  Quantum embedding theories are promising approaches to investigate
strongly-correlated electronic states of active regions of large-scale
molecular or condensed systems. Notable examples are spin defects in
semiconductors and insulators. We present a detailed derivation of a quantum
embedding theory recently introduced, which is based on the definition of
effective Hamiltonians. The effect of the environment on a chosen active space
is accounted for through screened Coulomb interactions evaluated using density
functional theory. Importantly, the random phase approximation is not required
and the evaluation of virtual electronic orbitals is circumvented with
algorithms previously developed in the context of calculations based on
many-body perturbation theory. In addition, we generalize the quantum embedding
theory to active spaces composed of orbitals that are not eigenstates of
Kohn-Sham Hamiltonians. Finally, we report results for spin defects in
semiconductors.
