these states. Localization of CB states is much weaker, and hence their dispersion is much stronger. Therefore, the average height of the DOS is smaller in comparison with the VB states. It is interesting to report in detail the energy band gaps calculated within the two different GW schemes proposed in the present publication, namely $G_{0}W_{0}$ and scQP-GW. As it is clear from Table \[tab:mygap\] a nearly rigid shift of about 1 eV appears going from $G_{0}W_{0}$ to scQP-GW transition energies. Therefore, with respect to $CdF_{2}$, considering Fig. \[fig:1\] and Table \[tab:mygap\], the action of the self-consistency upon the eigenvalues operates a kind of ”second order scissor operator rule” on the conduction bands to obtain good accordance with experiments. The valence bandwidths after the two approximations of *XC*, respectively $G_{0}W_{0}$ and scQP-GW, agree within 0.5 eV. Thereby, the scQP-GW value of 22.99 eV corresponds excellently to an experimental estimate of 23 eV reported by Raisin.[@raisin]

Two-particles properties and dielectric function {#sec:2particle}
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The treatment of excitons and resulting optical spectra, i.e., the full excitonic problem, requires to set up and diagonalize an electron-hole Hamiltonian Ĥ. Within Hedin’s GW scheme and a restriction to static screening this two-particle Hamiltonian reads for 