In Figure and Figure , we plot the vertical excitation energies using a (2,8) active space as was used in reference 32. The corresponding numbers are reported in Table . The active orbitals are reported in the SI. The excitation energies obtained from various non-embedding correlated theories exhibit closer agreement with one another in the larger (2,8) active space, providing a means of evaluating the performance of DME-PDFT for both smaller (2,2) and larger (2,8) active spaces. For the (2,8) active space, all pDME-tPBE and tPBE0 excitation energies agree to within 0.05 eV of the non-embedding references, whereas NEVPT2-DMET shows a higher (although not very significant) sensitivity to the impurity cluster. To quantify the sensitivity of the excitation energies to the embedding space we report the slopes for all the linear extrapolations in Tables and of the SI. Next, we investigate the S 0 → S 1 and S 0 → T 1 excitations of the M-center, which is a neutral oxygen di-vacancy (OOV) on the (100) monolayer of MgO. This defect is also known as the M centre. Here, the removal of two neutral oxygen atoms leaves four electrons in the cavity created by the two missing oxygens. In the singlet ground state these electrons occupy the two defect-localized states present between the VBM and the CBM. Experimentally, Kramer et al. tentatively assigned the 1.0 eV and 1.3 eV adsorption peaks to the M center on thin films of MgO. The computational model is shown in Figure . We consider four impurity clusters as shown in Figure . We show the five active natural orbitals forming the minimal (4,5) active space in Figure . In Figure , we present the vertical excitation energies for the OOV system. The corresponding numbers are reported in Table . Although the excitation energies calculated using pDME-tPBE0 for the three larger fragments OOV+Mg 6 O 2 , OOV+Mg 6 O 6 and OOV+Mg 6 O 10 are within 0.07 eV of the corresponding non-embedded calculations, the smallest fragment OOV+Mg 6 deviates by 0.14 eV for the S 0 → T 1 gap. This highlights the inadequacy of the smallest impurity cluster (OOV+Mg 6 ) in providing an accurate approximation of the overall system densities. Therefore, when extrapolating to the non-embedding limit, only the three larger fragments are taken into account. The excitation energies for the OOV+Mg 6 impurity cluster clearly fall outside the range of the linear extrapolation, as indicated by the detailed analysis presented in Section S01 of the Supporting Information, which includes R 2 values for the linear fits. The results for the OOV system appear to be slightly more sensitive, as indicated by the slopes of the linear extrapolations in Table of the SI, compared to Table : Vertical excitation energies (in eV) of the oxygen divacancy on the MgO(100) surface obtained using CAS-DMET, NEVPT2-DMET, pDME-tPBE and pDME-tPBE0, with an active space of 4 electrons in 5 orbitals. The extrapolated energies from linear regression of the last three points are labeled as "Extrap". "Reference" here indicates the non-embedded Γ-point CASSCF, NEVPT2, tPBE and tPBE0 calculations. Next, we explore electronic excitations in the oxygen mono-vacancy on MgO surfaces con- In the three-layer case, like in the example above, the non-embedding calculations are pro-