  We consider the optical conductivity of carriers scattered by local sombrero
potentials associated with hindered planar rotators in solids. Examples for
hindered rotators are provided by off-center substitutional Li ions in colored
alkali halides and off-site apical oxygens O(A) in layered perovskites. We
obtain off-center displacements by solving for the vibronic mixing of
electronic states at the displacing ion site. The eigenvalue equation of a
hindered rotator is solved in 2D by Mathieu's periodic functions, as the
rotation is quantized in rotational bands. We discuss the optical
conductivities pertaining to each of the exemplified cases.
