Model of electron tunneling coupled to torsional vibrations: Exact solution and study of performance of approximation methods

A two dimensional model for the electron interaction with molecular vibrations in molecular junctions is proposed. Alternatively the model can be applied to tunneling through a cylindrical nano-structure. The transmission function is calculated accurately numerically. The exact results are then compared with various approximations: (1) completely frozen vibrations for very light molecule, (2) Chase approximation for very heavy molecule, and (3) discrete-state-in-continuum model in resonant regime. The validity of these approximations is discussed in terms of the characteristic time-scales and coupling strengths. The excitation of the vibrational degree of freedom and the emergence of prominent threshold structures in the strong coupling regime are discussed in more details.