  The high-frequency Raman-active phonon modes of metallic single-walled carbon
nanotubes (SWNTs) are thought to be characterized by Kohn anomalies (KAs),
which are expected to be modified by the doping-induced tuning of the Fermi
energy level $\epsilon_F$, obtained through the intercalation of SWNTs with
alkali atoms or by the application of a gate potential. We present a
Density-Functional Theory (DFT) study of the phonon properties of a (9,9)
metallic SWNT as a function of electronic doping. For such study, we use, as in
standard DFT calculations of vibrational properties, the Born-Oppenheimer (BO)
approximation. We also develop an analytical model capable of reproducing and
interpreting our DFT results. Both DFT calculations and this model predict, for
increasing doping levels, a series of EPC-induced KAs in the vibrational mode
parallel to the tube axis at the $\mathbf\Gamma$ point of the Brillouin zone,
usually indicated in Raman spectroscopy as the $G^-$ peak. Such KAs would arise
each time a new conduction band is populated. However, we show that they are an
artifact of the BO approximation. The inclusion of non-adiabatic (NA) effects
dramatically affects the results, predicting KAs at $\mathbf\Gamma$ only when
$\epsilon_F$ is close to a band crossing $E_{X}$. For each band crossing a
double KA occurs for $\epsilon_F=E_{X}\pm \hbar\omega/2$, where $\hbar\omega$
is the phonon energy. In particular, for a 1.2 $nm$ metallic nanotube, we
predict a KA to occur in the so-called $G^-$ peak at a doping level of about
$N_{el}/C=\pm 0.0015$ atom ($\epsilon_F\approx \pm 0.1 ~eV$). Furthermore, we
predict that the Raman linewidth of the $G^-$ peak significantly decreases for
$|\epsilon_F| \geq \hbar\omega/2$.
