  A phonon frequency shift of the radial breathing mode for metallic single
wall carbon nanotubes is predicted as a function of Fermi energy. Armchair
nanotubes do not show any frequency shift while zigzag nanotubes exhibit phonon
softening, but this softening is not associated with the broadening. This
chirality dependence originates from a curvature-induced energy gap and a
special electron-phonon coupling mechanism for radial breathing modes. Because
of the particle-hole symmetry, only the off-site deformation potential
contributes to the frequency shift. On the other hand, the on-site potential
contributes to the Raman intensity, and the radial breathing mode intensity is
stronger than that of the $G$ band. The relationship between the chirality
dependence of the frequency shift of the radial breathing mode and the $\Gamma$
point optical phonon frequency shift is discussed.
