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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 essential for a first principles description of both the electronic- and the vibrational properties of TiSe$_2$, M. Hellgren et al. [Phys. Rev. Lett. 119, 176401 (2017)]. The $GW$ approximation provides a parameter-free description of screened exchange but is usually employed perturbatively ($G_0W_0$) making results more or less dependent on the starting point. In this work, we develop a quasi-self-consistent extension of $G_0W_0$ based on the random phase approximation (RPA) and the optimized effective potential of hybrid density functional theory. This approach generates an optimal $G_0W_0$ starting-point and a hybrid exchange parameter consistent with the RPA. While self-consistency plays a minor role for systems such as Ar, BN and ScN, it is shown to be crucial for TiS$_2$ and TiSe$_2$. We find the high-temperature phase of TiSe$_2$ to be a semi-metal with a band structure in good agreement with experiment. Furthermore, the optimized hybrid functional agrees well with our previous estimate and therefore accurately reproduces the low-temperature charge density wave phase.