  Motivated by recent results of $N$th order muffin-tin orbital (NMTO)
implementation of the density functional theory (DFT), we re-examine
low-temperature ground-state properties of the anti-ferromagnetic insulating
phase of vanadium sesquioxide V$_2$O$_3$. The hopping matrix elements obtained
by the NMTO-downfolding procedure differ significantly from those previously
obtained in electronic structure calculations and imply that the in-plane
hopping integrals are as important as the out-of-plane ones. We use the NMTO
hopping matrix elements as input and perform a variational study of the ground
state. We show that the formation of stable molecules throughout the crystal is
not favorable in this case and that the experimentally observed magnetic
structure can still be obtained in the atomic variational regime. However the
resulting ground state (two $t_{2g}$ electrons occupying the degenerate $e_g$
doublet) is in contrast with many well established experimental observations.
We discuss the implications of this finding in the light of the non-local
electronic correlations certainly present in this compound.
