Infrared spectroscopy of polyatomic molecules

The infrared spectra of polyatomic molecules involve vibrational transitions along with rotational transitions just like in diatomic molecules. However, especially in the case of low-resolution spectra of polyatomic molecules, the rotational fine structure is lost. The peaks in the infrared spectrum are assigned as a fundamental vibrational transition. Hence, the 

The vibrations of a molecule set up a potential called a force field. The force field is determined for a set of internal coordinates. If the force field is completely known, then the vibrations of the molecule are known. The force field can be obtained from absorptions in infrared spectra. The simplest force field model is the harmonic oscillator, and this will be used in the analysis here of the pure vibration of polyatomic molecules. 

The Hamiltonian for a harmonic potential of a polyatomic molecule can be transformed from atomic displacement coordinates of the individual nuclei to normal coordinates whereby separability of the harmonic vibrations is achieved. The normal coordinates correspond to the actual vibrational noodes that the molecule will undergo. 

Each vibrational mode represented by i is separable resulting in multiple vibrational Shroedinger equations mathematically equivalent to that for a diatomic molecule. The eigenfunctions and the vibrational energy eigenvalues will have the same form as for a diatomic molecule. 

The selection rules for vibrational transitions of polyatomic molecules are the same as for diatomic molecules. The selection mle results for diatomic molecules can be applied to the harmonic model for vibrations of polyatomic molecules because the separation of variables achieved in the result states that each normal mode for vibration is regarded as a 1-dimensional oscillator. For the harmonic model, it was found that the quantum number changes by one. Also the dipole moment of the molecule must also change in the course of the transition in order for it to be an allowed transition. A molecule such as does not have an allowed fundamental infrared transition whereas HQ does. 

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