ground-state calculations without spoiling accuracy too much. In our case we use a cutoff of 475 eV.

The matrix elements of the excitonic Hamiltonian have been evaluated by means of HSE03 wave functions. Since the HSE03+$G_{0}W_{0}$ QP eigenvalues still leads to a gap underestimate (see Table \[table:2\]), HSE03+$scQPGW$ eigenvalues are employed instead of HSE03+$G_{0}W_{0}$ eigenvalues on the main diagonal of the Hamiltonian. It has been carefully checked that no change of band ordering occurs when going from PBEsol to HSE03.

In Fig. \[fig:3\], we report the principal result of the present calculations: the imaginary part of the dielectric function $Im ~\varepsilon (\omega)$ determined within the BSE approach by using the method of Schmidt *et al.*.[@smith] The real part has been obtained by a Kramers-Kronig (KK) transform afterwards. Also the experimental results after Bourdillon[@bourdillon] are reported, which are obtained with synchrotron radiation technique. The agreement both in the positions and intensities of the peaks between theory and experiment should be stressed for the imaginary part of the dielectric function (e.g., the first main three peaks are located at 7.7, 13.1, and 15.1 eV, respectively). As shown in the top panel of Fig. \[fig:3\], the good agreement turns out for the full 