title: Fourier projection algorithm for calculating phonon dispersion relations of crystals
author: Wang, Shaoqing
description: We propose a Fourier projection algorithm and demonstrate its superiority for calculation of the phonon dispersion relations of 2D and 3D crystals based on the atomic coordinate trajectories from supercell molecular dynamics/first-principle molecular dynamics simulations. The lattice vibration states are described in a six-dimensional phase space composed of lattice and wave vectors. Phonon dispersion information in the first Brillouin zone sampled with a k-mesh grid of N×N×N is generated by Fourier projection of the lattice vibration frequencies in the supercell by N×N×N primitive cells, the dispersion relations along high-symmetry paths are retrieved therefrom. Our algorithm is physically intuitive, computationally convenient, and universally applicable. As examples of application, the successful dispersion relation calculations of 2D graphene, 3D BCC-Fe and FCC-Cu are presented. This algorithm shows the correspondence between real and reciprocal lattices, that is, the spatial information in the states of the lattice vibration were preserved. The high-symmetry paths of the first Brillouin zone in the reciprocal lattice primitive cells of the hexagonal, BCC, and FCC lattice were also given in detail. In combination with first-principles molecular dynamics, as long as the interatomic force is accurately known, reliable and accurate dispersion relation can be achieved by this Fourier projection algorithm.