Molecular Mechanics (Statics)

Given a force field, one can compute U(q) and can characterize any configuration. In the limit that one configuration is so favored as to be truely representative, then for a property F, it is accurate to write:

<F>= F_config

All combinatorial properties are neglected (i.e. entropy S = 0). A = U - TS = U_config. Alternatively, it is sometimes argued that MM corresponds to a situation where T=0, but this may not be strictly true, since thermal behavior may be (and usually is) already incorporated in the force field parameterization itself.

The Method:

Determine q_most_probable such that U(q_mp) = U(q)_min. In numerical terms, this corresponds to find U(q) such that all of the first derivatives of U with respect to q are zero and all of the second derivative os U with respect to q are greater than zero (therefore a minimum). The numerical methods commonly used for this purpose come from the usual collection of tried-and-true numerical minimization algorithms, such as steepest descents, conjugate gradients, and variable metric or modified Newton methods. One starts with the best initial guess one has available for the configuration (i.e. all coordinates q of the atoms of the system or molecule) and computes the energy U of that initial configuration. Then the coordinates are systematically changed according to one of the aforementioned algorithms and the energies (and sometimes also the first and/or second derivatives of energy) of the new configurations computed using the force field so that the configuration is marched downhill in energy to a nearby local minimum.

A planar cyclohexane ring is a good example of this method. One needs to know little more than the chemical formula of the ring (C6H12) to build a planar hexagonal molecule as a starting guess. After a brief molecular mechanics calculation, one of the more commonly recognized conformations of cyclohexane is immediately apparent.

This ester molecule illustrates another example of molecular mechanics.

Use of Periodic Boundary Conditions for Simulation of Bulk Systems

For bulk liquids or solids, boundary conditions in such a system must be properly handled to avoid the strong influence of surface effects. Consider, for example, the situation where surface effects are assumed to influence only a region within 10 angstroms of the surface of a sample. In a real, macroscopic sample, this is a vanishingly small fraction of the material. In an atomistic level simulation which may be limited to a model system in a cube of only 30 angstroms on a side, this surface region constitutes 96% (26/27 `ths) of the material.

In order to overcome this limitation in bulk simulations, it is common to use periodic, or cyclic, boundary conditions (PBC's). The simulation cell is treated as though is were surrounded on all sides by cells which are its exact image. This periodicity imposes a crystal-like structure on the model for lengths greater than the size of the parent cell, which may be realistic for some materials and not for others. The use of PBC's creates the problem of multiple interactions with a single particle (and its images). Unless the system under study is truely periodic (e.g. a crystal) one counts only the most important (i.e. the nearest) interaction with a particle or one of its images. This is the so-called Minimum Image Convention for simluations of non-crystalline bulk systems. Interactions between particles are only counted up to distances of L/2, where L is the sidelength of the (cubic) simulation cell. (Through careful accounting, this upper limit can be extended to one half the length of the body diagonal of the cube: L(3/4)^0.5.) Notice, however, that the Minimum Image Convention causes a truncation of the force field at L/2. This does not mean that interactions between particles which are more distant than L/2 are negligible. Instead, a long range correction, called a "tail correction" is usually required.

Three common methods for computing this tail correction term are use of

Ewald summation (for truely periodic systems)
Reaction field (for polarizable materials)
Uniform Pair Distribution: g(r)=1 for r>r_c (for amorphous systems)

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