  We predict the existence of vibrational Andreev bound states in deformable
magnetic molecules on superconducting surfaces. We discuss the Anderson
impurity model with electron-phonon coupling to a realistic anharmonic
vibrational mode that modulates the tunneling barrier and show that the
vibronic features are spectroscopically most visibile near the transition point
between the Kondo-screened singlet and the unscreened doublet ground state. We
find competing tendencies between phonon hardening due to anharmonicity and
softening due to coupling to electrons, contrary to the Anderson-Holstein model
and other models with harmonic local phonon mode where the vibrational mode is
always softened. In addition, we find that the singlet and doublet many-body
states may experience very different effective phonon potentials.
