of 7.7, 7.8, and 8.38.5 at 0 K, 80 K, and 300 K, respectively)[@young] together with phonon data,[@deligoz; @cribier] which lead toward values in the range of 2.32.5 for $\varepsilon_{\infty}$. Hence all these values from direct measurements are in excellent agreement with our computed value. Furthermore the real part of the dielectric function $Re ~\varepsilon (\omega)$, as shown in the second panel of Fig. \[fig:3\], is in close agreement with the experimental data after Ref. .

In Fig. \[fig:4\], we show the imaginary part of the dielectric function $Im ~\varepsilon (\omega)$ within the different approximations proposed here, starting from the DFT reference which is the PBEsol one. Going from this approximation to the gKS HSE03 one an opening of the gap results. Both GW schemes further enlarge the gap energies. Finally the BSE results are disposed which reproduce the experiment at best in terms of oscillator strengths, peak positions, and onset behavior.

Optical and loss functions {#sec:optical}
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In Fig. \[fig:5\], the imaginary and real part of the refractive index in comparison with experiment after Ref. are shown. A fair agreement results with respect to the three main peaks at 7.8, 13.2, and 15.2 eV and to the high-energy region 