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
The Born-Oppenheimer surface is then a multidimensional surface describing
the energy of the molecule in terms of nuclear positions.
Energy = f(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
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
E = Es + Eb + Ew + Enb
where Es is the energy involved in the deformation of a bond, either
by stretching or compression, Eb is the energy involved in angle
bending, Ew is the torsional angle energy, and Enb
is the energy involved in interactions between atoms that are not directly
bonded. We will take a look at each of these individually below.
Bond Potential Function, Es
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 lo, 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
Es = Sumbonds 1/2 kl (l - lo)2
where kl is the force constant of the particular bond (related to
the strength of the bond), l is the bond length at the deformed structure,
lo is the strain free bond length ('equilibrium' bond length), and
the summation is over all the bonds in the molecule.
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 Csp3 - Csp3o=1.54 angstroms,
and the force constant for this bond is given as, kl = 65 kcal/mol.
Insert these into the boxes and press run for the result.