  In this paper we explore whether the quantum oscillation signals recently
observed in ortho-II YBa$_2$Cu$_3$O$_{6.5}$ may be explained by conventional
density functional band-structure theory. Our calculations show that the Fermi
surface of YBa$_2$Cu$_3$O$_{6.5}$ is extremely sensitive to small shifts in the
relative positions of the bands. With rigid band shifts of around 30 meV small
tubular pockets of Fermi surface develop around the Y point in the Brillouin
zone. The cross-sectional areas and band masses of the quantum oscillatory
orbits on these pockets are close to those observed. The difference between the
bandstructure of YBa$_2$Cu$_3$O$_{6.5}$ and YBa$_2$Cu$_4$O$_{8}$ are discussed.
