Modern Polymer Spect..

3.2 The Dyriamical Case qf Simll arid Symnietric Molecziles 91
elements of Lx are simply the numbers needed to directly pro-iect on a screen the
atomic vibrations for each Q,,.
3. The treatment of vibrational infrared and Raman intensities provides quantities
such as (7p/dx, and d((a))/ax; (,where p is the total molecular dipole electric
dipole moment and ((a)) is the total niokcular polarizability tensor). These
quantities have a direct physical meaning and are directly calculated ah iizitio or
sometimes by extended programs of molecular mechanics calculations [4-61.
4. When lattice dynamic calculations for one- or three-dimensional systems need
to be carried out, the Cartesian space is more suitable in order to describe both
inter- and intra-molecular vibrations [20].
When the chemical reality of an isolated molecule needs to be considered through
its vibrational spectrum, new sets of ‘chemical’ coordinates can be defined [3]. Let
R = BX (3-10)
define a set of internal displacement coordinates where B is the matrix of geometrical
coefficients and
where
xvib 1 AR (3-1 1)
For a discussion of the A matrix, see [21]. For simplicity in this discussion
we assume that a nonredundant set of 3N - 6 Rs has been defined [22]. The Rs
describe bond stretching, valence angle bendings, and torsions. The vibrational
potential can be rewritten in terms of Rs as
2V = R’FRR (3-13)
and the kinetic energy matrix as
where
2T = R’(GR)-’R (3-14)
GR = BM-~B’ (3-15)
G contains all information on equilibrium geometry and masses. The potential
energy matrix is written as
FR = A‘F,A (3-16)
Equation (3-16) is the one to be used when F, is calculated by (16 iizitio methods and