DFT as we used it so far, is meant for the ground state: it makes predictions about crystals at 0 K. If thermal energy is absorbed by a crystal, it will be move to an excited state. Thermal energy can be absorbed by the electrons. However, you would need very high temperatures before this has a significant effect on the properties of the crystal. That high, that the crystal would have been molten already. Therefore, it is a good approximation to treat the electrons always as ‘cold’. How does a crystal mainly absorb thermal energy? By the nuclei of the atoms. Thermal energy is used to let the nuclei move (oscillate). When the Born-Oppenheimer approximation is used, the position of the nuclei is something that is given. You can in principle suggest an infinite number of positions for the nuclei, and calculate by ground state DFT the total energy for all sets of positions. This will give a potential energy landscape in a high-dimensional space. With this, we’re back at classical mechanics: you can study the oscillations of the crystal as a motion on a known potential energy surface. These collective motions of the nuclei are the so-called ‘phonons’ (every phonon represent a particular oscillation pattern). The fact that nuclei move, can significantly affect the properties of a crystal. The higher the temperature, the more they move — and the more the properties are affected. Making temperature-dependent predictions therefore boils down to studying phonons.
The present chapter provides you with the basics of phonons, it gives you a guide on how the full phonon spectrum can be calculated with an auxiliary code to your DFT code (which requires quite some computing resources), and it shows how so-called soft modes can be used to predict structural phase transitions.
Not all of the available material is fine-tuned, but it should be sufficiently clean to meet its purpose.