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Considerable insight can be gained by comparing CI and LDA+DMFT calculations. Our LDA+DMFT cal- culations, performed using N $^{th}$-order muffin-tin orbital (NMTO) downfolded Hamiltonian for the 1.1% Cr-doped V$_{2}$O$_{3}$ and Hirsch-Fye Quantum Monte Carlo as impurity solver, confirm the above mentioned tendency (we obtain a mixing of 50:50 and 35:65 for the ( e π $_{g}$, a$_{1}$$_{g}$ ):( e π $_{g}$, e π $_{g}$) oc- cupation in the PM and PI phases). For LDA+DMFT we have chosen the same interaction parameter U = 4 . 2 eV for the PI phase as in Ref. [9] and followed their sug- gestion to slightly decrease its value in the PM phase (we assume U = 4 . 0 eV)[23]. Remarkably the simple struc- ture of the core hole potential in the K -edge spectroscopy allows us to associate the pre-edge spectrum with the k − integrated spectral function above the Fermi energy calculated by LDA+DMFT. The electron–addition part of the spectral function shows three main features in PM phase: a coherent excitation at the Fermi level and a double peak associated to the incoherent electronic ex- citations i.e. the upper Hubbard band (UHB), similarly to the undoped compound. In the PI phase obviously, only the latter survives. Comparison with the experi- mental spectra clearly shows that the pre-edge features have to be related to the “incoherent” part of the spectral function only. The physical reason is that the core hole potential localizes the electrons destroying the coher- ent quasiparticle excitations and making the XAS spec- trum atomic-like. All the “incoherent” LDA+DMFT, CI, and experimental spectra shown in Fig. 2 agree in many aspects, especially as for the splitting of the first two peaks by ≈ 2.0 eV ( ≈ 1.8 eV in experiment) which originates in LDA+DMFT from the Hund’s exchange J in the Kanamori Hamiltonian. This further validates the choice of J = 0 . 7 eV used in our calculations con- trasting with larger values assumed in previous stud- ies [8, 10, 23], and also clarifies the mismatch between XAS and LDA+DMFT spectra reported in the undoped V$_{2}$O$_{3}$ compound [8]. Moreover, the ratio between A and B peak displays the same trend in the PM-PI transition as the CI (or experimental) data. The quantitative dif- ference between the two calculations is attributed to the lack of matrix elements in LDA+DMFT. 

The intensity ratio of the first two incoherent excita- tions peaks A and B (associated to the quartet and dou- blet states in the oversimplified picture) thus appears as the key spectral parameter to understand the differences between PM and PI. Even in a powder sample, this ratio is still sensitive to the a$_{1}$$_{g}$ orbital occupation of the initial state. Indeed, due to the trigonal distortion a consider- able spectral weight transfer from the peak B to higher energies (corresponding to final states with two a$_{1}$$_{g}$ elec- trons in the limit of large Δ$_{trig}$) can take place for the ( e π $_{g}$, a$_{1}$$_{g}$ ) but not for the ( e π $_{g}$, e π $_{g}$) initial state. Therefore, the K pre-edge XAS can serve as a direct probe of the a$_{1}$$_{g}$ orbital occupation in the ground-state. As a rule of thumb, the larger the ratio between the SW of A and B, 

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the larger the a$_{1}$$_{g}$ orbital occupation. 

Hard X-ray absorption also provides a unique way to explore the “pressure” pathway across the MIT, from which relevant information can be extracted by applying our interpretation scheme. Fig. 3(a) shows the XAS pow- der spectra of the P-induced MIT with the corresponding spectra for the T- and doping-driven transition (cf. the loci in the phase diagram, Fig. 3). To ease the com- parison, the pressure dataset (P) obtained in transmis- sion mode has been deconvolved from the 1 s Lorentzian lifetime broadening (1 eV FWHM) using the GNXAS code [24] to match the improved resolution of datasets ( x ) and (T) measured by PFY-XAS. Together with the PM − PI differences in Fig. 3(b), Fig. 3(a) clearly evi- dences that, besides a rigid shift of the first two peaks of ∼ +0.13 eV and contrary to the doping- or T-driven tran- sition, very small changes in spectral shapes and weights are observed in the P-driven MIT. In the light of the arguments discussed above, our finding proves that the metallic state reached by applying pressure is character- ized by a much lower occupation of the a$_{1}$$_{g}$ orbitals com- pared to the metallic state reached just by changing tem- perature or doping. Importantly, the spectra measured through the doping induced MIT are identical within the experimental uncertainty to those measured through the T-driven transition. The temperature-doping equiv- alence is confirmed by our photoemission data displayed in Fig. 3(c,d) [15] and is consistent with the very similar lattice parameter changes across the transition [25]. The x and T equivalence is also borne out by the observa- tion from XAS at the L -edges in doped V$_{2}$O$_{3}$ [16] that the a$_{1}$$_{g}$ occupation within both the PM or PI phases is mostly independent of the doping level. Hence, the local incoherent excitations probed by XAS at the V L edge or K pre-edge are not directly affected by disorder [26]. 

