  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.
