Origin of Vibrational Spectra

The solution of this problem must of course be approached quantum mechanically. When this is done, however, it can be shown that, in the 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2. qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements  

By choosing the equilibrium position as the zero energy, V0 can be set equal to zero. Also, since the energy of the molecule is a minimum at the 

The normal coordinates are required to be combinations of the qt such that the total energy, when expressed in terms of these coordinates, becomes the sum of the energies of individual harmonic oscillators. This means that no cross terms should appear in V and T when written in normal coordinates. For an oscillator of mass 1, wc would therefore have 

This transformation will exist if there is a linear transformation expressing the Q s in terms of the q s, and equations (8) and (9) will result by suitably choosing the coefficients chi. By introducing equations (8) and (9) into the Lagrangian form of the equations of motion, viz., 

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Molecular spectroscopy is now a mature field of study. It is, however, difficult to find references superior to the classic treatise written by Herzberg nearly 50 years ago (1). The origin of vibrational spectra is usually considered in terms of mechanical oscillations associated with mass of the nuclei and interconnecting springs (9). Vibrational spectroscopy considers the frequency, shape, and intensity of internuclear motions due to incident electromagnetic fields. In the harmonic approximation, the vibrational bands are associated with transitions between nearest vibrational states. When higher order transitions, resonance, and coupling between vibrational motions require analysis, quantum mechanical treatment is mandated (1). Improvements and advancements in poljuner spectroscopy are driven by the many problems of interest in the polymer community. 

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