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A quantitative description of the excited electronic states of point defectsand impurities is crucial for understanding materials properties, and possibleapplications of defects in quantum technologies. This is a considerablechallenge for computational methods, since Kohn-Sham density-functional theory(DFT) is inherently a ground state theory, while higher-level methods are oftentoo computationally expensive for defect systems. Recently, embeddingapproaches have been applied that treat defect states with many-body methods,while using DFT to describe the bulk host material. We implement such anembedding method, based on Wannierization of defect orbitals and theconstrained random-phase approximation approach, and perform systematiccharacterization of the method for three distinct systems with currenttechnological relevance: a carbon dimer replacing a B and N pair in bulkhexagonal BN (C$_{\text{B}}$C$_{\text{N}}$), the negatively chargednitrogen-vacancy center in diamond (NV$^-$), and an Fe impurity on the Al sitein wurtzite AlN ($\text{Fe}_{\text{Al}}$). For C$_{\text{B}}$C$_{\text{N}}$ weshow that the embedding approach gives many-body states in agreement withanalytical results on the Hubbard dimer model, which allows us to elucidate theeffects of the DFT functional and double-counting correction. For the NV$^-$center, our method demonstrates good quantitative agreement with experimentsfor the zero-phonon line of the triplet-triplet transition. Finally, weillustrate challenges associated with this method for determining the energiesand orderings of the complex spin multiplets in $\text{Fe}_{\text{Al}}$.

Quantum embedding methods for correlated excited states of point defects: Case studies and challenges