The Vibration-Rotation Problem

The kinetic energy of a molecule, treated as an ensemble of nuclei and electrons, is determined with reference to a Cartesian axis system with its origin in its center of mass. With this choice, the translational motion of the whole molecule is factored apart. The rotational motion of a vibrating molecule is defined by the rotational Eckart conditions, and separated from the deformation (vibrational) motions in a first approximation. Then normal coordinates are used for the vibrational motions, to simplify the problem. The whole expression, inclusive of vibration-rotation and electron orbit-rotation terms, is transformed to the classical Hamiltonian form in terms of momenta, and the quantum mechanical Hamiltonian is eventually obtained.