Essentials of Computational Chemistry

2.2 POTENTIAL ENERGY FUNCTIONAL FORMS 35
In general force fields, stretch–stretch terms can be useful in modeling systems characterized
by π conjugation. In amides, for instance, the coupling force constant between CO and CN
stretching has been found to be roughly 15% as large as the respective diagonal bond-stretch
force constants (Fogarasi and Balázs, 1985). Stretch–bend coupling terms tend to be most
useful in highly strained systems, and for the computation of vibrational frequencies (see
Chapter 9). Stretch–torsion coupling can be useful in systems where eclipsing interactions
lead to high degrees of strain. The coupling has the form
U(rBC,ωABCD) = 1
2 kBC,ABCD(rBC − rBC,eq)[1 + cos(jω + ψ)] (2.28)
where j is the periodicity of the torsional term and ψ is a phase angle. Thus, if the term
were designed to capture extra strain involving eclipsing interactions in a substituted ethane,
the periodicity would require j = 3 and the phase angle would be 0. Note that the stretching
bond, BC, is the central bond in the torsional linkage.
Other useful coupling terms include stretch–stretch coupling (typically between two adjacent
bonds) and bend–bend coupling (typically between two angles sharing a common central
atom). In force fields that aim for spectroscopic accuracy, i.e., the reproduction of vibrational
spectra, still higher order coupling terms are often included. However, for purposes
of general molecular modeling, they are typically not used.
In the case of non-bonded interactions, the discussion in prior sections focused
on atom–atom type interactions. However, for larger molecules, and particularly for
biopolymers, it is often possible to adopt a more coarse-grained description of the overall
structure by focusing on elements of secondary structure, i.e., structural motifs that
recur frequently, like α-helices in proteins or base-pairing or -stacking arrangements in
polynucleotides. When such structural motifs are highly transferable, it is sometimes possible
to describe an entire fragment (e.g., an entire amino acid in a protein) using a number of
interaction sites and potential energy functions that is very much reduced compared to what
would be required in an atomistic description. Such reduced models sacrifice atomic detail
in structural analysis, but, owing to their simplicity, significantly expand the speed with
which energy evaluations may be accomplished. Such efficiency can prove decisive in the
simulation of biomolecules over long time scales, as discussed in Chapter 3. Many research
groups are now using such coarse-grained models to study, inter alia, the process whereby
proteins fold from denatured states into their native forms (see, for example, Hassinen and
Peräkylä 2001).
As a separate example, Harvey et al. (2003) have derived expressions for pseudobonds
and pseudoangles in DNA and RNA modeling that are designed to predict base-pairing and
-stacking interactions when rigid bases are employed. While this model is coarse-grained, it
is worth noting that even when a fully atomistic force field is being used, it may sometimes be
helpful to add such additional interaction sites so as better to enforce elements of secondary
structure like those found in biopolymers.
Finally, for particular biomolecules, experiment sometimes provides insight into elements
of secondary structure that can be used in conjunction with a standard force field to more
accurately determine a complete molecular structure. The most typical example of this
approach is the imposition of atom–atom distance restraints based on nuclear Overhauser