Modern Polymer Spect..

onds which hold atoms together through directional forces generated by the interactions
of valence electrons. Since the vibrational molecular potential is unknown it
is approximated by a Taylor expansion about the equilibrium geometry in terms of
a set of suitably chosen coordinates. Let x,, y, and z, be the instantaneous Cartesian
coordinates of the a-th atom of the vibrating molecule and let x,", y: and z," be the
corresponding equilibrium coordinates when the molecule is at rest. For simplicity
let x, (i = 1-3N) label the vibrational displacements of the type xu - x:, yr - y: and
z, - z," for the N atoms. The generally unknown vibrational potential can be
approximated by the following Taylor expansion about the equilibrium structure
2V = 2Vo + 2 C (CJV/~X~)~X~
1 'J
+ C (8sV/8~,d~J)o~,~,
+ . . . (3-1)
The zero-th order term is removed by a suitable shift of the origin, the first-order
term is zero because the forces (dV/ax,),, vanish at equilibrium, the second-order
term defines a quadratic harmonic potential. In most cases of chemical interest the
expansion is truncated at the second order. The analysis of the higher-order terms is
outside the scope of our discussion. The truncation at the second order implies that
the restoring forces are assumed to be linear with the infinitessimal displacements
from the equilibrium position.
The quadratic potential can be written in matrix notation as
2V = x'Fxx (3-2)
where x is the vector (and x' its transpose) of the 3N Cartesian displacement coordinates
and F, is the matrix of the quadratic force constants:
Correspondingly, the kinetic energy can be written as:
2T = X'MX (3-4)
where M is the diagonal matrix of the atomic masses. Only 3N - 6 of the xi are
independent because the Cartesian coordinates are related by the Eckart-Sayvetz
conditions [3].
With such a model, atoms are allowed to perform very small harmonic oscillations
about their equilibrium positions and the dynamical treatment describes the
normal modes Q, of a set of coupled harmonic oscillators.
When the kinetic and potential energies are introduced into the Lagrange equation
the solutions are of the form:
where A, are called frequency parameters and define the frequency of oscillation of