  We present a theoretical investigation of the electronic structure of rutile
(metallic) and M$_1$ and M$_2$ monoclinic (insulating) phases of VO$_2$
employing a fully self-consistent combination of density functional theory and
embedded dynamical mean field theory calculations. We describe the electronic
structure of the metallic and both insulating phases of VO$_2$, and propose a
distinct mechanism for the gap opening. We show that Mott physics plays an
essential role in all phases of VO$_2$: undimerized vanadium atoms undergo
classical Mott transition through local moment formation (in the M$_2$ phase),
while strong superexchange within V-dimers adds significant dynamic intersite
correlations, which remove the singularity of self-energy for dimerized
V-atoms. The resulting transition from rutile to dimerized M$_1$ phase is
adiabatically connected to Peierls-like transition, but is better characterized
as the Mott transition in the presence of strong intersite exchange. As a
consequence of Mott physics, the gap in the dimerized M$_1$ phase is
temperature dependent. The sole increase of electronic temperature collapses
the gap, reminiscent of recent experiments.
