addition, such a DOS qualitatively resembles that of a three-dimensional lattice system. In fact, it has a finite bandwidth given by $4t$ defined in equation . As in previous works, we adopt as unit of energy the half bandwidth $D=2t=1$. We should emphasize here that the physics of models treated within DMFT in general do not depend on specific geometry of the lattice, but on the nature of the quantum impurity problem. As we mentioned already, the realistic VO$_2$ lattice within DFT+CDMFT has the same type of quantum dimer impurity as in the present case. Our model has the additional simplification of considering one orbital at each site instead of the 3 in the realistic case. We may see in Fig. \[fig0\] schematic “rutile” and “monoclinic” lattices. From our previous discussion, the key feature is that in the former case there is a single site in the unit cell, while in the latter case there is a dimer. The “rutile” lattice can be qualitatively associated to a conventional single-band Hubbard model [@georges1996_dynamical_mean_field_theory_strongly]. While the “monoclinic” one can be thought of as two copies of Hubbard models coupled at every unit cell site by the intra-dimer hopping $t_\perp$. In the limit 