Calculation of Vibrational Frequencies, and Band Assignments

Computational Calculation of Vibrational Frequencies, and Band Assignments 

The established method for calculating the vibrational frequencies of molecules is the Wilson GF method.27 In this method, the potential energy of a molecule is defined in terms of the force constants by a matrix F, and the kinetic energy, which depends on the geometry of the molecule, is defined by a matrix G. Using the methods of classical mechanics, the following equation may be derived. 

The second step consists in constructing the matrices G and F. Although G and F are symmetric, the GF product found in the secular equation is unsymmetric, requiring that G and F be separately diagonalized. A procedure that yields a symmetric, secular equation was proposed by Hannon 

It has been shown142 that, for long-chain polymers in an ordered conformation, the calculation of the normal modes is reduced by symmetry arguments to the determination of the vibrations of the repeat unit. The vibrations in a chemical unit are related to those in adjacent units by the secular equation through a phase angle 6, so that the form of the secular equation used in the previous calculations29-30 was 

The number of force constants calculated in the general potential function is very large. For molecules having no symmetry, such as carbohydrates, this number is equal to 1 + 2+3 + - - + (3 N-7) + (3 N-6) = (l/2)(3 N—6)(3 N — 5), so that it can be determined for small molecules only. For such large and complex molecules as sugars and their derivatives, additional information may be obtained from studies employing isotopes and model molecules. 

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