Knowledge Bank

Mathematical complexities preclude attempts to solve the Schr\”{u}dinger equation H $\Phi$ = E $\Phi$ for molecular vibration-rotation energies or wave functions in a single perturbation procedure. It is the purpose of this paper to justify the following two-step procedure, which is in common use: (i) One solves the Schr\”{u}dinger equation, assuming the rotational state to be fixed. Perturbation theory, with or without the contact trasformation, yields energies in terms of vibrational quantum numbers and angular momenta $Px$, $Px$ and $Px$ provided that the order of these factors in each additive term is preserved throughout the calculation. (ii) One uses the results of (i) to formulate the hamiltonian for a non-rigid rotator in a fixed state of vibration. Matrix algebra or perturbation theory is commonly employed to determine rotational energies. The justification is based upon the complete separation of the vibrational and rotational parts of the zero order hamiltonian and the fact that one knowns the commutation relations among the angular momentum operators.