Modern Polymer Spect..

96 3 Vihratioiial Spectru as a Probe qf Structural Order
This issue has rarely been faced by molecular spectroscopists for several reasons.
All those who searched for chemically useful characteristic frequencies aimed at
modes strongly localized within the functional group [23-261. In contrast, those
who dealt with large systems generally considered oligomeric or polymeric molecules
as structurally perfect systems, often with translational periodicity, and
studied 'phonons' which, by definition, are collective phenomena [ 17-1 91. The
problem cannot be overlooked when the vibrations of large and structurally complex
molecules must be understood.
The first theoretical approach seeded by the explosive development of chemical
spectroscopic group frequency correlations has been presented by King and Crawford
who gave the dynamical conditions under which a normal mode can be defined
as 'localized' [38].
111 correlative vibrational spectroscopy, a group of chemically similar molecules
shows characteristic group frequencies [23-261 vi (say, the stretching of the carbonyl
group >C=O, very strong in infrared) which exhibit systematic changes that chemical
spectroscopists would like to ascribe to inductive and/or mesomeric effects by
the various substituents placed in the molecules either at short or large distances
from the functional group of interest. On the other hand, the observed wavenumber
shifts may also originate from changes of mass and/or geometry.
Use is made here [38] of the dynamical quantities presented in Chapter 2. Let
Go, Fo and La-' be the kinetic, potential, and normal coordinate transformation
matrices of one molecule of the series taken as reference [39]. In going from one
molecule to a chemically similar one within the same class, one may expect small
changes in the geometry (AG) or in the force constants (AF), or in both, which may
be the cause of the observed changes of vi. The corresponding matrices of the
molecule so modified can be written:
G=Go+AG
F = Fo + AF
L-I = L~-' + A(L-')
(3-29)
(3-30)
(3-31)
If the perturbations of G and F matrices are small it can be assumed that the
eigenvectors do not change and the zeroth-order values in Eq. (3-31) are used.
When the eigenvalue Eq. (3-17) is expanded in terms of the quantities in equations
(3.29) through (3-31) in terms of the first-order changes AG and AF and the zerothorder
eigenvectors, the final expression for the (group) frequency parameter turns
out to be:
in which (lo), is the frequency parameter of the i-th group-frequency mode of the
reference unperturbed molecule, AG,1 and AFml are the changes of those elements
of the kinetic and potential energy matrix whose contributions to the changes of A,
are scaled by the related elements of the Lo and L,' matrices.