  We calculate the lattice dielectric function of strongly anharmonic rutile
$\mathrm{TiO}_2$ from ab initio anharmonic lattice dynamics methods. Since an
accurate calculation of the $\Gamma$ point phonons is essential for determining
optical properties, we employ the modified self-consistent approach, including
third-order anharmonicity as well as fourth-order anharmonicity. The resulting
optical phonon frequencies and linewidths at the $\Gamma$ point much better
agree with experimental measurements than those from a perturbative approach.
We show that the four-phonon scattering process contributes as much as the
third-order anharmonic term to phonon linewidths. Furthermore, incorporating
the frequency dependence of phonon linewidth reveals that experimentally known
but unidentified peaks of the dielectric function are due to two-phonon
process. This work emphasizes the importance of a self-consistent approach in
predicting the optical properties of highly anharmonic materials.
