Modern Polymer Spect..

88 3 Vibvationul Spectru us a Pvobe of Structurcil 0rdt.vlDisordt.r
(molecules can always be made to ‘wiggle’ on a screen), another matter is to account
quantitatively for frequencies, amplitudes, infrared (dipole derivatives), and Ranian
or hyper-Raman intensities (polarizability derivatives at various orders).
The latter attempt to obtain vibrational force constants is a combination of
the two last methods, i.e., ab initio methods are used to obtain the parameters of
two-body or many-body interactions generally adopted in semiempirical molecular
mechanics calculations [15].
The size of the molecules that can be treated using the above-mentioned methods
is quite large (20-30 atoms) and is not a problem if periodicity [ 16- 181 or symmetry
[3] are intrinsic properties of the molecule. However, the problem becomes more
complex when very large and irregular molecular objects need to be studied.
In this chapter we do not wish to dwell on the problem of the usefulness and
limitations of the various methods for calculating vibrational force constants.
Rather, we will deal with the problems of carrying out normal-mode calculations on
very large molecular systems which have no symmetry are of irregular shape, and
chemically are highly disordered [19]. It appears obvious that none of the abovementioned
methods of calculation can be applied to such huge and structurally
complex systems.
In nature, chemical systems with chemical and/or structural irregularities or disorder
are very common and their relevance in natural phenomena is of fundamental
importance. Biological as well as synthetic oligoiners and polymers form a large
class of compounds whose structure and properties need to be characterized qualitatively
and quantitatively. Vibrational infrared and Rarnan spectroscopy is one of
the few physical tools available to study the microstructure of these systems. Many
empirical assignment and correlative and analytical studies have been carried out,
but the possibility of a theoretical understanding of the vibrational properties (and
the derived infrared and Raman spectra) have not been yet thoroughly explored.
In this chapter we present and discuss the dynamics and vibrational spectra of
long-chain molecules with disordered structure, and discuss the current possibilities
of calculating the vibrational infrared, Raman and neutron-scattering spectra of
these systems.
3.2 The Dynamical Case of Small and Symmetric
Molecules
Let us briefly define the elementary parameters and review the basic equations for
the treatment of molecular and lattice dynamics. The concepts presented here form
the background for a further discussion later in the chapter for the case of large
molecular systems.
In classical molecular spectroscopy, a vibrating molecule is always represented by
a set of balls connected by weightless springs. The springs represent the chemical