A brief survey of the methods of applying various kinds of symmetry coordinates for the calculation of normal vibration frequencies for symmetrical molecules with special reference to the more recent method of constructing the symmetry coordinates in terms of the internal coordinates of the molecular system. The importance of forming redundant symmetry coordinates when there are more internal coordinates than the vibrational degrees of freedom has been given. The present vibrational analyses have been applied to the diborane molecule after giving a short account regarding the mole cillar structure.
1. SYMMETRY COORDINATES
In order to factor the secular equation for the normal vibration requencies of symmetrical molecules, so-called symmetry coordinates that form the basis for a completely reduced representation of the molecular symmetry group. Each symmetry coordinate belongs to one of the irreducible representations of this group i.e., to one of the symmetry species into which the normal modes of vibration may be divided.
Symmetry coordinates were introduced by Brester (1) who first showed how symmetry can be utilized to classify molecular vibrations. Wigner (2) treated this problem with the help of group theory. Rosenthal and Wilson (s), Mannebeck (4). Redlich (s), Redlich and Tompa (6) and others (7) have developed methods of applying various kinds of symmetry coordinates to calculate normal vibration frequencies for symmetrical molecules.
At first the symmetry coordinates were constructed as linear combinations of the 3n cartesian coordinates that describe small displacements of the n atomic nuclei in the molecule. If 3n such symmetry coordinates are used, the complete secular equation will have six zero roots.