Vibrational Analysis

In 1930 Wigner showed that symmetry offers efficient method for systematic classification of normal vibrational modes, which are valuable tool in many physical problems (e.g., Raman or infrared activity, vibronic (in)stability). This task has been completed for small molecules (with the help of the point groups), quasi-one-dimensional systems (using line groups), and layers (utilizing diperiodic groups). For three-dimensional crystals (space groups) only partial results exist in literature. Here we discuss normal modes of the systems with line group symmetry. The results are also applicable to three-dimensional crystals, since some of the line groups are subgroups of the relevant space groups. Classification of normal modes will be performed for all the orbits of the line groups. In fact, there are only several orbit types for each line group, and any system with a line group symmetry consists of such simple subsystems. Symmetry assignation of normal modes is achieved through reduction of the dynamical representation of the system onto its irreducible components. This representation is sum of the dynamical representations of the constituting orbits. Therefore, with decompositions of the dynamical representations of the orbits presented here, the result for any system can be easily obtained by summation over the orbits included in the considered system.