Modeling the self-energy and wavefunction relaxation in the orbitals

The strong boundary normalized condition of wavefunction for fully occupied semicore 3d orbitals leads the linear response DFT+U on such metal oxide to have an insurmountable obstacle in Hubbard U determination. We treated the orbital self-energy and orbital relaxation as components of eigenvalues with respective orbital occupation number that follows the Fermi-Dirac distribution. By self-consistently solving the second partial deviation of total energy based on the most simple local density formalism with Hubbard U correction, we found the local density exchange-correlation potential functional can only give a minimized residue of the self-energy and orbital relaxation on the focus orbital if the Janak theorem maintained. Such residue turns to well counteracted in the fully occupied orbitals and non-zero the partially occupied orbitals. With keeping the validation of Janak theorem on localized orbitals, the self-consistent cycle by local density functional with Hubbard U correction cannot find out a set of orbital occupation that simultaneously offsets the orbital self-energy and relaxations in the empty and partially filled shell, but returns a unique set of the occupation for fully occupied shell. The band gap calculations on fully occupied orbital based compounds are thus improved and the relaxed lattices are also shown based on minimization of the self-energy error, which shows a possible route for accurate excited state studies.