MOLECULAR MODELING AND MOLECULAR MECHANICS 165
hybridization about the central atom since a commercial program not written
for inorganic molecules may not recognize metal – ligand bonds. In summary,
the signifi cant advantage of molecular mechanics calculations is that they are
relatively rapid; however, caution is needed in interpretation of molecular
mechanical results produced by empirical force fi eld calculations. A good
analogy is to that of the neural network; that is, the molecular mechanical
result is completely dependent on the facts the system has been taught.
The types of input and output accepted by the computer program as well
as algorithms used to achieve energy minimization are described here briefl y.
Generally, an energy minimization routine produces an optimized structure
(conformer) most closely related to the input coordinates. That is, the routine
falls into theclosest energy minimum, which may or may not be the global
energy minimum of a system. One carries out a conformational analysis by
deriving the energy surface that results from changing a particular rotational
value, often a set of dihedral angles. Conformations are then submitted to
energy minimizing routines that include: (1) simplex (only the potential energy
function is used), (2) gradient or fi rst derivative methods (steepest descent),
(3) conjugate - gradient methods where the history of the search can infl uence
direction and step size (Fletcher – Reeves, Polak – Ribi è re), (4) second -
derivative methods (Newton – Raphson and block - diagonal Newton – Raphson),
and (5) least squares methods (Marquardt). In applying molecular mechanics
methods, particularly for those structures that are predominant in solution, it
is important to fi nd the lowest - energy structure, that is, the global energy
minimum. Several methods exist for carrying out the minimization, but only
the torsional Monte Carlo method will be discussed here. The Monte Carlo
method is a stochastic search in which the variation from one starting confor-
mation to another is limited in magnitude — for instance, by limiting the start-
ing geometries to those that conform to some energy requirement (perhaps
15 kJ/mol from the energy minimum). Using the internal coordinate of
torsional angle causes signifi cant differences in possible structure conforma-
tions — in contrast to changes in bond distances or bond angles, which do not.
In each Monte Carlo step, a random number of torsional angles are varied
by a random amount generating a new starting geometry that can be
minimized.
Molecular dynamics involves the calculation of the time - dependent move-
ment of each atom of a molecule, achieved by solving and applying Newton ’ s
equations of motion. Structures for starting geometries are sampled as a
function of time or geometry during a molecular dynamics run of a few nano-
seconds. Usually, an elevated temperature is used to favor faster and more
complete molecular dynamics searching. A good conformational search sys-
tematically (deterministically) scans the entire potential energy surface,
generates starting geometries, and then minimizes them. A combination of the
above - described methods — deterministic, stochastic, and molecular dynamics,
illustrated in Figure 4.1 — screens the entire potential energy surface and pro-
duces the best possible results.