Biomedical Engineering Reference
In-Depth Information
eigenvalue of
H
means that one has to be much more careful with the step taken in an optim-
ization. For example, it is always possible to take a steepest descent step in a minimum
search, which will lower the energy or leave it unchanged. Such a step is not appropriate
for transition state searching; it is therefore harder to find a transition state than a minimum
and many Newton-Raphson type procedures need to start the optimization fairly close to
the true transition state structure.
The key concept in our discussion is therefore the Hessian matrix
H
. This is a real
symmetric matrix and the eigenvalues and eigenvectors can be found by standard numer-
ical methods. Simons
et al.
(1983) showed that each Newton-Raphson step is directed
in the negative direction of the gradient for each eigenvector that has a positive Hessian
eigenvalue, and along the positive direction of the gradient for each eigenvector that has a
negative eigenvalue.
For a transition state search, if you are in a region of themolecular potential energy surface
where the Hessian does indeed have one negative eigenvalue, the Newton-Raphson step
is appropriate. If you have landed on some region of the surface where the Hessian does
not have this desired property, then you must somehow get out of the region and back to a
region where the Hessian has the correct structure of one negative eigenvalue.
For minima, qualitative theories of chemical structure are a valuable aid in choosing
starting geometries whilst for transition states one only has a vague notion that the saddle
point geometry must lie somewhere between the reactants and products. Molecular mech-
anics (MM) structures are often good as starting structures for geometry optimizations but
MM cannot handle bond making and bond breaking.
One of the earliest algorithms that could take corrective action when the wrong region of
the molecular potential energy surface was chosen was that due to Poppinger (1975). This
author suggested that the lowest eigenvalue of the Hessian should be followed 'uphill'. The
technique has come to be known as
eigenvector following
. There is a good discussion of
eigenvector following in Baker's (1986) paper, which has the following abstract.
An algorithm for locating transition states designed for use in the ab initio program package
GAUSSIAN-82 is presented. It is capable of locating transition states even if started in the
wrong region of the energy surface, and, by incorporating the ideas on Hessian mode following
due to Cerjan andMiller, can locate transition states for alternative rearrangements/dissociation
reactions from the same initial starting point. It can also be used to locate minima.
Baker refers to eigenvector following as 'Hessianmode following'. The big problemwith
eigenvector following is that the remaining directions along the potential energy surface
are left in isolation until the initial fault is corrected.
If we think of a chemical reaction (either reactants or products) as minima on the molecu-
lar potential energy surface, then there is no unique way of moving uphill on the surface
since all directions go uphill. The
linear synchronous transit
(LST) algorithm searches
for a maximum along a path between reactants and products. It frequently yields a struc-
ture with two or more negative Hessian eigenvalues, and this is not a transition state. The
quadratic synchronous transit
(QST) method searches along a parabola for a minimum in
all directions perpendicular to the parabola. The following is the abstract of Halgren and
Lipscomb's (1977) original reference for synchronous transit.
In the synchronous-transit method, a model linear synchronous transit pathway is first con-
structed and is then refined by optimizing one or more intermediate structures subject to the
