The full symmetry groups for all single-and multi-wall carbon nanotubes are found. As for the single-wall tubes, the symmetries form nonabelian nonsymorphic line groups, enlarging the groups reported in literature. In the multi-wall case, any type of the line and the axial point groups can be obtained, depending on single-wall constituents and their relative position. Several other consequences are discussed: quantum numbers and related selection rules, electronic and phonon bands, and their degeneracy, application to tensor properties. 61.46.+w,02.20

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Various properties of the energy band structures (electronic, phonon, etc.), including systematic band degeneracy, sticking and extremes, following from the full line group symmetry of the singlewall carbon nanotubes are established. The complete set of quantum numbers consists of quasi momenta (angular and linear or helical) and parities with respect to the z-reversal symmetries and, for achiral tubes, the vertical plane. The assignation of the electronic bands is performed, and the generalized Bloch symmetry adapted eigen functions are derived. The most important physical tensors are characterized by the same set of quantum numbers. All this enables application of the presented exhaustive selection rules. The results are discussed by some examples, e.g. allowed interband transitions, conductivity, Raman tensor, etc.

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The symmetry‐based study of MS2 (M=Mo, W) single‐wall nanotubes (SWNTs) is reviewed. First, the structure and symmetry of MS2 NTs is determined. Then, conserved quantum numbers and general forms of potentials are derived. The valence force‐field method implemented into the POLSym code is used to calculate phonon dispersions. Phonons characterized by a zero angular‐momentum quantum number are studied in detail. The functional dependence of the frequency of rigid layer modes on NT diameter and chirality are found, and Raman‐ and infrared‐active modes are singled out. Electronic band structure calculations are performed by the symmetry‐based density functional tight‐binding (DFTB) method. Changes in the band‐gap type and size with NT chirality and diameter are evaluated. Optical absorption spectra of individual NTs are calculated using DFTB wave functions for exact transition matrix element calculations. Diffraction patterns of MS2 are predicted and NT characterization by different diffraction methods is discussed.

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