# Extensive Benchmarking of DFT+\(U\) Calculations for Predicting Band Gaps
## 1 Introduction  
Density-functional theory (DFT) [1; 2] with approximate exchange-correlation (xc) functionals--e.g., local-density approximation (LDA) or generalized-gradient approximation (GGA)--has been remarkably successful in predicting ground-state properties of a large variety of systems, such as crystal structure and thermodynamic stability. However, these DFT calculations have known limitations, including the underestimation of the band gap (on the order of \(\sim\)40% in semiconductors and insulators [3]) due to self-interaction errors (SIE) inherent to approximatexc functionals [4; 5]. Variousxc functionals and methods were developed to alleviate this problem. One practical solution is to usehybrid functionals, which employ a fraction of the exact (Fock) exchange energy to reduce SIE [6; 7; 8; 9; 10; 11]. The drawback of this method is its very high computational cost (which prevents its wider application to high-throughput computational studies [12]), and difficulties in determining the magnitude of the screening parameter for solids (that controls the amount of the exact exchange), although there has been considerable progress in this direction in recent years (see, e.g., Refs. [13; 14; 15; 16; 17; 18]). In parallel, a range of computationally efficient meta-GGA functionals have emerged [19], including the strongly constrained and appropriately normed semilocal density functional (SCAN) [20]. The SCAN functional uses a xc potential which depends on Kohn-Sham (KS) wavefunctions via a kinetic energy density; however, SCAN still contains significant SIE (especially when applied to transition-metal compounds [21; 22]) and exhibits some potential limitations in describing magnetic systems [23; 24]. An alternative approach is provided by Koopmans-compliant (spectral) functionals [25]. The main idea of these is to enforce the piecewise linearity of the total energy with respect to all orbitals occupations; this class of functionals has proven to be effective in improving band gaps in finite and extended systems [26; 27; 28; 29]; however, this method is not straightforward to implement in practice, though recent progress in this direction has been achieved [30]. A very popular extension of DFT is the DFT+\(U\) approach [31; 32; 33], in which the Hubbard \(U\) correction acts selectively on a subset of states in the system (typically, of \(d\) or \(f\) character) by imposing the piecewise linearity in the energy functional as a function of the occupations of this subset [34]. DFT+\(U\) is only marginally more expensive than DFT within LDA or GGA, while significantly improving various properties of materials such as transition-metal compounds. An extension of DFT+\(U\) to take into account inter-site Hubbard \(V\) interactions (DFT+\(U\)[35]) was introduced, showing success in describing materials with strong inter-site electronic hybridizations (i.e., covalent interactions) [35; 36; 37; 38; 39; 40; 41]. Finally, there are various methods beyond DFT, including many-body perturbation theory (MBPT) [42] (e.g., GW approximation [43; 44; 45; 46]) and dynamical mean field theory (DMFT) [47; 48; 49; 50; 51], which are widely used for predicting optical properties of (strongly) correlated systems. In contrast to MBPT and DMFT, the scope of DFT and DFT+\(U\) is limited to ground-state properties, including electronic band gaps but excluding optical band gaps [28; 52]. The optical band gap of a material is the minimal energy required for absorbing an incident photon (corresponding to a neutral excitation), whereas the electronic band gap is the difference between the occupied and unoccupied states (corresponding to charged excitations) [52]. In most inorganic semiconductors, the electronic band gap and the optical band gap are approximately equal, but for organic semiconductors, the difference between these two energies (the exciton binding energy) may be substantial.