==========
Exercise 5
==========

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Nudged Elastic Band calculations
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.. contents::
.. section-numbering::
.. include:: ../charents.txt





Self-diffusion on the Al(110) surface
=====================================

In this exercise, we will find minimum energy paths and transition
states using the "Nudged Elastic Band" method.

.. figure:: Al-Al110.gif
   :width: 200
   :align: center
   :alt: Al/Al(110)
   :figwidth: 200

   Al adatom on a Al(110) surface. 

Look at the Al(110) surface in the file `<Al-Al110.pdb>`__ with
Rasmol.  The adatom can jump along the rows or across the rows.

* Which of the two jumps do you think will have the largest energy
  barrier?

The template script `<neb1.py>`__ will find the minimum energy path for a jump
along the rows.  Fill in the missing information (marked with ``??``)
and run the script.

* Make sure you understand what is going on (make a good sketch of the
  110 surface).

* What is the energy barrier?

* Copy the script to ``neb2.py`` and modify it to find the barrier for
  diffusion across one of the rows.  What is the barrier for this
  process?

* Can you think of a third type of diffusion process?  Hint: it is
  called an exchange process.  Find the barrier for this process, and
  compare the energy barrier with the two other ones.



Analytical problem on transition state theory (TST)
===================================================

The second problem of today's exercise is mostly an analytical one. This
problem is developed by Hannes Jonsson and deals with the potential
energy surface (PES) of a hydrogen atom adsorbed on a (100) crystal
surface of an FCC metal.

What are the four basic assumptions of TST?  What should the shape of
the potential energy surface be like in order for TST to give a good
approximation to the rate constant of a transition?

A hydrogen atom adsorbed on the surface of a metal crystal can diffuse
by hopping from one binding site to another. The atom can be
considered to be a particle moving on a periodic potential surface
(PES) while the metal atoms can be taken to be stationary. This is a
good first approximation because the metal atoms are so much heavier
than the hydrogen atom.  For a hydrogen atom on the (100) surface of
an FCC metal, the potential energy of the hydrogen atom can be
approximated by the function 

  *V*\ (*x, y, z*) = *V*\ :sub:`s`\ [exp(-cos(2\ |pi|\ *x*\ /*b*)
  - cos(2\ |pi|\ *y*\ /*b*) - 2\ |alpha|\ *z*) - 2exp(-|alpha|\ *z*)]

For the parameters of the potential, we take *V*\ :sub:`s` = 0.2 eV,
*b* = 3 |angst|, |alpha| = 2 |angst|\ :sup:`-1`.

Identify the minima, maxima and saddle points of the PES. You may do this
by either looking at the plot or taking the partial derivatives. You will need
the derivatives anyway at a later point.

Plot the function with the
plot program of your choice. In the following, it is described, how
you can plot the potential with Mathematica. Type ``mathematica`` to
start the program.  Then, type::
 
  f[x_, y_, z_] := Vs * (Exp[-Cos[2 Pi x/b]-Cos[2 Pi y/b] -2 alpha z]
                         -2 Exp[-alpha z])

To make mathematica read and evaluate this statement, press
SHIFT + ENTER. You have to do this after each statement to make mathematica
evaluate your statement and give you output. Then you can set the parameters
by the statements::
 
  Vs = 0.2 
  b = 3 
  alpha = 2
 
Then you would like a 3D contour
plot of the function and type::

  Plot3D[f[x, y, -1], {x, -3, 3} {y, -3, 3}]

or::

  Plot3D[f[x, 0, z], {x, -3, 3} {z, -1.5, 3}]

Then again, press SHIFT+ENTER. Now you should see a pretty plot of the
potential.

Evaluate the activation energy for diffusion hops. Then Taylor expand
the potential energy about the minimum and saddle point to find the
frequency of the vibrational modes and evaluate the prefactor.  Add a
correction to the activation energy due to zero point energy.

What is the average length of time in between diffusion hops at room
temperature and at 400 K?





