  The electronic band structure of V$_2$O$_5$ is calculated using an
all-electron quasiparticle self-consistent (QS) $GW$ method, including
electron-hole ladder diagrams in the screening of $W$. The optical dielectric
function calculated with the Bethe-Salpeter equation exhibits excitons with
large binding energy, consistent with spectroscopic ellipsometry data and other
recent calculations. Sharp peaks in the direction perpendicular to the layers
at high energy are found to be an artifact of the truncation of the numbers of
bands included in the BSE calculation of the macroscopic dielectric function.
The $\varepsilon_1(\omega=0)$ gives indices of refraction in good agreement
with experiment. The excitons are charge transfer excitons with the hole
primarily on oxygen and electrons on vanadium, but depending on which exciton,
the distribution over different oxygens changes. The exciton wave functions
have a spread of about 5-15\AA, with asymmetric character for the electron
distribution around the hole depending on which oxygen the hole is fixed at.
The monolayer quasiparticle gap increases inversely proportional to interlayer
distance once the initial interlayer covalent couplings are removed which is
thanks to the long-range nature of the self-energy and the reduced screening in
a 2D system. The optical gap on the other hand is relatively independent of
interlayer spacing because of the compensation between the self-energy gap
shift and the exciton binding energy, both of which are proportional to the
screened Coulomb interaction $\hat{W}$. Recent experimental results on very
thin layer V$_2$O$_5$ obtained by chemical exfoliation provide experimental
support for an increase in gap.
