Chemistry Reference
In-Depth Information
optimization calculation, the number of iterations needed to achieve conver-
gence can be reduced if the quality of the initial estimate of the solution is
improved. For constructing initial estimates for the images along an MEP,
this often means that it is worthwhile to adjust the estimates with some phys-
ical or chemical intuition about the process being described. In the present
example, we might anticipate that the Ag atom will be slightly higher on the
surface as it goes over the bridge site than it is in the fourfold sites. To construct
initial images that include this simple idea, we took the configurations defined
by linear interpolation and increased the height of the Ag atom by 0.1
˚
for
images 2 and 7, 0.15
˚
for images 3 and 6, and 0.2
˚
for images 4 and 5.
The energies associated with these initial states are shown with open diamonds
in Fig. 6.9. Repeating the NEB calculation beginning from this initial estimate
yielded a converged result in 24 iterations. Although this is a relatively modest
improvement in computational effort, it illustrates the idea that putting a
little effort into improving upon simple linear interpolation can improve the
convergence rate of NEB calculations.
Comparing Fig. 6.9, which shows the MEP determined from NEB calcu-
lations, with Fig. 6.3, which shows the MEP determined from symmetry con-
siderations, points to an important shortcoming of our NEB results: none of the
images in the NEB calculation lie precisely at the transition state. Image 4 and
image 5 both lie 0.35 eV above the end points of the calculated MEP, a result
that slightly underestimates the true activation energy. For the particular
example we have been using, this problem could easily have been avoided
by using an odd number of images in the NEB calculation since we know
from symmetry that the transition state must lie precisely in the middle of
the reaction coordinate. We chose not to do this, however, to highlight the
important point that in almost all situations of practical interest, the location
of the transition state
cannot
be determined by symmetry alone.
Because finding a precise value for the activation energy is so important in
using transition state theory (see the discussion of Fig. 6.5 above), NEB
calculations performed as we have described them need to be followed by
additional work to find the actual transition state. There are at least two
ways to tackle this task. First, variations of the NEB method have been devel-
oped in which the converged result not only spaces images along the MEP but
also has one of the images at the transition state. Information about this
method, the climbing image NEB method, is given in the further reading at
the end of the chapter. Another simple approach relies on the fact that some
geometry optimization methods will converge to a critical point on an
energy surface that is a transition state or saddle point if the initial estimate
is sufficiently close to the critical point. Said more plainly, geometry optimiz-
ation can be used to optimize the geometry of a transition state if a good
approximation for the geometry of the state can be given.
