Knowledge Bank

Several molecules, mostly amines, are known to show large amplitude internal motions of the inversion type. The prominent feature is normally the tunneling of a hydrogen atom through a potential barrier. The simplified model for such a molecule consists of a rigid frame and a hydrogen atom or a group of atoms which are bound to the frame to move on a circular path between the two equilibrium positions. The classical Hamiltonian of the over-all rotation and inversion is set up and converted to the correct quantum-mechanical Hamiltonian for this model. It is shown that the matrix elements of the Hamiltonian can rigorously be calculated in a basis composed of the symmetric rotor wave functions and the trigonometric functions. The form of the potential function was chosen as to reproduce the characteristics of a double minimum potential but to facilitate the calculation of the matrix elements.