Modern Polymer Spect..

5.2 FOYCC Fields 241
such calculations can only be done for relatively small molecules, and the force
constants generally need to be scaled so that calculated frequencies agree with
experimental data. Nevertheless, valuable information can be obtained by this
approach.
The MM method will be seen to be the most useful one for obtaining force constants
that can be used for different conformations of the molecule. In this approach
the force constants are obtained from the second derivatives of an assumed potential
energy function consisting of quadratic bonded terms and non-quadratic nonbonded
terms. Although present MM functions are too crude to be spectroscopically
reliable, a new method for deriving such functions holds the promise of
providing a reliable vibrational force field for the polypeptide chain.
5.2.1 Empirical Force Fields
The most general force field of a molecule would include anharmonic as well as
harmonic terms. However, with the limited experimental information generally
available for refining an empirical force field for complex molecules, the harmonic
approximation is the only feasible one at present. This means that, for the isolated
molecule, we need to know the force constants, FQ, in the quadratic term of the
Taylor series expansion of the potential energy, V:
where r is, for example, an internal displacement coordinate and IZ = 3N - 6, N
being the number of atoms in the molecule. If interactions with other molecules are
involved, analogous intennolecular interaction energy terms must be included.
Since the maximum number of observable frequencies is n, whereas the general
number of F,s is n(n + 1)/2, it is necessary to find a physically meaningful model
for T7 that brings the number of empirically determined F,Js into reasonable relation
to the number of observable frequencies, even including those of isotopic derivatives.
Two main models have been used to accomplish this: the Urey-Bradley
force field (UBFF) and the general valence force field (GVFF). Both models incorporate
the same diagonal (i.e., Fll) valence-type ternis, involving bond stretch, angle
bend, and torsion coordinates, but differ in how they treat the off-diagonal (i.e., F,,)
terms.
In the unmodified UBFF, elaborated by Shimanouchi [lo], off-diagonal terms in
T’ are represented by atom pair 1,3 non-bonded interactions. Because this introduces
inherent redundancies in the coordinates, the equilibrium configuration of the
molecule represents a state with internal tensions, and therefore linear terms must
be included in the force field. By introducing the redundancy condition, the linear
terms can be eliminated from V, although the coefficients of the quadratic terms
then contain ‘intramolecular tension’ as well as the valence and nnn-bonded force
constants. The important characteristic of such a UBFF is that only a limited