  A first-principles atomic orbital-based electronic structure method is used
to investigate the low index surfaces of rutile Titanium Dioxide. The method is
relatively cheap in computational terms, making it attractive for the study of
oxide surfaces, many of which undergo large reconstructions, and may be
governed by the presence of Oxygen vacancy defects. Calculated surface charge
densities are presented for low-index surfaces of TiO$_2$, and the relation of
these results to experimental STM images is discussed. Atomic resolution images
at these surfaces tend to be produced at positive bias, probing states which
largely consist of unoccupied Ti 3$d$ bands, with a small contribution from O
2$p$. These experiments are particularly interesting since the O atoms tend to
sit up to 1 angstrom above the Ti atoms, so providing a play-off between
electronic and geometric structure in image formation.
