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Electronic and vibrational properties of $\textup{TiSe}_2$ in the charge-density wave phase from first principles

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.