  We have synthesized Gd$_{2}$FeCrO$_{6}$ (GFCO) double perovskite which is
crystallized in a monoclinic structure with P2$_{1}/$n space group. The
UV-visible and photoluminescence spectroscopic analyses confirmed its direct
bandgap semiconducting nature. Employing experimentally obtained structural
parameters in first-principles calculation, we report the spin-polarized
electronic band structure, charge carrier effective mass, density of states,
electronic charge density distribution and optical absorption property of the
GFCO double perovskite. The effects of on-site d-d Coulomb interaction energy
(U$_{eff}$) on the electronic and optical properties were investigated by
applying a range of Hubbard U$_{eff}$ parameters from 0 to 6 eV to the Fe-3d
and Cr-3d orbitals within the generalized gradient approximation (GGA) and
GGA+U methods. When we applied U$_{eff}$ in the range of 1 to 5 eV, both the
up-spin and down-spin band structures were observed to be direct. The charge
carrier effective masses were also found to enhance gradually from U$_{eff} =$
1 eV to 5 eV, but, these values were anomalous for U$_{eff} =$ 0 and 6 eV.
These results suggest that U$_{eff}$ should be limited within the range of 1 to
5 eV to calculate the structural, electronic and optical properties of GFCO
double perovskite. We observed that considering U$_{eff} =$ 3 eV, the
theoretically calculated optical band gap 1.99 eV matched well with the
experimentally obtained value 2.0 eV. The outcomes of our finding imply that
the U$_{eff}$ value of 3 eV most accurately localized the Fe-3d and Cr-3d
orbitals of GFCO keeping the effect of self-interaction error from the other
orbitals almost negligible. Therefore, we may recommend U$_{eff} =$ 3 eV for
first-principles calculation of the electronic and optical properties of GFCO
double perovskite that might have potential in photocatalytic and related solar
energy applications.
