Modern Polymer Spect..

3.2 The Dyrinniical Case of Sinnll and Sjwimetric Molecules 93
The scalar quantities V and T in Eqs. (3-13) and (3-14) can be re-expressed in
terms of symmetry coordinates
2V = S’UFRU’S = S’FsS (3-22)
2T = S’U(G,)p’U’S = S’(Gs)p’S (3-23)
The new matrices Fs and Gs are factored into blocks with dimensions identical
to the structure of the irreducible representation [3]. The corresponding secular
equation becomes separated into blocks of smaller size. Numerical calculations
carried out separately for each symmetry block enable to identify the normal modes
which belong to a given symmetry species. This is essential when the vibrational
assignment needs to be carried out [2]. Band shape analysis (in IR and Raman)
of gaseous samples, Raman depolarization ratios (for gaseous or liquid/solution
samples) and dichroic ratios in IR or Raman experiments in polarized light on
single crystals or stretch-oriented solid samples [28] provide the experimental data
supporting group theoretical predictions.
The numerical solution of the eigenvalue Eq. (3-17) is obviously greatly simplified
when symmetry factoring can be carried out for molecules which possess some
symmetry elements. Symmetry factoring was essential in the past (even for small
molecules) when numerical processes were necessarily cumbersome and time consuming
because of primitive computing technologies. In spite of the explosive development
of computational tools this problem cannot yet be fully overlooked. The
molecular systems to be treated have become large and the size of the secular determinant
can still pose some problems.
A typical example for a modern textbook is the case of the molecule of fullerene
(C~O) which shows a very high symmetry (point group Ih): its 174 normal vibrations
are separated into smaller symmetry blocks. Since icosahedral symmetry gives rise
to a large number of degenerate modes, only 46 distinct mode frequencies are
expected:
The application of group theory for the determination of the structure of the
irreducible representation (for the prediction of optical selection rules for infrared
and Ranian spectra) is generally easy and straightforward [2, 31. On the contrary
the construction of symmetry coordinates (i.e., the construction of the U matrix,
Eq. (3-21)) is sometimes diflicult and cumbersome for large systems especially when
degenerate species occur (e.g., adaniantane and fdlerene). An automatic method
for the construction of symmetry coordinates using computers based on the diagonalization
of the GR matrix has been proposed [29]. The method is based on the
fact that GR contains in itself all the information on the symmetry of the problem
and on the redundancies, if some or all have not been previously removed. The
diagonalization of GR provides directly the elements ofthe U matrix (Eq. 13-21)) as