Hartree-Fock (HF) calculations, which show as expected deviations in the opposite direction (i.e., an unrealistic overestimate of the gaps).[@hl; @strinati] As the last step, in order to open that gap further, we performed a ”self-consistent quasiparticle energy” calculation within the GW approximation (namely, scQP-GW) which is a kind of ”iterated $G_{0}W_{0}$” just updating the eigenvalues only but keeping the HSE03 wave functions fixed. This procedure gives almost perfect gaps with respect to experiment.[@smith; @caruso]

The main outcomes of the present paragraph are reported in Table \[table:2\] and in Figs. \[fig:1\] and \[fig:2\]. More details are also discussed in Ref. . The QP values for the fundamental gaps $\Gamma-\Gamma$ (direct) and $W-\Gamma$ (indirect) obtained within different approximations are also reported in Table \[table:2\].

     data in eV       HF     PBEsol   HSE03   $G_{0}W_{0}$   scQP-GW              Exp.
  ----------------- ------- -------- ------- -------------- --------- -----------------------------
   $\Gamma-\Gamma$   16.02    3.41    5.50        7.64        8.58     8.4[@orlowski],8.7[@raisin]
     $W-\Gamma$      15.30    2.91    4.91        6.90        7.83           7.8[@orlowski]

  : QP energies for the fundamental gaps of $CdF_{2}$, between lowest CB and highest VB at high symmetry points, are computed with several approximations and compared with available experiments.[]{data-label="table:2"}

        Direct bandgaps         $G_{0}W_{0}$   scQP-GW
  ---------------------------- -------------- ---------
        $L\rightarrow L$           12.11        13.08
   $\Gamma\rightarrow\Gamma$        7.64        8.58
       $ X\rightarrow X $          12.57        