In this exercise the structure of MgO will be optimised with respect to the free energy at a number of temperatures. The free energy is computed within the quasi-harmonic approximation.

A free energy optimisation is performed as follows.

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

Click on optimisation

Click on Optimisation opts and select Optimise Gibbs free energy.

Click on General opts and select Phonon DOS.

Select suitable shrinking factors for the k-space sampling based on your previous investigations.

Set the Temperature

Run GULP and examine the log file.

These calculations will take a little while to complete. GULP is computing the internal energy and phonons at a sequence of geometries as it seeks to minimise the free energy with respect to the structure.

Vary the temperature from 0K to 1000K in steps of 100K and compute the variation in the free energy and optimal lattice constant by optimising the structure at each temperature.


  • Plot the free energy against temperature
  • Plot the lattice constant against temperature
  • Comment on the shape of these curves.
  • Compute the coefficient of thermal expansion for MgO
  • How does this compare to that measured ? Find a measurement in the literature or on the web - at what temperature was the measurement made ?
  • What are the main approximations in your calculation ?

An opportunity to speculate ...

  • What is the physical origin of thermal expansion ?
  • As the temperature approaches the melting point of MgO how well do the phonon modes represent the actual motions of the ions ?
  • In a diatomic molecule with an exactly harmonic potential would you expect the bond length to increase with temperature ? Why does it increase in the solid when we are using an quasi-harmonic approximation ?

Next exercise: Molecular Dynamics

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