  In this paper, we provide a detailed expression of the vibrational potential
for the lattice dynamics of the single-wall carbon nanotubes (SWCNT) satisfying
the requirements of the exact rigid translational as well as rotational
symmetries, which is a nontrivial generalization of the valence force model for
the planar graphene sheet. With the model, the low frequency behavior of the
dispersion of the acoustic modes as well as the flexure mode can be precisely
calculated. Based upon a comprehensive chiral symmetry analysis, the calculated
mode frequencies (including all the Raman and infrared active modes),
velocities of acoustic modes and the polarization vectors are systematically
fitted in terms of the chiral angle and radius, where the restrictions of
various symmetry operations of the SWCNT are fulfilled.
