Modern Polymer Spect..

3.5 Towmu" Larger Molecules: F~om Oligoi?iers to Polyriers 105
a
-1 0 +I
Figure 3-4. (a) Chemical
structure of trarzs-polyacetylene
and ib) model
assumed in the calculation
reported in the text. The
model gives the numbering
of the one-dimensional
repeat unit cell and the
labeling of the internal
coordinates.
GR (y) for a simplified chain of planar trans-polyacetylene, (-CH=CH-), where
each CH group is taken as a point mass. Only in-plane motions are considered.
From the viewpoint of chemistry and materials science, polyacetylene is a prototypical
relevant material in molecular electronics and shows a low (HOMO-
LUMO) band gap (= 1.4 eV) because pz orbitals of the carbon atoms in the sp2
hybridization are strongly delocalized along the molecular chain (large conjugation).
The distance of interaction between 7c bonds is yet unknown and is the subject
of active research [40-421.
Figure 3-4 shows the drawing of the actual molecule as well as of the simplified
model with the labeling (1) of the sites of the translational repeating unit and the
labeling of the internal coordinates of the 1 = 0 unit and the equivalent ones at sites
+/- s. The unit cell is made up by two inasses (p = 2) and the in-plane internal
coordinates are 2 x 2 = 4 which is also the size of the secular equation. We expect
to calculate four branches in the dispersion relation, two of which are optical and
two acoustical.
We calculate explicitly the algebraic expressions of the elements of the GR(~)
matrix which enters Eq. (3-43). The following simplifications have been accepted in
such a simple calculation: (i) the groups C-H are taken as point masses of in = 13
daltons; (ii) C=C and C-C bond lengths are equal; and (iii) bond angles are taken
equal to 120".