Force Field Methods
In force field or molecular mechanics (MM)
methods the molecules are treated as mechanically connected systems of atoms. In contrast to quantum
mechanical methods, electrons are not treated explicitly but together with the nuclei as effective atoms.
The energy expressions used in MM methods are chosen such that the system is well described at typical
bonding distances only. The description of reactions or transformations in which
bond breaking and making occurs is therefore not possible with most approaches.
A given force field usually contains a number of different interaction terms describing
different types of strain possible in a molecular system. The following is a list of components
that are typically considered:
The overall (strain) energy of the system is then given as the sum of all these terms
over all occurences in the system:
EMM = Vstr + Vbend + Vtor + Voop + Vcross + VvdW + VES
Van der Waals and electrostatic parameters are usually not calculated for atoms that are
at a distance of one or two bonds in a given molecule. The closest interacting centers for
these types of interactions are separated by at least three bonds and relative to each other
in 1-4 position. These types of interactions are therefore often referred to as 1,4-interactions.
While the smallest interacting unit in all atom (AA) force fields
are the single atoms of the system, groups of atoms are used as interacting units in
united atom (UA) force fields. In particular for the modeling of
large systems (such as proteins or polymers) it is quite sufficient to treat, for example, CH3 -
groups as one effective interaction center that is slightly larger than a single carbon atom.
Even in all atom force fields, however, different parameters are used to describe centers of
different bonding characteristics. Typically there will be different parameters for
sp3-, sp2-, and sp-type carbon atoms and additional ones for
carbon atoms in carbonyl groups or aromatic systems.
Typical energy expressions for some of the enery terms are as follows:
Most force fields use a simple quadratic (Hook's law) energy expression with only one term.
In order to evaluate this energy expression for a given structure, the reference bond
distance (for an unstrained bond) and a force constant describing the ease of deformation
are needed. These expressions are used for example by the AMBER or CHARMM force fields.
Somewhat more elaborate bond stretching functions containing cubic and quartic terms are
used by the MM3 hydrocarbon force field.
A similar functional form is used for the angle bending deformation:
Again an unstrained reference bond angle is assumed, whose deformation is possible with
force constant kijk.
The functions describing the torsional potential energies are different from the previous
terms as they must be periodic with respect to rotation by 360 degrees:
The use of all three potential parameters V1, V2, and V3
is not necessary in all cases.
There are several possibilities for the description of van der Waals interactions between
non-bonded atoms, the Lennard-Jones 6-12 potential being the most commonly used:
Here the interaction between two non-bonded atoms are described as slightly attractive
at longer distances, but strongly repulsive at closer distances. Two potential parameters
are needed for each non-bonded interaction type.
The energy expression for the electrostatic interactions usually follows a simple Coulomb
law, which is always attractive for partial charges of opposite sign and always repulsive
for partial charges of equal sign:
The parameters needed for evaluation of this expression are the partial charges q for
the interacting centers i and j as well as an effective dielectric constant. As calculation
of non-bonded interactions becomes quite time consuming for extended systems,
most programs use a cutoff value of between 9 and 16 Angstroms, beyond which the
interactions are tailed off to zero over an additional short distance.
In evaluating, deriving and comparing force field parameters for any of these
potential energy functions it is important to remember, that all of these parameters
are only valid within a given force field, that is, a set of other parameters and
potential energy functions. A single force field parameter has not to be meaningful by itself!
And there are many different parameters and functions that yield the same final
result (strain energy)!
last changes: 11.11.2005, HZ
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