Force field methods

• You should read Chapter 2 of Goodman's 'Chemical Applications of Molecular Modelling'

     about this

• If two atoms are connected by a spring of equilibrium length r_bond then there will be some potential energy in the spring if it is pulled out or compressed to some other length r :
E_bond = k_bond(r - r_bond)²
• (For the interested, this comes by integrating Hooke's Law, the equation for a spring balance:
Force = dE/dr = kr)
• For a particular kind of bond, e.g. 4-coord. carbon to 2-coord. oxygen, we would need to know the stretching constant k_bond and the unperturbed length r_bond
• The method supposes that these are constants over all occurrences of these elements with these coordination numbers
• There can be the same kind of energy expression for angle deformation:
E_angle = k_angle(q - q_angle)²
• The constants k_angle and q_angle need to be known for the angle at atom B in all possible combinations of atom types ABC
• The constants (force field parameters) have been worked out statistically by fitting models to known structures, and are supplied built in to the program purchased
• In force fields in use, more elaborate expressions may be used, to obtain better reproduction of observed measurements, and there are at least three more energy terms:
• E_{Van der Waals}  - non-bonded interactions of pairs of atoms
• E_{bond rotation}   - correction to account for rotamer stability beyond what can be explained as Van der Waals repulsion, e.g. the tendency for 3-coord., p-bonded carbon to be planar
• E_charge            - charges are assigned to atoms, based on electronegativities, then energy of attraction or repulsion of all pairs of atoms is calculated
• Altogether we have a total force field energy:
E_total = E_bond + E_angle + E_VdW + E_rotation + E_charge +  ...??
• All the energy terms and their parameters form a self-consistent set
• You can see this if you consider four atoms ABCD:
• Bond length BC depends on the Van der Waals interaction of A with D, as well as on the atoms types of B and C
• Therefore the average of all known bond lengths BC is not necessarily the best value for r_bond , because the modeller is going to use both E_bond and E_VdW  in predicting the bond length.  If this value of r_bond were used, the predicted bond lengths would come out too long, because the Van der Waals interaction AD is normally repulsion, to make the BC bonds longer.
• The parameters used in the E_bond expression depend on the parameters used in the E_VdW expression, etc.
• If extra correction terms are added, e.g. for H bonds, to improve the model, everything has to change
• The whole set of energy terms and all their parameters, for the whole set of atom types supported, is called a Force Field
• Names for some commonly used force fields are:

AMBER, CHARM, COSMIC, MM2, SYBYL, MMFF, DREIDING
• It is useful to be able to recognise these as force-field methods, in reading papers and in using modelling programs, where it may not be obvious which of the provided methods are force-field and which are semi-empirical
• Because the number of parameters to be supplied with the program increases rapidly with the number of atom types allowed, force fields are always restricted to certain elements in certain common environments, where there are sufficient literature structures to base the parameters on
• In general, they work well for ordinary organic compounds
• Some work better for some classes of compound, e.g. strained hydrocarbons, than others
• If (in later life) you are about to buy a modelling program, it is best to find out first whether its supported force fields will work for your kind of molecule.  Manufacturers sometimes supply trial versions, or trial periods using the real thing, before purchase.
• It may be possible to add an atom type in some of the programs, but it is very difficult to get the right values for the parameters, and to do enough tests to validate the method
• Many more elements can be dealt with by semi-empirical methods, and all can be done by ab initio methods, if they have few enough atoms
• Force field methods are generally not accurate enough to calculate reaction energies.  If it is at all practicable, an electronic method should be used after the force-field geometry optimisation.

Atom types

• Different parameters are needed for 3-coord carbon than for 4-coord. carbon, etc., since the bonds are of different strength and are at different angles
• Modelling programs try to guess the atom type from its connectivity, but do not always succeed
• It is wise to go round a molecule you have constructed, checking that all the atom types are sensible
• Modelling programs normally provide some way of finding out atom types and changing them
• It may not be obvious to the user what the atom type designation means:  e.g. one program has N sp³, N sp²,  N trigonal as possibilities.   Which should you use for N in an organic amide?  Reading the manual may help.