3-D off-center Li+ rotors in alkali halides: solving for the eigenvalue problem

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We are dealing with the long-standing problem of the eigenstates and eigenvalues of 3-D rotators by off-center impurity ions in alkali halides. The ion runs along the brim of a sombrero-like vibronic potential. The quantum-mechanical motion depending on two angular coordinates, we consider the motion along theta at averaged phi, and vice versa. We find the off-center ion performing two rotations: one in (x,y) plane along the azimuth phi and another one along the position angle theta, respectively, which are both described as Mathieu's non-linear oscillators. The motion along z is complemented by an anharmonic vibrational mode.