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Vanadium Sesquioxide (V$_2$O$_3$) is an antiferromagnetic insulator below$T_{\mathrm N}\approx$ 155 K. The magnetic order is not of C- or G-type as onewould infer from the bipartite character of the hexagonal basal plane in thehigh-temperature corundum structure. In fact, the N\'eel transition isaccompanied by a monoclinic distortion that makes one bond of the honeycombplane inequivalent from the other two, thus justifying a magnetic structurewith one ferromagnetic bond and two antiferromagnetic ones. We show here thatthe magnetic ordering, the accompanying monoclinic structural distortion, themagnetic anisotropy and also the recently discovered high-pressure monoclinicphase, can all be accurately described by conventional electronic structurecalculations within GGA and GGA+U. Our results are in line with DMFTcalculations for the paramagnetic phase, which predict that the insulatingcharacter is driven by a correlation-enhanced crystal field splitting between$e^\pi_g$ and $a_{1g}$ orbitals that pushes the latter above the chemicalpotential. We find that the $a_{1g}$ orbital, although almost empty in theinsulating phase, is actually responsible for the unusual magnetic order as itleads to magnetic frustration whose effect is similar to anext-nearest-neighbor exchange in a Heisenberg model on a honeycomb lattice.

Low-temperature magnetic ordering and structural distortions in Vanadium Sesquioxide (V$_2$O$_3$)