  The common interpretation of the LDA band structure of V$_{2}$O$_{3}$ is that
the apparent splitting of the $a_{1g}$ band into a low intensity structure deep
below the Fermi energy and a high intensity feature above it, is due to the
bonding-antibonding coupling of the vertical V-V pair. Using tight-binding
fitting to --as well as first-principles NMTO downfolding of-- the spin-up
LDA+U $a_{1g}$ band, we show that there are other hopping integrals which are
equally important for the band shape as the integral for hopping between the
partners of the pair.
