.. _stress:

=================================
Plane wave mode and Stress tensor
=================================

The major advantage of running DFT calculations on a real space grid, is a
very efficient parallelization scheme when dealing with large systems.
However, for small systems it is often faster to use a plane wave basis set
instead. In this case all quantities are represented by their Fourier
transforms on the periodic super cell and periodic boundary conditions are
required. In grid mode, the key convergence parameter is the grid spacing
`h`, whereas in planewave mode the corresponding parameter is
`E_{cut}=G_{cut}^2/2` (in atomic units). `G_{cut}` determines the maximum
size of reciprocal lattice vectors to be included in the plane wave
expansion.


Converging the plane wave cutoff
--------------------------------

As a simple example we perform a calculation on bulk Si using the plane wave
basis. The following script tests the convergence of the total energy with
respect to the cutoff parameter `E_{cut}`.

.. literalinclude:: con_pw.py

Note that we use a rather rough k-point sampling and the default Fermi
smearing of 0.1 eV. These parameters should of course be converged, but for
now we will just keep them fixed in order to speed up the calculations. At
`E_{cut}` = 400 eV, the total energy is seen to be converged to within 5 meV.
Now edit the script such that the calculation is performed in grid mode with
various values of the grid spacing (comment out the lines :samp:`for x in
[100, ...` and :samp:`mode=PW(x)` and uncomment the lines :samp:`for h in
[0.24, ...` and :samp:`h=x`).

What grid spacing is needed in order to converge the total energy to within 5
meV? What is the timing of this calculation compared to the one at 400 eV in
plane wave mode? Compare the number of grid points and plane waves used in
these two calculations respectively (look in the output file
'convergence_400.txt' and the corresponding one for the grid calculation.)
You can see how long a calculation took by looking at the times recorded in
the .txt file.


Optimizing the unit cell
------------------------

**Warning**: due to difficulties in optimizing cell and positions
simultaneously :class:`ase.constraints.UnitCellFilter` may produce incorrect
results. Always verify obtained structures by means of performing separate
cell (see :class:`ase.constraints.StrainFilter`) and positions optimizations
(see :mod:`ase.optimize`). Consider a much more tight fmax than the one used
in this tutorial!

In the :ref:`aluminium_exercise` exercise the lattice constant of bulk Al was
found by calculating the total energy at various lattice distances. A nice
feature e of the plane wave mode is that it allows a simple implementation of
the stress tensor, which can be used to optimize unit unit cells of periodic
systems directly. The following script performs such an optimization for bulk
Si.

.. literalinclude:: stress.py

The calculation uses 12 iterations to find the optimal lattice constant and
the relaxation can be viewed with the command line tool :program:`ase`:

.. highlight:: bash

::

  $ ase gui stress.txt

Since we know the experimental lattice constant, we could probably have
calculated the PBE lattice constant faster by fitting a parabola to five
points in the vicinity of the expermental lattice constant. However, for
complicated unit cells with more than one lattice parameter, the stress
tensor becomes a highly valuable tool.
