Modern Polymer Spect..

5.2 Force Fields 247
ture with conformation, In terms of conveniently designated components of the
energy of the molecule, which of course the ah initio calculation does not provide,
we can conceive of these changes as arising from changes in the charge distribution
with conformation, from changes in the dispersion energies as interatomic distances
change, and from any iiitrinsic valence force constant variations with conformation.
While we can show from ab initio calculations that the force field is not independent
of molecular conformation, it is much more difficult to specify the explicit
nature of this dependence. This is precisely because the method provides only a
total energy for a given structure. If we wish to infer the varying contributions of
different physical components, we must assume a model for the potential energy
and strive to make its behavior mimic the ah initio results as closely as possible. It is
toward this goal that some present efforts are directed.
5.2.3 Molecular Mechanics Force Fields
An MM energy function, whose second derivatives at a minimum are the spectroscopic
force constants, is generally represented as a sum of quadratic and nonquadratic
terms. The foimer encompass the valence-type deformations, such as
bond stretching and angle bending, while the latter involve torsions, dispersion
interactions, electrostatic interactions, and possible hydrogen-bond terms.
In order to serve as a spectroscopic force field, such an MM function would at
least have to reproduce vibrational frequencies to spectroscopic standards, viz., to
root-mean-square (rms) errors of the order of 5-10 cnir'. Some early MM potentials
came close to this mark [35], but since frequency agreement was not their primary
goal (but rather reproduction of structures and energies), subsequent efforts
resulted in functions with unacceptable spectroscopic predictability (rms errors of
>50 cm-'). One of the reasons for this was the lack of proper attention to crossterms
in the potential. Initial efforts to improve such potentials have led to only
marginally better frequency agreement. For example, MM3 gives an rms error for
frequencies below 1700 cnir' of 38 cm-' for butane [36a] (reduced to 22 cm-' in
MM4 [36b]) and 47 cm-l for NMA [37]; a Hessian-biased force field [38] gives a
similar rms error of 29 cn1-l for polyethylene [39]; for CFF93, the ims error for n-
butane is 25 cmrl 1401 and is 34 cnrl for a group of four acyclic and three cyclic
hydrocarbons; and a second generation version of AMBER gives 43 cm-' for
NMA [41]. Some force fields have been specifically designed to give improved
spectroscopic predictability and these [42, 431 indeed show better agreement, although
improvement is still desirable: the rms error for the above NMA frequencies
is 24 cn-l for SPASIBA [44].
What is clearly needed is an approach that systematically incorporates spectroscopic
agreement in the initial stages of the optimization of the MM parameters.
This has been achieved by a procedure designed to produce a so-called spectroscopically
determined force field (SDFF) [45, 461.
The elements in the SDFF methodology are the following. First, a form is selected
for the MM potential, one that is hopefully inclusive enough to incorporate all the