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.
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