Biomedical Engineering Reference
In-Depth Information
4.3 Force Fields
I have been vague so far about which variables are the 'correct' ones to take. Chemists
visualize molecules in terms of bond lengths, bond angles and dihedral angles, yet this
information is also contained in the set of Cartesian coordinates for the constituent atoms.
Both are therefore 'correct'; it is largely a matter of personal choice and professional
training. I should mention that there are only 3
N
- 6 vibrational coordinates, and so one has
to treat the 3
N
Cartesian coordinates with a little care; they contain three translational and
three rotational degrees of freedom. More of this technical point will be discussed later.
Spectroscopists are usually interested in finding a set of equilibrium geometric paramet-
ers and force constants that give an exact fit with their experimental data. This is harder than
it sounds, because for a molecule comprising
N
atoms and hence
p
=
3
N
−
6 vibrational
degrees of freedom, there are
1
/
2
p
(
p
1) force constants (diagonal and off-diagonal). In
order to measure the individual force constants, the spectroscopist usually has to make
experimental measurements on all possible isotopically labelled species. It usually turns
out that there are many more unknowns than pieces of experimental information. Spectro-
scopists usually want a
force field
(comprising force constants, equilibrium quantities and
every other included parameter) that is specific for a given molecule. They want to match
up 'theory' with their incredibly accurate measurements.
Many of the 'off-diagonal' force constants turn out to be small, and spectroscopists have
developed systematic simplifications to the force fields in order to make as many as possible
of the small terms vanish. If the force field contains only 'chemical' terms such as bond
lengths, bond angles and dihedral angles, it is referred to as a
valence force field
(VFF).
There are other types of force field in the literature, intermediate between the VFF and the
general force field discussed above.
−
4.4 Molecular Mechanics (MM)
Molecular modellers usually have a quite different objective; they want a force field that can
be transferred frommolecule to molecule, in order to predict (for example) the geometry of
a new molecule by using data derived from other related molecules. They make use of the
bond concept, and appeal to traditional chemists' ideas that a molecule comprises a sum
of bonded atoms; a large molecule consists of the same features we know about in small
molecules, but combined together in different ways.
The term
molecular mechanics
was coined in the 1970s to describe the application of
classical mechanics to determinations of molecular equilibrium structures. The method
was previously known by at least two different names: the
Westheimer
method and the
force-field
method. The name and acronym MM are now firmly established.
The idea of treating molecules as balls joined by springs can be traced back to the
work of Andrews (1930). A key study on the development of MM was that by Snyder and
Schachtschneider (1965), who showed that transferable force constants could be obtained
for alkanes provided that a few off-diagonal terms were retained. These authors found that
off-diagonal terms are usually largest when neighbouring atoms are involved, and so we
have to take account of nonbonded interactions but only between next nearest neighbours.
