Quantum-quantum and quantum-quantum-classical schemes with projection-based-embedded $GW$-Bethe--Salpeter Equation

We present quantum-quantum and quantum-quantum-classical schemes based on many-body Green's functions theory in the $GW$ approximation with the Bethe--Salpeter equation ($GW$-BSE) employing projection-based-embedding (PbE). Such approaches allow defining active and inactive subsystems of larger, complex molecular systems, with only the smaller active subsystem being explicitly treated by $GW$-BSE offering significant computational advantages. However, as PbE can modify the single-particle states in the ground state calculation and screening effects from inactive region are not automatically included in $GW$-BSE, results from such PbE-$GW$-BSE calculations can deviate from a full-system reference. Here, we scrutinize in detail, e.g., the individual and combined effects of different choices of active regions, the influence of screening from the inactive region, and strategies for basis set truncation on frontier orbital and near-gap electron-hole excitation energies. As prototypical systems, we consider a diketopyrrolopyrrole bicyclic ring including side-chains, a polarity-sensitive dye (prodan) in aqueous environment, and a $\pi$-stacked dimer of benzene and tetracyanoethylene in water, respectively, covering a variety of excitation characters in molecular systems with complex chemical environments and photoinduced processes.