Electronic structure of TiSe$_2$ from a quasi-self-consistent G$_0$W$_0$ approach

In a previous work it was shown that the inclusion of exact exchange is crucial for a first principles description of both the electronic- and the vibrational properties of TiSe$_2$ [Phys. Rev. Lett. 119 456325 (2017)]. The GW approximation provides a parameter-free description of screened exchange but is usually employed perturbatively (G$_0$W$_0$) making results more or less dependent on the starting point. In this work we study the GW band structure of TiSe$_2$ and show that it is a case extremely sensitive to the choice of G$_0$. We then develop a quasi-self-consistent scheme based on the random phase approximation (RPA) and the optimized effective potential of hybrid density functional theory. This approach generates an optimal G$_0$W$_0$ starting-point and a hybrid exchange parameter consistent with the RPA. The high-temperature phase of TiSe$_2$ is found to be semi-metallic, in agreement with most experimental results. Furthermore, the optimized hybrid functional accurately reproduces the low-temperature charge density wave phase.