Modern Polymer Spect..

90 3 Vihrntimzal Spectra as n Probe qf Strzrcturol Or-cler
the atoms during the j-tli normal mode Q,. During each Q, all atoms move in phase
(i.e., they go through their equilibrium positions at the same time) with frequency v,
(in cm-') = (Eb,/4n-c-)
7 7 112 and with amplitudes L,.
Frequency parameters and vibrational amplitudes can be calculated by solving
the secular equation
M-'F,L, = L,A (3-6)
Six of the 2, vanish because of the Eckart-Sayvetz conditions and describe three
rigid translations and three rigid rotations of the molecule; each of the remaining
3N - 6 nonvanishing A, is associated to the normal mode Q,. Depending on the
shape (symmetry) of the molecule, degenerate symmetry species may occur and
some of the nonvanishing may turn out to be equal (doublets, triplets, and even
multiplets with very high symmetrical molecules, e.g., fullerene). Accidental degeneracy
may occur for large and asymmetric molecules when intramolecular coupling
is small or zero. When all 3N solutions are considered together, the relation between
Cartesian displacements and normal modes is as follows:
x = L,Q (3-7)
The matrix L, is nonorthogonal because M-'F, is not symmetric. The normalization
conditions are
L,L', = M-' (3-5)
which ensures that
2T = Q'Q (3-921,)
2V = Q'AQ
(3-9b)
i.e., normal modes are mutually independent and can be described as isolated
harmonic oscillators. The shape of the vibrational motions sought in any work of
vibrational assignment is described precisely by the solution of Eq. (3-6), once the
intramolecular potential and the geometry of the molecule is suitably chosen.
A few observations on the treatment of molecular and lattice dynamics in cartesian
coordinates are necessary.
1. When molecular dynamics is applied to the understanding of chemical intramolecular
phenomena the use of chemical internal displacement coordinates as
initially defined by Wilson et al. [3] are much more useful and may have a direct
chemical meaning (see below).
2. When molecular vibrations are studied by quantum chemical or molecular mechanics
methods, the problem is first treated in Cartesian coordinates and possibly
later transformed into internal coordinates. These programs generally
provide the numerical values of each element of the F, and L, matrices; the