The Molecular Potential Energy Surface - Molecular Modelling - page 69

engineers are concerned with related problems. The field of study is now an established part
of applied mathematics, and is referred to as optimization theory . An optimization problem
involves minimizing a function of several variables, possibly subject to restrictions on the
values of the variables. There is no single universal algorithm for optimization, and it is
necessary for us to spend a little time dealing with the different methods of interest.
I should tell you that one speaks of optimization and less commonly of minimization or
maximization; the latter two are equivalent, for maximization of function f is the same as
minimization of the function

f .
The first MM optimization seems to have been carried out by Westheimer (1956), who
did the calculations 'by hand'. The first computer calculations seem to have been done by
Hendrickson (1961). Neither of their methods was generally applicable to any molecule.
Many algorithms have been developed over a number of years for the location of station-
ary points, some of which are suitable for MM calculations, some of which are suitable for
quantum mechanical calculations and many of which are not particularly suited to either.
In MM we tend to deal with large molecules and consequently the molecular potential
energy function will depend on hundreds if not thousands of variables. On the other hand,
evaluation of the potential energy at each point on the hypersurface is simple.
Different considerations apply to quantum chemical calculations, to be dealt with later in
the text. Calculation of the energy at points on the surface is far from easy but the number
of variables tends to be smaller.
As stated above, transition states are special cases of saddle points; they are stationary
points on the surface where the Hessian has just one negative eigenvalue. Transition state
searching is a hot topic in chemistry, and a number of specialist algorithms have been
proposed. I will deal with transition states in Chapter 22.

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5.5 Multivariate Grid Search

This is the oldest method for finding minima, and it has a long history. What we do is the
following.

1. Choose a suitable grid for the variables.
2. Choose a starting point A on the grid.
3. For each variable q 1 , q 2 , ..., q p evaluate the molecular potential energy U at the two
points surrounding A (as determined by the grid size).
4. Select the new point for which U is a minimum, and repeat steps 3 and 4 until the local
minimum is identified.

The method is intuitive, and it is apparent that a local minimum will eventually be found.

5.5.1 Univariate Search

This is sometimes called the cyclic search, for we perform successive one-dimensional
searches for each of the variables in turn.

1. Choose a starting point A.
2. Minimize U ( q ) for each variable q 1 , q 2 , ..., q p in turn.
3. Repeat the cycle as necessary.