Chemistry Reference
In-Depth Information
The images on the MEP calculated by the NEB calculations above provide a
straightforward way to approximate the geometry of the transition state. Fitting
a smooth curve through the NEB-calculated images such as the data shown in
Fig. 6.9 gives a well-defined maximum in energy. In our simple example, this
maximum occurs exactly half way between images 4 and 5 because of the
symmetry of the problem. If we create a configuration by linear interpolation
between the two images nearest the maximum in the fitted curve at the value of
the reaction coordinate defined by the maximum, this will typically give a
good estimate for the transition-state geometry. Performing a geometry optim-
ization starting from this configuration converges quickly to a transition state
with a corresponding activation energy of 0.36 eV. We reiterate that this
process of refining the configuration associated with the transition state is
much more important in examples where symmetry cannot be used to locate
the transition state.
6.4 FINDING THE RIGHT TRANSITION STATES
Using the techniques described in this chapter, you may identify the geometry
of a transition state located along the minimum energy path between two
states and calculate the rate for that process using harmonic transition state
theory. However, there is a point to consider that has not been touched on
yet, and that is: how do you know that the transition state you have located
is the “right” one? It might be helpful to illustrate this question with an
example.
A process that is very important for phenomena such as crystal growth and
epitaxy is the diffusion of metal atoms across a metal surface. When the atom
that is diffusing across the surface is of the same type as the atoms that make up
the crystal, this process is known as self-diffusion. The conventional view of
self-diffusion is of an atom that hops from local minimum to local minimum
on top of the surface; in this picture the diffusing atom is called an adatom.
This process, illustrated in Fig. 6.10, is the reaction path we assumed for an
Ag atom moving about on a Cu(100) surface. We can also adopt this hopping
reaction path to describe diffusion of a Cu atom on the Cu(100) surface.
Performing NEB calculations for Cu hopping on Cu(100) in the same way
as the Ag
Cu(100) calculations described above predicts that the activation
energy for this process is 0.57 eV.
The simple hopping mechanism, however, is not the only possible way for
atoms to move on Cu(100). Figure 6.11 illustrates another possible diffusion
process known as an exchange mechanism. In this mechanism, the adatom
(shown in dark grey), replaces an atom that is initially a surface atom
(shown in light grey) while the surface atom pops up onto the surface,
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