2 0 . 69 eV for C$_{60}$ and 0 . 80 eV for the nanotube, and is thus off-resonance in both cases. For C$_{60}$, seven of the ten Raman-active modes were appreciably excited: modes A$_{g}$ (1), A$_{g}$ (2), H$_{g}$ (1), H$_{g}$ (4), H$_{g}$ (5), H$_{g}$ (6) and H$_{g}$ (8), with periods of 59 . 7 fs, 20 . 6 fs, 125 fs, 38 . 5 fs, 27 . 0 fs, 23 . 0 fs, and 19 . 0 fs respectively. For the (3 , 3) nanotube only an E$_{g}$ mode with a period of 182 fs was observed, and this is consistent with the experimental difficulty of observing the radial breathing mode in the Raman spectrum [31]. As in Ref. [12], it is natural to characterize the strength of the vibrational response of a specific mode by its max- imum kinetic energy K max k . In the harmonic approxi- mation, after completion of the laser pulse, the veloc- ity ˙ Q$_{k}$ ( t ) is proportional to cos ( ω$_{k}$t + δ ), so K$_{k}$ is pro- portional to cos $^{2}$( ω$_{k}$t + δ ) = [1 + cos (2 ω$_{k}$t + 2 δ )] / 2. A numerical Fourier transform of the total kinetic energy therefore shows a peak at 2 ω$_{k}$ , with a strength propor- tional to the response of the normal mode with angular frequency ω$_{k}$ . Figure 1 shows our results for the Fourier transform of the total kinetic energy for C$_{60}$ following a 12 fs, 0 . 69 eV, 5 . 0 V/nm pulse. The total energy ( ˙ Q 2 k + ω 2 $_{k}$Q 2 k ) / 2 in vibrational mode k is therefore E$_{k}$ is, of course, equivalent to the maximum kinetic en- ergy K max . (1) Time-dependent Schr¨ odinger equation in a nonorthogonal basis: k The maximum response is then given by the extremum of the function in brackets, which occurs at r$_{k}$ ≈ 0 . 8375, or 2 τ ≈ 0 . 8375 T$_{k}$ . We have tested this prediction by performing inde- pendent supercomputer simulations for C$_{60}$ and a small carbon nanotube, using the density-functional-based approach of Frauenheim and co-workers [34, 35], to- gether with semiclassical electron-radiation-ion dynamics (SERID), which is defined by the following equations [36]: E$_{k}$ = 1 32 [ α ′ k ω$_{k}$ sin ( r$_{k}$π ) 1 − r 2 k ] 2 E 4 0 , t ≥ 2 τ . (7) i ℏ ∂ ψ ( t ) /∂t = S − 1 · H · ψ ( t ) . (8) With 60 atoms and a minimal basis set, the matrices are 240 × 240. A time step of 50 attoseconds was used. The simulation time after completion of the pulse was 2000 fs for C$_{60}$ and 1000 fs for the nanotube. (2) Ehrenfest’s theorem (in a nonorthogonal basis): M d $^{2}$X dt 2 = − 1 2 ∑ n ψ † n $_{·}$( ∂ H ∂X − i ℏ ∂ S ∂X ∂ ∂t )$_{·}$ ψ$_{n}$ + h.c. − ∂U$_{rep}$ ∂X (9) where X is any nuclear coordinate. The Hamiltonian matrix H , overlap matrix S , and effective ion-ion repul- sion U$_{rep}$ were determined by the methods and results of Refs. [34] and [35] and later work by this group. (3) Coupling of electrons to the radiation field through the time-dependent Peierls substitution where q = − e . A is the vector potential, which in the present simulations was taken to have the form H ( ℓ $^{′}$, ℓ ) = H$_{0}$ ( ℓ $^{′}$, ℓ ) e iq A ( t ) · ( X $^{′}$− X ) / ℏ c (10) A ( t ) = A$_{0}$ sin ( πt/ 2 τ ) cos( ωt ) , 0 ≤ t ≤ 2 τ (11) with ω ≫ π/τ , so that E ( t ) ≈ ω A$_{0}$ sin ( πt/ 2 τ ) sin( ωt ), although Eq. (11) was actually used. The polarization vector was taken to lie along the x -axis, with the z -axis pointing down the axis of a nanotube. Within the present DFT-based model, the energy gap for electronic excitations is 1 . 80 eV for C$_{60}$ and 1 . 56 eV for a (3 , 3) nanotube with a periodicity length of 5 unit cells. (This model nanotube, with only 60 atoms, has a substantial gap because small wavenumbers k$_{z}$ are not al- lowed. An infinitely long (3,3) nanotube would be metal- lic [30].) The laser pulse photon energy was chosen to be Figure 2 shows that the detailed simulations agree with Eq. (7) regarding the dependence of the maximum kinetic energy K max k of mode k on the field amplitude E 0 (for the range of amplitudes considered here) when the pulse duration is fixed: K max k ∝ E 4 $_{0}$. When E 0 is fixed, on the other hand, K max k is predicted to be proportional to the square of the function involving r$_{k}$ in Eq. (7), and is maximal when r$_{k}$ ≈ 0 . 84. Figure 3 shows, for example, that the H$_{g}$ (8) mode in C$_{60}$ domi- FIG. 1: Numerical Fourier transform of the total kinetic en- ergy of C 60 after being subjected to a laser pulse with a FWHM duration τ of 12 fs, photon energy ℏ ω of 0 . 69 eV, and electric field amplitude E$_{0}$ of 5 . 0 V/nm. Seven Raman-active modes are clearly observed, each with a frequency ω$_{k}$/ 2 π which is half the frequency exhibited in the figure by the cor- responding kinetic energy K$_{k}$ . 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