Vibrational Anharmonicity and Spectra

The most realistic vibrational potentials of molecules are not strictly harmonic. For a diatomic molecule, the stretching potential s dependence on the separation distance may turn out to be cubic, quartic, and so on. A potential is harmonic if it has only a quadratic dependence. Any higher-order dependence is an anharmonicity in the potential. Vibrational anharmonicity refers to the effects on energy levels and transitions of an otherwise harmonic oscillator arising from anharmonic potential terms. 

The first term is a harmonic element. For as 1, the typical situation, the terms diminish in size beyond the harmonic term. Even for as 1, the terms diminish beyond the cubic term. The cubic and quartic terms are the leading sources of anharmonicity near the equilibrium of a Morse oscillator, and it is very often the case that a cubic potential element is the most important contributor to anharmonicity effects in real molecules. 

To incorporate anharmonicity effects in a description of a diatomic, a third-order polynomial function is a starting point for the potential, and we can express this generally as 

This is a starting point because higher-order terms or other functional forms might be employed for still greater precision in representing a true potential. The cubic term (and any higher-order terms) may be well treated as a perturbation of the harmonic oscillator. The first-order corrections for all states turn out to be zero. The second-order corrections arising from the cubic term gs are 

The states of a vibrating-rotating diatomic molecule are distinguished by three quantum numbers, n,, and M. Under the assumption of rigid rotation and a harmonic stretching potential, the selection rules derived for a harmonic oscillator and a rigid rotator apply to a diatomic molecule. An, the change in n for an allowed transition, is +1 (absorption) or -1 (emission). A/ is +1 or -1, and AM = 0. To see this, we need to make explicit use of the wavefunctions, which are products of radial and spherical harmonic functions  

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