Modern Polymer Spect..

92 3 J,’ibrationrrl Spectra as a Probe of Structural Order
the problem requires a treatment in terms of internal displacement coordinates R.
Care must be taken in the construction of A [21].
The eigenvalue equation in internal coordinates becomes
and the relation to normal coordinates is given by
(3-17)
R = LRQ
(3-18)
with the iiornialization condition
LL’ = GR (3-19)
As discussed in Section 3.4, many norinal modes are characterized by the fact
that only a few atoms belonging to a specific chemical functional group are moving,
while the other do not feel the intramolecular geometrical and electronic changes
which occur at a specific site of the molecule. Such group uihrutiorzs are then characteristic
of a given functional group and allow its identification in a chemically
unknown material. The existence of ‘localized’ group vibrations showing specific
and characteristic group frequencies have fueled the development of spectroscopic
correlations which have made infrared (and recently also Raman) spectra a valuable
physical tool for chemical and structural diagnosis (23-261. These correlations
form the basis of many automated computing programs for the chemical identification
of unknown compounds.
For the above reason, researchers felt the need to describe normal modes in terms
of ‘group coordinates’ (or local symmetry coordinates) which can be defined by an
orthogonal transformation
X=CR (3-20)
The expressions for Gx and Fx and for the corresponding secular equations are
the results of a straightforward similarity transformation such as in Eqs. (3-13) and
(3-14). The Japanese School of Shimanouchi has provided many force fields and
vibrational assignment in terms of group coordinates [27].
When all redundancies between internal coordinates are removed, the size of the
eigenvalue equation to be solved is equal to the number of normal modes (3N- 6)
expected for the n~olecule under study. If the molecule has some symmetry it
belongs to a given symmetry point group g; group theory provides the structure
of the irreducible representation of g, i.e., the number of normal modes in each
symmetry species r,. By a suitable linear and orthogonal transformation
S=UR 01 (S=UC‘) (3-21)
(with U’ = U-’) it is possible to define a new set of symmetry coordinates S which
foiin the basis of an irreducible representation of the point group g 131.