Polyatomic molecules Morse oscillator

The Morse potential provides a useful model for the anharmonic stretching vibrations of a polyatomic molecule. It is superior to a harmonic oscillator perturbed by cubic and quartic anharmonicity terms in terms of both convergence (a V(r) that includes an r3 term cannot be bound, thus cannot have any rigorously bound vibrational levels) and the need for a smaller number of adjustable parameters to describe both the potential energy curve (a and De for Morse frr, frrri and frrrr for the cubic plus quartic perturbed harmonic oscillator) and the energy levels. 

The model described here is based on a series of approximations, but it has nonetheless been found to be quantitatively successful in fitting the observed vibrational levels of a considerable number of small molecules [6,7,9,15,16,17,18]. The most obvious approximations concern the use of perturbation theory and the implicit use of harmonic basis functions in deriving the vibrational hamiltonian matrices, and the neglect of matrix elements coupling different values of the total vibrational excitation quantum number V. However when the same approximations are applied to a diatomic Morse oscillator they yield the exact solution, as observed by Mills and Roblette [7], and some similar cancellation of approximations holds for the polyatomic examples. Another more serious problem concerns the mixing with other vibrations this has been discussed here only in connection with Fermi resonance, but its importance in other ways is only just being investigated. 

The local mode description with only one bond such as X-H being in motion at a time leads us to consider the polyatomic molecule as a single diatomic molecule M-H where H is hydrogen and M denotes the full rest of the molecule. The main advantage of this description is the dramatic reduction of the number of coupling constants. The anharmonic oscillator potential adopted in the local mode model is the Morse potential, which takes account of the dissociation energy De (53). 

R. Wallace. A theory of nuclear motion in polyatomic molecules based upon the Morse oscillator. Chem Phys 11 189-199, 1975. 

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