Local Vibrational Mode Analysis of Phonon Dispersion Relations in Crystals

We present a general framework for performing local vibrational mode analysis of vibrations in crystalline materials at arbitrary wavevectors throughout the Brillouin zone. The approach enables phonon dispersion relations to be interpreted in terms of chemically meaningful interatomic interactions and structural motifs, providing direct insight into the microscopic origins of the phonon behavior in periodic systems. We demonstrate the methodology for representative one-, two-, and three-dimensional materials including polymeric chains, graphene, and prototypical rock-salt and perovskite crystals. Across these systems, the analysis reveals how specific bonding patterns and structural features govern phonon dispersion relations. This framework provides a quantitative tool for the chemically intuitive analysis of phonon spectra and offers a pathway toward the rational design of phonon-dependent properties in crystalline materials.