When it is of interest to study structure and bonding in very large molecular systems comprising hundreds or even thousands of atoms, many of the more sophisticated molecular modeling techniques are too demanding of computer time resources to be of general use. Molecular mechanics calculations, also known as force field calculations, can be of considerable use in the qualitative descriptions in these cases. In these cases, we concentrate on the structural aspect and not on the electronic and/or spectroscopic properties. In essence, we describe the potential energy surface without invoking any quantum mechanical calculations or descriptions. The Born-Oppenheimer approximation, fundamental to our molecular description, states that the Schrodinger Equation for a molecule can be separated into a part describing the motions of the electrons and a part describing the motions of the nuclei and that these two motions can be studied independently. This can be interpreted in one of two manners, one of which allows the study of the electronic structure, one of which allows the study of the molecular mechanics structure.

*Electronic Structure:* It is common practice to establish the
positions of the nuclei of the system by some method and then to study the
electronic structure using fixed nuclear positions.

*Molecular Mechanics (Vibrational Spectroscopy):* The motions of the
nuclei are studied and the electrons are not explicitly examined at all,
but are assumed to find an optimal distribution about the
nuclei.

The Born-Oppenheimer surface is then a multidimensional surface describing
the energy of the molecule in terms of nuclear positions.

Based on our history of physical and chemical experiences, we can think of molecules as mechanical assemblies made up of simple elements like balls (atoms), rods or sticks (bonds), and flexible joints (bond angles and torsion angles).

The method then treats a molecule as a collection of particles held together by simple harmonic forces. These various types of forces are described in terms of individual potential functions which in sum constitute the overall molecular potential energy or steric energy of the molecule.

In its simplest representation, the molecular mechanics equation is

where E_{s} is the energy involved in the deformation of a bond, either
by stretching or compression, E_{b} is the energy involved in angle
bending, E_{w} is the torsional angle energy, and E_{nb}
is the energy involved in interactions between atoms that are not directly
bonded. We will take a look at each of these individually below.

In terms of a ball and spring description, a bond deformation will involve
energy changes that can be approximated via Hooke's Law. If the strain free
energy of a particular bond length is taken as l_{o}, then any
deviation from this value will lead to an increase in the potential energy.
The mathematical form of this potential energy function describing the
deformation is

Let's look at an example of this functional form for a couple of bonding
situations. In the following exercise, we can choose values of the force
constant as well as the equilibrium bond distance and plot the resulting
potential energy description for deformations from equilibrium. For
example, let us consider a C*sp*^{3} - C*sp*^{3}o=1.54 angstroms,
and the force constant for this bond is given as, k_{l} = 65 kcal/mol.
Insert these into the boxes and press run for the result.

- General Harmonic
Force Fields (GHFF)
- All elements of the F matrix are considered in the analysis
- Good for very small molecules with high symmetry

`Constrained Force Fields(CFF)`

- More appropriate for larger molecules
- Systematic simplifications are made of the complete force field
- Basic assumption: off-diagonal elements and can be set to zero.(i.e., higher freq. stretch will not interact much with a low freq.. bend.)

__Direct constraints Force Field__- Fixes values of all interaction force constants from vibrations separated by at least two noncommon atoms.

- Valence Force Field
- All interaction force constants between stretching and bending of different bonds set equal to zero.
- No cross terms included.

Too simplistic for quantitative application. A slight relaxation is to allow some stretch-stretch and stretch-bend terms. __Central Force Fields__- Steric effects considered in a very limited way
- Describes angle bending in terms of nonbonded interaction distances. Rarely used. Like VFF, this is too simplistic to explain molecular frequencies. Relaxation of some constraints improves the situation by allowing interactions between interbond parameters.

__Urey-Bradley Force Field: Adds 1,3 interactions longer ranged nb terms__- In
addition to conventional stretching and bending force constants, all
interactions that are stretching/stretch-bend coordinated are defined in
terms of interactions between nonbonded atoms.

Well established shortcomings. Inadequacies are limited by allowing stretch-stretch and stretch-bend interactions missing in the simple model (Modifies Urey-Bradley Force Field) - Nonbonded terms can eliminate need for cross terms (eliminates force constants; therefore replaced w/ others ~ same # of parameters on Valence Force Field and Urey-Bradley Force Field.

__Orbital Valence Force Field__- Like the UBFF in its treatment of interactions in terms of repulsions between nonbonded atoms.
- Unlike UBFF, treats angle distortions in terms of maximizing orbital overlap.

- In
addition to conventional stretching and bending force constants, all
interactions that are stretching/stretch-bend coordinated are defined in
terms of interactions between nonbonded atoms.

**Molecular Mechanics Force Fields/Empirical Force Fields (EFF)**- Modified Urey-Bradley Force Fields
- Central Force Fields
- Valence Force Fields (most popular)

Note:- The MM representation is a gross oversimplification, justified only by history of research.
- Authors of various EFF's partition the component terms in vastly
different ways:
- Some have every conceivable kind of interaction treated.
- Others have only 5 - 10 terms,>li>Missing terms accounted by judicious parameterization

- The numerical value of V has no inherent physical meaning.
- Directly dependent on potential functions and parameters.
- Difference between energies of isomers is usually predicted well implying that MM is usually well suited for confrontational analysis.