Quantum-Quantum and Quantum-Quantum-Classical Schemes for Near-Gap Excitations with Projection-Based-Embedded -Bethe-Salpeter Equation.

We present quantum-quantum and quantum-quantum-classical schemes based on many-body Green's functions theory in the approximation with the Bethe-Salpeter equation (-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 -BSE offering significant computational advantages. However, as PbE can modify the single-particle states in the Kohn-Sham (KS) ground state calculation and screening effects from the inactive region are not automatically included in -BSE, results from such PbE--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 omitting the 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 π-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. Our results suggest that to obtain agreement of approximately 0.1 eV between near-gap excitation energies from embedded and full calculations, the active region should be chosen based on the Mulliken population of the full highest-occupied molecular orbital and that careful benchmarking should be done on the KS level before the actual -BSE steps when basis set truncation is used. We find that PbE--BSE offers significant reductions in computation times and, more importantly, memory requirements, making calculations for considerably larger systems tractable.