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.

 

Using molecular mechanics potential functions

 

·     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.

 

Molecular mechanics energy terms