Molecular vibrational energy levels

The bonds between the atoms in a molecule are subject to stretching and (sometimes) bending degrees of freedom that may contribute to IR spectra. The simplest case is for a diatomic molecule composed of atoms of mass mx and m2. To a first approximation we may consider the binding forces between the atoms to be similar to a spring of stiffness k. Such a system, for which the restoring force is proportional to the displacement, will vibrate with simple harmonic motion and we may use quantum mechanics to show that the allowed energy levels of such an oscillator are (e.g., Rae, 1985)

where p is the reduced mass p =-; and v is the vibrational integer quantum number. m1 ^ m]

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