Modern Polymer Spect..

3.9 Moving Towards Reality: Froin Order to Disorder 121
It becomes apparent that as the translational symmetry of the chain is removed
by the existence of defects, any periodicity is lost and Eq. (3-431 cannot be used.
Ratter, Eq. (3-17) must be used which corresponds to the dynamical case of a finite
molecule. Since in this case our molecular model is huge and has no symmetry, the
size of the secular equation to be solved becomes extremely large. Moreover, such
types of study require the freedom to try many models with different kinds, concentrations,
and distributions of defects. Last, but not least, in order to play a game
as close as possible to reality in these studies, molecular models must be composed
of as many monomer units as possible.
In solid-state physics, the vibrations of very simple lattices containing a small
concentration of simple defects have been treated with sophisticated analytical
treatments using Green’s fiinctions 1921. Even if some authors have bravely tackled
an analytical solution of the dynamical problem of disordered polymers [93], they
were forced to introduce into the molecule such drastic structural simplifications
that the flavor of chemistry has been lost and the theoretical molecular models have
become, again, too unrealistic.
For our complex molecular systems the problem must be solved by numerical
methods. The problem of the solution of a huge secular equation producing thousands
of vibrational frequencies can be solved by the application of the so-called
‘Negative Eigenvalue Theorem’ (NET) originally proposed by Dean [79] who
aimed at the calculation of the vibrational spectra of disordered ice. We think that
the original work by Dean did not receive due acknowledgement; indeed, his
methods has been widely applied (with little due reference to Dean) in solid-state
physics and in molecular theories whenever the eigenvalues corresponding to vibrational
or electronic states of huge and disordered systems had to be calculated. (As
an example of the calculation on the electronic states of disordered systems, see
[941).
Dean’s method calculates the density of states (vibrational or electronic) in a
given energy range. The whole energy range can be spanned by the calculation and
the complete g(v) is obtained and can be plotted as an histogram to be compared
with the experimental spectrum. The accuracy of the histogram can be improved by
narrowing the steps (Av) in the energy interval in each cycle of the calculation (i.e.,
by improving the resolution of the numerical experiment).
Dean’s method works well for band-matrices which are always found in the case
of polymer chains or in the case of tridimensional lattices with short-range interactions.
The reader is referred to the original papers for a discussion of the method,
its advantages and limitations [91. 951.
Once the histogram of g(v) is plotted, one needs to find the eigenvalues for such
huge matrices corresponding to the approximate eigenvalues comprised in a given
energy range. The problem has been solved with the application of the Wilkinson’s
inverse iteration method [96] (hereafter referred as to IIM; see for instance [97]).
The comparison of the calculated g(v) with the actual experimental infrared,
Ranian and neutron-scattering spectra requires some care and possibly more refined
calculations with improved resolution. g(v) gives one information, namely how the
very many normal modes are clustered in a given frequency range. From IIM we
may also learn the shape of the normal modes at a chosen frequency. Nothing is