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The crystal lattice is never rigid. Due to temperature, external fields or pressure, the nuclei vibrate, the lattice distorts, and instabilities can induce phase transitions. We review the basic concepts of density-functional perturbation theory, a computational method especially suited to determine from first-principles the microscopic parameters governing such behaviour. Then, we present the additional formalism leading to the following properties of minerals: the infra-red and Raman spectra; the prediction of (meta)stability or instability of a crystalline phase, based on the phonon spectrum; the computation of thermodynamics quantities like the free energy, entropy, specific heat; the atomic temperature factors. For each property, examples are given. When appropriate, we mention the computation of related properties, like dielectric tensor and Born effective charges that are needed to get infra-red spectra. Finally, we discuss briefly, on one hand, other applications of the density-functional perturbation theory, and, on the other hand, an alternative technique, the finite-difference computation of dynamical matrices.

First-principle studies of the lattice dynamics of crystals, and related properties