi_{0}^{-1}-\chi^{-1}\right)_{II}, \tag{5}\]  
where \(\chi_{0}\) and \(\chi\) are the susceptibilities which measure the response of atomic occupations to shifts in the potential acting on individual Hubbard manifolds. In particular, \(\chi\) is defined as  
\[\chi_{II}=\sum_{\sigma m}\frac{dn_{mm}^{Ie}}{d\alpha^{I}}\, \tag{6}\]where \(\alpha^{I}\) is the strength of the perturbation on the \(J\)th site. While \(\chi\) is evaluated at self-consistency of the linear-response KS calculation, \(\chi_{0}\) (which has a similar definition) is computed before the self-consistent re-adjustment of the Hartree and exchange-correlation potentials [34].