Example:
calculating the energy of propane with a molecular mechanics potential function
BONDS: Propane (C3H8)
has 10 bonds: two C-C bonds (symmetrically equivalent), eight C-H bonds. Two
C-H bonds are in the CH2 group, 6 in CH3 groups.
Typically, the same parameters (k and l0) would be used for all these C-H
bonds, for simplicity. This is an example of the transferability of molecular
mechanics parameters – the same parameters can be used for similar groups.
VALENCE ANGLES: One C-C-C angle,
ten C-C-H angles, seven H-C-H angles. All angles are included, even though they
are not all independent.
TORSIONS: 18 torsion angles to consider
altogether: twelve H-C-C-H torsions and six H-C-C-C torsions. Each modelled by
a cosine series with minima at trans and gauche conformations.
NON-BONDED INTERACTIONS: There are
27 non-bonded interactions to calculate (i.e. between atoms separated by at
least 3 bonds)
Altogether, therefore, there are 73 energy terms to calculate. This is
quite a large number (even for a small molecule like propane), but much less
than the number of integrals that would have to be calculated in an ‘ab initio’
electronic structure calculation on this molecule.
·
Many different
molecular mechanics methods use the same, or closely related, functional forms
for the potential function (similar to equation 1 above).
· This is not enough, by itself, to define a molecular mechanics potential function: to do this, the parameters must be specified.
·
Parameters for
the same molecule may be very different in different force fields, but each may
give good results for e.g. its structure.
· There is no ‘right answer’ for the form or parameters of a molecular mechanics forcefield, they are purely empirical, and can be judged only on their results.
·
One important
factor is that for energy minimization and molecular dynamics simulations, the
first derivative of the energy is needed (and preferably second derivative
also), making a simple mathematical form desirable.
· Well-known molecular mechanics methods include CHARMM and AMBER (for modelling proteins and other biological macromolecules), and MM3 (for modelling small molecules).
·
Parameterization
of a molecular mechanics method is a laborious task, requiring fitting of large
amounts of data for many molecules.
·
Usually atoms will be defined as
falling into a particular ‘type’, according to its hybridization and chemical
environment in the molecule.
·
For example, all carbon atoms in
methyl groups (CH3) would be assigned one atom type; carbon atoms in
carbonyl groups a different atom type, etc. It is desirable to keep the number
of atom types to a minimum (to keep the numbers of parameters to a minimum
(also making the parameters transferable between different groups), while at
the same time representing different chemical groups accurately.
·
Each atom type would have
Lennard-Jones parameters associated with it. All bond, angle, and torsion terms
would also be listed in terms of the atoms types.