  We study the charge-density wave phase in $\textup{TiSe}_2$ by using first
principle calculations. We show that, regardless of the local functional used
and as long as the cell parameters are in agreement with the experiment,
density-functional calculations are able to reproduce not only the structural
instability of $\textup{TiSe}_2$, but also the effective distortion observed in
the experiments. We study the electronic structure evolution of the system
under the charge-density wave deformation. In particular, we show that the
energy bands for the distorted superstructure, unfolded into the original
Brillouin zone, are in reasonable agreement with angle-resolved photoemission
spectroscopy (ARPES) data taken at low temperature. On the contrary, the energy
bands for the undistorted structure are not in good agreement with ARPES at
high temperature. Motivated by these results, we investigate the effect of the
correlation on the electrons of the localized Ti-$d$ orbitals by using the
LDA+$U$ method. We show that within this approximation the electronic bands for
both the undistorted and distorted structure are in very good agreement with
ARPES. On the other hand, the $U$ eliminates the phonon instability of the
system. Some possible explanations for this counter intuitive result are
proposed. Particularly, the possibility of taking into account the dependence
of the parameter $U$ from the atomic positions is suggested.
