A phonon is a collective excitation of a set of atoms in condensed matter. These excitations can be decomposed in different modes each being associated with an energy that corresponds to the frequency of vibration. The different energies associated with each vibrational mode constitute the phonon vibrational spectra (or phonon band structure). The vibrational spectra of materials plays an important role in physical phenomena such as thermal conductivity, superconductivity, ferroelectricity and carrier thermalization.
There are different methods to calculate the vibrational spectra from first-principles using the density functional theory formalism (DFT). It can be obtained from the Fourier transform of the trajectories of the atoms on a molecular dynamics run, from finite-differences of the total energy with respect to atomic displacements or directly from density functional perturbation theory (DFPT). The latter method is the one used in the calculations on the Materials project page.
In the density functional perturbation theory formalism the derivatives of the total energy with respect to a perturbation are directly obtained from the self-consistency loop  For a generic point q in the Brillouin zone the phonon frequencies and eigenvectors are obtained by solving of the generalized eigenvalue problem
where labels the atoms in the cell, and are cartesian coordinates and are the interatomic force constants in reciprocal space, which are related to the second derivatives of the energy with respect to atomic displacements. These values have been obtained by performing a Fourier interpolation of those calculated on a regular grid of q-points obtained with DFPT.
The vibrational density of states is obtained from the integration over the full Brillouin zone
where is the number of atoms per unit cell and is the number of unit cells. The expressions for the Helmholtz free energy , the phonon contribution to the internal energy , the constant-volume specific heat and the entropy can be obtained in the harmonic approximation 
where is the Boltzmann constant and is the largest phonon frequency.
The PBEsol  semilocal generalized gradient approximation exchange-correlation functional (XC) was used for the calculations. This functional was proven to provide accurate phonon frequencies compared to experimental data . The pseudopotentials are norm-conserving  and taken from the pseudopotentials table Pseudo-dojo version 0.3 .
The plane wave cutoff is chosen based on the hardest element for each compound, according to the values suggested in the Pseudo-dojo table. The Brillouin zone has been sampled using equivalent k-point and q-point grids that respect the symmetries of the crystal with a density of approximately 1500 points per reciprocal atom and the q-point grid is always Γ-centered .
All the structures are relaxed with strict convergence criteria, i.e. until all the forces on the atoms are below 10-6 Ha/Bohr and the stresses are below 10-4 Ha/Bohr3.
The primitive cells and the band structures are defined according to the conventions of Setyawan and Curtarolo .
Guido Petretto, Shyam Dwaraknath, Henrique P. C. Miranda, Donald Winston, Matteo Giantomassi, Michiel J. van Setten, Xavier Gonze, Kristin A. Persson, Geoffroy Hautier, Gian-Marco Rignanese, High-throughput density functional perturbation theory phonons for inorganic materials, Scientific Data, 5, 180065 (2018) doi:10.1038/sdata.2018.65.
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