Using line group theory for the symmetry assignment of the phonons of single walled carbon nanotubes

Calculating the phonon dispersions of an arbitrary single walled carbon nanotube became cheap in the numerical sense by exploiting the screw axis symmetry. The eigenvectors of the dynamical matrix are the irreducible basis vectors of the representation of the symmetry group of the nanotubes: L(2n)n/mcm for achiral and Lqp22 for chiral tubes. We developed a numerical code which can solve the eigenvalue problem of the dynamical matrix produced by a density functional theory code (VASP), in the helical Brillouin zone. The code represents the symmetry elements of the line group in the helical unit cell. After decomposing the matrix representation we obtain one to one correspondence between the vibrational modes and the irreducible representations of the line group. Thus we obtain first principles level phonon dispersions accompanied by a full line group based symmetry assignment for all the phonon branches