3 eV gap error) can be considered as an improvement also for fluorides.

In the present work QP bands of $CdF_{2}$ are determined by an iterative solution of the QP equations given in Eq.(\[eq:2\]) with the *XC* self-energy in the GW approximation. The iteration starts with the gKS equations, with a self-energy derived from the nonlocal HSE03 hybrid functional (zeroth order).[@hse03] Subsequently, the GW corrections are calculated in first-order perturbation theory, i.e., within the one-shot $G_{0}W_{0}$. Thereby, the full frequency dependence of the RPA dielectric function entering the screened Coulomb interaction ($W_0$) is taken into account sampled on a frequency grid with 128 points up to $\hbar\omega\approx$350 eV, i.e., no plasmon pole models[@Stankovski] or model dielectric functions[@cappellinissc92] have been involved. We present here QP eigenvalues calculated in the $G_{0}W_{0}$ approach on top of the HSE03 ground-state electronic structures.

The strategy to calculate QP corrections then follows: The band structure and DOS have been calculated first with the PBEsol GGA, then a HSE03 calculation is performed, and finally we apply the $G_{0}W_{0}$ scheme on top of HSE03. Even after this step, the QP energies are of the order of $1$ eV smaller than the experimental ones. For comparison we made also 