## Computing the Free Energy - The harmonic approximation

In this exercise the free energy of MgO will be computed
within the quasi-harmonic approximation.

In the quasi-harmonic approximation the
free energy is computed as a sum over the vibrational modes of the
infinite crystal - that is over all the **bands** and
**k**-lables. Numerically this sum is approximated
as a sum over a finite grid of **k**-points.
The same grid of points used to plot the phonon density of states
in the previous exercise.

The number of points used
in this grid will affect the accuracy of the calculation. The accuracy
would be perfect for an infinite grid but, unfortunately, the calculation
would take an infinite amount of CPU time to run !

### Selecting a suitable k-space grid

The compromise between accuracy and CPU time can be established empirically.
Here the free energy will be computed for ever increasing sizes of grid
and its convergence to the infinite grid value monitored. The convergence
of the density of states plot in the previous exercise is a useful guide
for choosing the optimal grid size.

A calculation is performed as follows.

Load an MgO structure and bring up the **Execute GULP** panel.

Click on **General opts** and select **Phonon DOS**.

Shrinking factors along **A**, **B** and **C** will be displayed;
these default to a 1x1x1 grid.

Set the **Temperature** to 300 Kelvin (the pressure will default to 0 GPa)

Run GULP and examine the log file. Take note of the reported Free Energy.

Repeat this calculation for 2x2x2, 3x3x3, 4x4x4 etc. grids.

### Questions

- How does the free energy vary with grid size ?
- Which grid size is appropriate for calculations accurate to 1
meV, 0.5 meV and 0.1 meV per cell ?

### An opportunity to speculate

- Would this optimal grid size for MgO be appropriate for a calculation on;
- a similar oxide (eg: CaO) ?
- a Zeolite (eg: Faujasite) ?
- a metal (eg: lithium) ?

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