Band gap
DFT: self-repulsion problem Standard DFT (LDA, PBE, and Pw91) includes Self-Coulomb Repulsion artifact implicit in DFT. The electron interacts with itself. This biases the QM toward extra delocalization of electron states to minimize the self-repusion. As a result band gaps are usually 1 to 3 eV too small. Thus CIGS and most III-V are predicted to have zero-bandgap (metals rather than insulators)
Empirical: add Hubbard U and adjust until get experimental value
Problem: not useful for design
Rigorous theory: GW calculate rigorous quasiparticle excitation spectrum using Green’s function (many-body perturbation theory)
Problem: full GW: 104 cost of PBE G0W0 (non-self-consistent) 103 cost of PBE
Hybrid functional: Including exact Hartree-Fock exchange into DFT functionals eliminates the Self-Repulsion Problem.
B3PW with VASP ~1000 times the cost of PBE because of using plane-wave basis set
B3PW with CRYSTAL 3.2 times cost of PBE from using localized Gaussian basis sets[1]
B3PW91 with Crystal code is the best among hybrid functionals including B3LYP and HSE.[2]
Mean absolute Deviation (MAD) for band gaps: B3PW91 (0.09 eV), B3LYP (0.19 eV), HSE (0.27 eV)
Including Spin-orbit coupling (or splitting) is necessary.
While excitation energy indicates the energy required for excitation between any two arbitrary energy levels provided that selection rules are not violated, the band gap energy is the energy required for excitation from HOMO (the highest occupied molecular orbital) into LUMO (the lowest unoccupied molecular orbital).[3]
reference
Nature of the Chemical bond, W. A. Goddard III
[1] Resolution of the Band Gap Prediction Problem for Materials Design, J. Phys. Chem. Lett. 2016, 7, 1198
[2] Accurate Band Gaps for Semiconductors from Density Functional Theory, J. Phys. Chem. Lett. 2011, 2, 212
[3] Band gap vs exciatatoin energy