Local normal modes and lattice dynamics

The calculation of phonon dispersion for crystalline solids with r atoms in a unit cell requires solving a 3r-dimensional eigenvalue problem. We propose a simplified approach to lattice dynamics which yields approximate analytical expressions and accurate numerical solutions to phonon dispersion without explicitly solving this eigenvalue problem. This is accomplished by a coordinate transformation to the normal modes of the isolated primitive unit cell, which is extended over the entire crystal by Fourier transformation, so each phonon branch is labelled by the irreducible representations of the symmetry group of the unit cell from which the atomic displacements can be readily identified from standard group theoretic methods. The resulting dynamical matrix is analyzed perturbatively, with the diagonal elements as the zeroth-order matrix and the off-diagonal elements as the perturbation. The zeroth-order matrix provides approximate analytical expressions for the phonon dispersions, the first-order terms vanish, with the higher-order terms converging to the exact solutions. We describe the application of this method to a one-dimensional diatomic chain, graphene, and hexagonal close-packed zirconium. In all cases, the zeroth-order solution provides reasonable approximations, while the second-order solutions already show the rapid convergence to the exact dispersion curves. This methodology provides insight into the lattice dynamics of crystals, molecular solids, and Jahn--Teller systems, while significantly reducing the computational cost. Similarities between our method and other techniques that use local basis sets for calculating electronic and vibrational properties of materials are discussed. We conclude by exploring extensions that widen the scope of our approach