The Use of a hierarchy scheme for the evaluation of normal modes of vibration

In vibrational analysis it is the common practice to initially approximate a set of force constants rather than to initially approximate a set of normal modes. This does not allow a flexible and easily analyzed computational system. A hierarchy scheme of approximation has been developed. Systematic testing of alternative hierarchies and geometries is essential since a force field may always be evaluated to match exactly observed and calculated frequencies for any set of approximate normal modes. Other physical requirements must therefore be considered to establish the correctness of an answer. A set of N linearly independent internal coordinates of a particular symmetry type are placed in a hierarchy and a set of N orthormal modes are developed in such a way that the nth normal mode does not change the internal coordinates of the n + 1 to Nth members of the hierarchy. Constraints on the form of the force field require iteration from these well-defined initial approximations. The hierarchy scheme offers a much more satisfying physical picture of characteristic group frequencies and lends itself readily to the use of perturbation theory.