Modern Polymer Spect..

106 3 Vibrutioricil Spectra as a Probe of Structural Order
(3-48)
where p is the reciprocal of the C-C distance and p is the reciprocal of the mass of
the CH unit taken as point mass.
It is immediately seen that nonzero elements occur in the diagonal block and only
in the two neighboring blocks which describe the interactions of the central unit
with the nearest neighbors in position +1 and -1, i.e., the distance of interaction is
s = 1, very short indeed. The construction of the GR(w) matrix through Eq. (3-44)
yields a 4 x 4 matrix where the generic element has the following form:
(3-49)
The opposite is the case of the potential energy matrix whose structure is synibolically
given in Table 16, p. 294 of [41].
The most relevant fact is that while the Gp, matrix is certainly truncated at the +I
and -1 units in the FR matrix the distance of interaction of the zero-th unit with its
s-th neighbour is yet unknown since the so-called 'delocalization length' needs to be
determined [41-431. The truncation at the site s of the long ribbon of interaction
submatrices is then arbitrary. Many ab iiiitio calculations are at present dealing with
this essential problem which is common to all poly-conjugated chains presently
available.
The truncation of the sum in Eq. (3-45) is then arbitrary, thus affecting the
validity of the physical conclusions which may be derived. It follows that any
addition or variation of terms or extension of the interactions (increase of delocalization
length) will affect the numerical values of the elements of the FR(~)
matrix,
thus changing the shape of the dispersion curves [41, 421. It is apparent that the
shape of the dispersion curves is a direct consequence of the electronic properties of
the poly-conjugated chains [58] (Figure 3-5).
An opposite case is found for trans-planar polyethylene taken as a prototype of
all exclusively o bonded chains for which conjugation is not predicted (the existence
of o-conjugated chains like polysilanes needs further studies). In the case of polyethylene,
experiments or calculations indicate that the electronic interactions between
o C-C bonds fall OR very quickly along the chain with s = 2 [59]. Indeed, as
shown by Schachtschneider and Snyder, the normal modes of finite n-alkanes lie
almost perfectly on the dispersion curves calculated for polyethylene [33-351. No
chain length effect in the FK(~) matrix is revealed for these classes of systems.
Analogously for o bonded tridimensional lattices like cubic diamond and silicon, a
valence force field limited to a few first-neighbor interactions nicely accounts for the
optical and neutron-scattering data available to date [60] (Figure 3-6).