We compute, from first-principles, the frequency of the phonons associated to the Raman G-bands in a graphene monolayer and in metallic nanotubes as a function of the charge doping. In both cases, the frequency displays an important measurable shift in the range of doping reached experimentally. As a consequence, Raman spectroscopy can be used as a direct probe of the doping of these systems. Calculations are done using (i) the adiabatic Born-Oppenheimer approximation and (ii) time-dependent perturbation theory to explore dynamic effects beyond this approximation. The two approaches provide very different results. The frequency shift of the G-band in graphene and of the G- in metallic tubes represent remarkable failures of the adiabatic Born-Oppenheimer approximation.
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