Calculating the Free Energy in 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.
- 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) ?
Next exercise: The Thermal Expansion of MgO