Regarding the comparison to MO formulations, the matrix WC as defined in Eqs. ( ) and ( ) only comprises occupied-occupied contributions while analogous MO formulations treat the combined occupied-virtual space. Contributions referring to the virtual space of Refs. ( ) are included in our formalism via Q projections, explicitly taken into account due to the reformulated Brillouin condition (Eq. ( )) and the WX constraint. Furthermore, the intermediate H stems from the KS matrix contributions of Eq. ( ) and thus the explicit formula depends on the chosen GS reference, given here for hybrid functionals with or without ADMM,