address the question of the physical crossover from a pure Peierls insulator, which is a band insulator of a lattice of dimers without correlations, to the pure (un-dimerized) correlated Mott insulator. We shall see that the behavior of the system across the crossover regime is non-trivial. One of our main results is that as the system evolves from the Mott to the Peierls insulator there are at least four different regimes, including an unprecedented state where the Hubbard bands have an electronic structure with mixed coherent and incoherent character. Interestingly, VO$_2$ seems to be in this peculiar regime, which might eventually be seen in spectroscopy experiments. On the other hand, we also investigated the correlated “monoclinic” metallic state and show that it can be understood in simple terms as a renormalized two-band heavy metal at low frequencies. Interestingly, indications of a monoclinic metal in VO$_2$ have been reported in several experimental studies [@kim2006_monoclinic_correlated_metal_phase_invo2as; @arcangeletti2007_evidence_pressure_induced_metallization_process; @qazilbash2011_nanoscale_imaging_electronic_structural_transitions_vanadium_dioxide; @tao2012_decoupling_structural_electronic_phase_transitions_invo2; @yoshida2014_ultrafast_photoinduced_transition_insulating_vo; @morrison2014_photoinduced_metal_like_phase_monoclinic; @laverock2014_direct_observation_decoupled_structural_electronic; @wegkamp2014_instantaneous_band_gap_collapse_photoexcited].

Consistent with the DFT+CDMFT studies, we find that VO$_2$ should be characterized as a realization of a Mott state in a dimer lattice rather than a renormalized Peierls band insulator. However, in a larger perspective, our systematic investigation of the 