Vibration spectra of benzene-like models with Hooke's law interactions

TL;DR

This paper analyzes the harmonic vibrations of benzene-like nanosystems modeled with Hooke's law, revealing exact spectral properties, mode hybridization, and generalizations to related molecules.

Contribution

It provides an exact analytical solution for the vibration spectra of benzene-like models with complex interactions, including generalizations to substituted benzene molecules.

Findings

Vibration frequencies follow the dispersion law of the lattice model.

Spectral branches are affected by hybridization and crossover effects.

All normal modes and frequencies are explicitly derived and depend on elastic constants.

Abstract

The harmonic oscillations of a spring-ball model of benzene-like nanosystems with Hooke's law interactions between nearest, second, and third neighbors are explored. We show that in the cylindrical coordinates the dynamics of this cyclic hexagonal system is described by the Lagrange equations similar to those of the one-dimensional two-component crystal model. We expose that the vibration frequencies of the hexagonal model lie on the branches of the dispersion law of the associated lattice model, and their positions are determined by the cyclic Born-Von Karman condition. The hexagonal model is generalized to one describing the benzene molecule and the fully deuterated and halogenated benzenes. The effect of hybridization of vibration modes and the pushing apart of spectral branches in the crossover situation is revealed. All the discrete frequency spectrum and normal modes of oscillations and their explicit dependencies on all the constants of elastic interactions are exactly found.