Vibration-rotation Hamiltonians classical mechanics

The first term on the right is the translational kinetic energy of the molecule as a whole this simply adds a constant to the total energy, and we shall omit this term. The second and third terms are ihe rotational and vibrational kinetic energies of the molecule. The final term is the energy of interaction between rotation and vibration. To get the classical-mechanical Hamiltonian function, we add the potential energy V to (5.2), where U is a function of the relative positions of the nuclei. 

Classical and Quantum-Mechanical Vibration—Inversion—Rotation Hamiltonian... 

Vibrational, rotational, and vibrational/rotational energy levels are found by first transforming the classical Hamiltonians described in the previous section to the appropriate quantum mechanical operator H. The eigenvalue equation... 

Thus the asymptotic quantum states are labeled by the vibrational and rotational quantum numbers, whereas the projection quantum number is treated classically. We have in the above Hamiltonian indicated that it depends upon time through the classical variables. Thus the quantum mechanical problem consists of propagating the solution to the TDSE... 

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