In a very crude sense molecular modeling treats a molecule as a collection of wieghts connected with springs, where the weights represent the nuclei and the springs represent the bonds.
Molecular modeling, also known as molecular mechanics, is a method to calculate the structure and energy of molecules based on nuclear motions. Electrons are not considered explicitly, but rather it is assumed that they will find their optimum distribution once the positions of the nuclei are known. This assumption is based on the Born-Oppenheimer approximation of the Schrödinger equation. The Born-Oppenheimer approximation states that nuclei are much heavier and move much more slowly than electrons. Thus, nuclear motions, vibrations and rotations can be studied separately from electrons; the electrons are assumed to move fast enough to adjust to any movement of the nuclei.
In a very crude sense molecular modeling treats a molecule as a collection
of wieghts connected with springs, where the weights represent the nuclei
and the springs represent the bonds.
A force field is used to calculate the energy and geometry of a molecule. It is a collection of atom types (to define the atoms in a molecule), parameters (for bond lengths, bond angles, etc.) and equations (to calculate the energy of a molecule). In a force field a given element may have several atom types. For example, ethylbenzene contains both sp3-hybridized carbons and aromatic carbons. sp3-Hybridized carbons have a tetrahedral bonding geomtery, while aromatic carbons have a trigonal bonding geometry. The C-C bond in the ethyl group differs from a C-C bond in the phenyl ring, and the C-C bond between the phenyl ring and the ethyl group differs from all other C-C bonds in ethylbenzene. The force field contains parameters for these different types of bonds. Some of these parameters are given below. The total energy of a molecule is divided into several parts called force potentials, or potential energy equations. Force potentials are calculated independently, and summed to give the total energy of the molecule. Examples of force potentials are the equations for the energies associated with bond stretching, bond bending, torsional strain and van der Waals interactions. These equations define the potential energy surface of a molecule.
Below are examples of some of the force potentials, and parameters one may find in a force field. These examples are from Allinger's MM2 force field [(a)"Conformational Analysis. 130. MM2. A Hydrocarbon Force Field Utilizing V1 and V2 Torsional Terms", Allinger, N. L., J. Am. Chem. Soc. 1977, 99, 8127. (b) Burket, U.; Allinger, N. L. Molecular Mechanics; American Chemical Society: Washington, DC, 1982.] and the MMX force field of PCMODEL ["PCMODEL", Gilbert, K., Serena Software: Bloomington, IN, 1993]
Whenever a bond
is compressed or stretched the the energy goes up. The energy potential
for bond stretching and compressing is described by an equation similar
to Hooke's law for a spring, except a cubic term is added. This cubic term
helps to keep the energy from rising too sharply as the bond is stretched.
As angles are bent from their norm the energy increases. The potential
function below works very well for bends of up to about 10 degrees. To
handle special cases, such as cyclobutane, special atom types and parameters
are used in the force field.
When a bond angle is reduced the two bonds forming the angle will stretch to alleviate the strain. To handle phenomena such as this, cross term potential functions are introduced. Cross term potential functions take into account at least two terms such as bond stretching and bond bending.
Intramolecular rotations (rotations about torsion or dihedral angles) require
energy. For example, it takes energy for cyclohexane to go from the chair
conformation to the boat conformation. The torsion potential is a Fourier
series that accounts for all 1-4 through-bond relationships.
The van der Waals
radius of an atom is its effective size. As two non-bonded atoms are brought
together the van der Waals attraction between them increases (a decrease
in energy). When the distance between them equals the sum of the van der
Waals radii the attraction is at a maximum. If the atoms are brought still
closer together there is strong van der Waals replusion (a sharp increase
in energy).
In some force fields electrostatic interactions are accounted for by atomic
point charges. In other force fields, such as MM2 and MMX, bond dipole
moments are used to represent electrostatic contributions. One can readily
see that the equation below stems from Coulomb's law. The energy is calculated
by considering all dipole-dipole interactions in a molecule. If the molecule
has a net charge (e.g., NH4+), charge-charge and
charge-dipole calculations must also be carried out.
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