The screened pseudo-charge repulsive potential in perturbed orbitals for band calculations by DFT+U

The conventional linear response overestimates the U in DFT+U calculations for solids with fully occupied orbitals. Here, we demonstrate that the challenge arises from the incomplete cancellation of the electron-electron Coulomb repulsion energy under external perturbation. We applied the second charge response, denoted as the "pseudo-charge" model, to offset such residue effects. Counteracting between these two charge response-induced Coulomb potentials, the U parameters are self-consistently obtained by fulfilling the conditions for minimizing the non-Koopmans energy. Moreover, the pseudo-charge-induced repulsive potential shows a screening behavior related to the orbital occupation and is potentially in compliance with the screened exact exchange-correlation of electrons. The resultant U parameters are self-consistent solutions for improved band structure calculations by the DFT+U method. This work extends the validity of the linear response method to both partially and fully occupied orbitals and gives a reference for estimating the Hubbard U parameter prior to other advanced methods. The U parameters were determined in a transferability test using both PBE and hybrid density functional methods, and the results showed that this method is independent of the functional. The electronic structures determined from the hybrid-DFT+U^hybrid approach are provided. Comparisons are also made with the recently developed self-consistent hybrid-DFT+U_w method.