  We present a detailed study of the vibrational properties of Single Wall
Carbon Nanotubes (SWNTs). The phonon dispersions of SWNTs are strongly shaped
by the effects of electron-phonon coupling. We analyze the separate
contributions of curvature and confinement. Confinement plays a major role in
modifying SWNT phonons and is often more relevant than curvature. Due to their
one-dimensional character, metallic tubes are expected to undergo Peierls
distortions (PD) at T=0K. At finite temperature, PD are no longer present, but
phonons with atomic displacements similar to those of the PD are affected by
strong Kohn anomalies (KA). We investigate by Density Functional Theory (DFT)
KA and PD in metallic SWNTs with diameters up to 3 nm, in the electronic
temperature range from 4K to 3000 K. We then derive a set of simple formulas
accounting for all the DFT results. Finally, we prove that the static approach,
commonly used for the evaluation of phonon frequencies in solids, fails because
of the SWNTs reduced dimensionality. The correct description of KA in metallic
SWNTs can be obtained only by using a dynamical approach, beyond the adiabatic
Born-Oppenheimer approximation, by taking into account non-adiabatic
contributions. Dynamic effects induce significant changes in the occurrence and
shape of Kohn anomalies. We show that the SWNT Raman G peak can only be
interpreted considering the combined dynamic, curvature and confinement
effects. We assign the G+ and G- peaks of metallic SWNTs to TO
(circumferential) and LO (axial) modes, respectively, the opposite of
semiconducting SWNTs.
