Computation and Theoretical Analysis of Vibrational Modes in Linear Carbon Clusters Using a Lie Algebraic Framework

Linear carbon clusters Cn represent one-dimensional, highly reactive assemblies of carbon atoms that serve as fundamental structural units for a wide range of carbon nanomaterials, including graphene and carbon nanotubes. Owing to their unique structural and electronic properties, the investigation of their vibrational characteristics—particularly the fundamental vibrational frequencies—plays a crucial role in understanding their stability, bonding, and chemical reactivity. These vibrational properties are commonly explored through experimental techniques such as Raman and infrared (IR) spectroscopy, as well as through theoretical frameworks. In the present study, the Lie Algebraic Model (LAM) is employed to investigate the stretching and bending vibrational modes of linear carbon clusters C3, C4, C5, C6, C7, and C8. Within this algebraic framework, a model Hamiltonian is constructed that explicitly incorporates the pseudo-Cnv ​dynamical symmetry associated with these molecular systems. The formulation utilizes a compact set of algebraic parameters capable of effectively describing the C–C stretching and bending dynamics. Furthermore, the invariant Majorana and Casimir operators are systematically derived to characterize the vibrational structure of the clusters. The results obtained from the algebraic treatment exhibit a high degree of consistency with available experimental observations and other theoretical calculations, demonstrating the reliability and effectiveness of the Lie algebraic approach in describing the vibrational behaviour of linear carbon clusters.