In the above, we have already conveniently ignored the frequency dependence, implying the use of the static limit, as was also done in GW-BSE-in-DFT embedding, for the whole frequency range. As an alternative, it was suggested also in Ref. to use the COHSEX approximation (cf., Eq. ( ) and Eq. ( )) to determine state-specific correction terms to the quasiparticle energies of an isolated system in a vacuum, ε QP i , as in