4f fine-structure levels as the dominant error in the electronic structures of binary lanthanide oxides

The ground-state 4f fine-structure levels in the intrinsic optical transition gaps between the 2p and 5d orbitals of lanthanide sesquioxides (Ln₂O₃, Ln = La...Lu) were calculated by a two-way crossover search for the U parameters for DFT + U calculations. The original 4f-shell potential perturbation in the linear response method were reformulated within the constraint volume of the given solids. The band structures were also calculated. This method yields nearly constant optical transition gaps between Ln-5d and O-2p orbitals, with magnitudes of 5.3 to 5.5 eV. This result verifies that the error in the band structure calculations for Ln₂O₃ is dominated by the inaccuracies in the predicted 4f levels in the 2p-5d transition gaps, which strongly and non-linearly depend on the on-site Hubbard U. The relationship between the 4f occupancies and Hubbard U is non-monotonic and is entirely different from that for materials with 3d or 4d orbitals, such as transition metal oxides. This new linear response DFT + U method can provide a simpler understanding of the electronic structure of Ln₂O₃ and enables a quick examination of the electronic structures of lanthanide solids before hybrid functional or GW calculations.