  Quasiparticle excitation energies and optical properties of TiO$_{2}$ in the
rutile and anatase structures are calculated using many-body perturbation
theory methods. Calculations are performed for a frozen crystal lattice;
electron-phonon coupling is not explicitly considered. In the GW method,
several approximations are compared and it is found that inclusion of the full
frequency dependence as well as explicit treatment of the Ti semicore states
are essential for accurate calculation of the quasiparticle energy band gap.
The calculated quasiparticle energies are in good agreement with available
photoemission and inverse photoemission experiments. The results of the GW
calculations, together with the calculated static screened Coulomb interaction,
are utilized in the Bethe-Salpeter equation to calculate the dielectric
function $\epsilon_{2}(\omega)$ for both the rutile and anatase structures. The
results are in good agreement with experimental observations, particularly the
onset of the main absorption features around 4 eV. For comparison to low
temperature optical absorption measurements that resolve individual excitonic
transitions in rutile, the low-lying discrete excitonic energy levels are
calculated with electronic screening only. The lowest energy exciton found in
the energy gap of rutile has a binding energy of 0.13 eV. In agreement with
experiment, it is not dipole allowed, but the calculated exciton energy exceeds
that measured in absorption experiments by about 0.22 eV and the scale of the
exciton binding energy is also too large. The quasiparticle energy alignment of
rutile is calculated for non-polar (110) surfaces. In the GW approximation, the
valence band maximum is 7.8 eV below the vacuum level, showing a small shift
from density functional theory results.
