MIT in the single band case, namely, that as the DOS($\omega=0$) becomes zero the Kondo effect can no longer be sustained and the impurities lose their respective Kondo screening clouds. In the single band (single-site) case, in the Mott insulator state one is left with almost free local moments. However, in the present situation a strong RKKY-like magnetic interaction between the two sites takes over and one has intra-dimer magnetic screening. This dramatic enhancement of the intra-dimer magnetic interaction translates into the sharp increase of the intra-dimer effective hopping $\tilde{t}_\perp = t_\perp + \Re\Sigma_{12}(0)$, which drives the opening of the gap (see Fig. \[fig:dimer\_transition\_spectra\]) [@najera2017_resolving_vo2_controversy]. Interestingly this sharp increase of the $\Re\Sigma_{12}(0)$ was also observed in DFT+CDMFT calculations in VO$_2$ by Brito et al. [@brito2016_metal_insulator_transition_in_vo]. They related this effect to the gap-opening by the B/AB band splitting of the a$_{1g}$ orbital in VO$_2$. It is interesting to see that our present model Hamiltonian does capture the same basic physical mechanism, albeit in a simplified scenario that makes its physical interpretation transparent. Thus, we see how the renormalization of the intra-dimer hopping that drives the MIT originates in the loss of lattice Kondo screening and the concomitant boost of the local 