  Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_3$-PbTiO$_3$ perovskite-based crystals attract
considerable scientific interest due to their interesting properties and
possible use in piezoelectricity and photovoltaics. To understand the local
structure and fundamental properties of such materials, in this work, we
focused on the study within the density functional theory of structural,
electronic, and optical properties of
Pb[(Mg$_{1/3}$Nb$_{2/3}$)$_{0.75}$Ti$_{0.25}$]O$_3$. Using GGA(PBEsol)
approximation for structure optimization gives a good agreement with
experimental data. Through the variation in Hubbard $U$ parameters to
GGA(PBEsol) functional, we achieve the bandgap for the
Pb[(Mg$_{1/3}$Nb$_{2/3}$)$_{0.75}$Ti$_{0.25}$]O$_3$ which is in good agreement
with the experimental results. The study of the bond populations showed that
the Mg-O bond demonstrates no covalency, whereas there is a significant Ti-O
and Nb-O covalent bonding. Such different bonding characteristics must be
responsible for the relaxor properties of
Pb[(Mg$_{1/3}$Nb$_{2/3}$)$_{0.75}$Ti$_{0.25}$]O$_3$ compound. In addition, the
investigations of the optical properties of the
Pb[(Mg$_{1/3}$Nb$_{2/3}$)$_{0.75}$Ti$_{0.25}$]O$_3$ by adopting Hubbard $U$
corrections, modifying the error of the GGA approximation, and confirming the
electronic analysis, were performed.
