Modern Polymer Spect..

94 3 Vibrational Spectro us a Probe oj Strzrctural Order
follows. Let GR be diagonalized by the unitary transformation D
GRD = Dl- (3-23)
where r is the diagonal matrix of the eigenvalues. The new set of coordinates
C = D’R (3-25)
forms an irreducible representation of the symmetry point group of the molecule,
thus the C, form a set of symmetry coordinates and D’ can replace U in Eq. (3-71).
The method is particularly useful for the automatic removal of all redundancies
in structurally complex systems. Indeed if in > 3N - 6 is the size of the eigenvalue
equation
r=m-(3N-6) (3-26)
is the number of zero eigenvalues in Eq. (3-34) which provide the required r
redundancy relations
D’~R = o (3-27)
between the coordinates used in setting up the starting B matrix (Eq. (3-10)).
A fully automated computer method for the handling of group theory has been
proposed and is recommended for extremely large and highly symmetrical systems
with many local and cyclic redundancies [30]. The numerical methods in treating
group theory and symmetry coordinates have been generally neglected, as few large
and highly syininetrical molecules have required the attention of spectroscopists. At
present, the need to understand the vibrations of fullerene, fulleroid, tubular molecules,
dendrites, etc., may revive the use of such numerical methods.
3.3 How to Describe the Vibrations of a Molecule
The main purpose of a dynaniical analysis of a molecule is to assign the vibrational
transitions observed in infrared and Rainan in terms of molecular motions. As
mentioned above, chemical group frequency correlations have been the justification
of most of the vibrational assignment. The description of the normal modes has
posed some problems throughout the years.
It would seem obvious that the only way to describe the atomic motion would be
that of plotting the Cartesian displacements of each atom during normal mode Q,
with frequency vJ. At present, with the availability of computer techniques and fully
automated computing programs, the Cartesian atomic displacements are a norinal
part of the output, while some programs even provide animated descriptions of the
molecular vibrations which can be viewed in all directions by appropriate use of