  Using an \textit{ab initio} approach based on the GW approximation which
includes strong local \textbf{k}-space correlations, the Metal-Insulator
Transition of M$_2$ vanadium dioxide is broken down into its component parts
and investigated. Similarly to the M$_{1}$ structure, the Peierls pairing of
the M$_{2}$ structure results in bonding-antibonding splitting which stabilizes
states in which the majority of the charge density resides on the Peierls
chain. This is insufficient to drop all of the bonding states into the lower
Hubbard band however. An antiferroelectric distortion on the neighboring
vanadium chain is required to reduce the repulsion felt by the Peierls bonding
states by increasing the distances between the vanadium and apical oxygen
atoms, lowering the potential overlap thus reducing the charge density
accumulation and thereby the electronic repulsion. The antibonding states are
simultaneously pushed into the upper Hubbard band. The data indicate that
sufficiently modified GW calculations are able to describe the interplay of the
atomic and electronic structures occurring in Mott metal-insulator transitions.
