  We analyze the effects on the structural and electronic properties of
vanadium dioxide (VO$_2$) of adding an empirical inter-atomic potential within
the density-functional theory$+V$ (DFT$+V$) framework. We use the DFT$+V$
machinery founded on the extended Hubbard model to apply an empirical
self-energy correction between nearest-neighbor vanadium atoms in both rutile
and monoclinic phases, and for a set of structures interpolating between these
two cases. We observe that imposing an explicit intersite interaction $V$ along
the vanadium-vanadium chains enhances the characteristic bonding-antibonding
splitting of the relevant bands in the monoclinic phase, thus favoring
electronic dimerization and the formation of a band gap. We then explore the
effect of $V$ on the structural properties and the relative energies of the two
phases, finding an insulating global energy minimum for the monoclinic phase,
consistent with experimental observations. With increasing $V$, this minimum
becomes deeper relative to the rutile structure, and the transition from the
metallic to the insulating state becomes sharper. We also analyze the effect of
applying the $+V$ correction either to all or only to selected
vanadium-vanadium pairs, and both in the monoclinic as well as in the metallic
rutile phase. Our results suggest that DFT$+V$ can indeed serve as a
computationally inexpensive unbiased way of modeling VO$_2$ which is well
suited for studies that, e.g., require large system sizes.
