Essentials of Computational Chemistry

48 2 MOLECULAR MECHANICS
by cut-offs in systems having such a high degree of localized charges (see, for instance,
Beveridge and McConnell 2000).
In aperiodic systems, another important contribution has been the development of the
so-called ‘Fast Multipole Moment’ (FMM) method (Greengard and Rokhlin 1987). In
essence, this approach takes advantage of the significant cancellations in charge–charge
interactions between widely separated regions in space, and the increasing degree to which
those interactions can be approximated by highly truncated multipole–multipole interactions.
In the most favorable case, FMM methods scale linearly with system size.
It should be remembered, of course, that scaling behavior is informative of the relative
time one system takes compared to another of different size, and says nothing about the
absolute time required for the calculation. Thus, FMM methods scale linearly, but the initial
overhead can be quite large, so that it requires a very large system before it outperforms PME
for the same level of accuracy. Nevertheless, the availability of the FMM method renders
conceivable the molecular modeling of extraordinarily large systems, and refinements of the
method, for example the use of multiple grids (Skeel, Tezcan, and Hardy 2002), are likely
to continue to be forthcoming.
An interesting question that arises with respect to force fields is the degree to which
they can be used to study reactive processes, i.e., processes whereby one minimum-energy
compound is converted into another with the intermediacy of some transition state. As noted
at the beginning of this chapter, one of the first applications of force-field methodology
was to study the racemization of substituted biphenyls. And, for such ‘conformational reactions’,
there seems to be no reason to believe force fields would not be perfectly appropriate
modeling tools. Unless the conformational change in question were to involve an enormous
amount of strain in the TS structure, there is little reason to believe that any of the
internal coordinates would be so significantly displaced from their equilibrium values that
the force-field functional forms would no longer be accurate.
However, when it comes to reactions where bonds are being made and/or broken, it is
clear that, at least for the vast majority of force fields that use polynomial expressions for
the bond stretching energy, the ‘normal’ model is inapplicable. Nevertheless, substantial
application of molecular mechanics to such TS structures has been reported, with essentially
three different approaches having been adopted.
One approach, when sufficient data are available, is to define new atom types and associated
parameters for those atoms involved in the bond-making/bond-breaking coordinate(s).
This is rather tricky since, while there may be solid experimental data for activation energies,
there are unlikely to be any TS structural data. Instead, one might choose to use structures
computed from some QM level of theory for one or more members of the molecular data set.
Then, if one assumes the reaction coordinate is highly transferable from one molecule to the
next (i.e., this methodology is necessarily restricted to the study of a single reaction amongst
a reasonably closely related set of compounds), one can define a force field where TS structures
are treated as ‘minima’ – minima in quotes because the equilibrium distances and force
constants for the reactive coordinate(s) have values characteristic of the transition state.
This methodology has two chief drawbacks. A philosophical drawback is that movement
along the reaction coordinate raises the force-field energy instead of lowering it, which