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(b) varying τ at constant E 0 should also be reasonably consistent. Table I shows the results for all of the promi- nent Raman-active modes. The two procedures do lead to generally consistent values, with the differences for the A$_{g}$ (2), H$_{g}$ (6), and H$_{g}$ (8) modes presumably arising from their short periods, so that the approximation of averag- ing out the ω oscillations in Eq. (2) is not valid. However, the prediction regarding optimum pulse duration works even for these modes, as can be seen in Fig. 3. 

It has long been suspected that there might be an op- timum pulse duration. For example, in Ref. [12] the pulse envelope was assumed to have a Gaussian form: |E ( t ) | 2 = A exp ( − 2 t $^{2}$/ ¯ τ $^{2}$) cos 2 ( ωt ) (with a slight change in notation). Simulations were then performed with a model that essentially lies between the analytical model of Eqs. (2)-(3) and our much more extensive density- functional-based simulations, defined by Eqs. (8)-(11). It was found that the optimum value of ¯ τ was about T/ 3 . 4. The corresponding FWHM duration then has an optimum value τ = √$_{2 ln 2 ¯}$ τ ≈ 0 . 35 T . In both our general model and DFT-based simulations, on the other hand, the optimal FWHM pulse duration is found to be given by τ ≈ 0 . 42 T . Both results seem to be consis- tent with the existing experiments [24–28], and both are larger than the na¨ıve prediction of τ = 0 . 25 T . In our result, the reason appears to be the phase lag apparent in Eq. (5). 

It should be emphasized that our result of τ ≈ 0 . 42 T is for the maximum response of this mode when the in- tensity of the laser pulse is kept fixed while the duration is varied. On the other hand, if the fluence is instead held 

fixed (while τ is varied), Eqs. (5) and (7) are changed to 

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and 

where 

is the energy in the pulse. If W is held constant, the maximum in Eq. (13) is achieved as r$_{k}$ → 0: 

so that 

Then r$_{k}$ = 0 . 84 gives E$_{k}$ (0 . 42 T ) /E$_{k}$ (0) = 0 . 385. How- ever, Eq. (15) holds for all modes, so there is no relative enhancement of any preferred mode. 

It should also be emphasized that the success of the simple analytical model of Eq. (3) results from the fact that the additional effects described below Eq. (3) are not very important for the pulse intensities, time dura- tions, and systems considered here. The various effects omitted in the model will lead to richer (or messier) be- havior in regimes where nonlinear effects, anharmonic- ity, damping, etc. become comparable in importance to the lowest-order effects included in the model. In future studies with density-functional-based simulations based on Eqs. (8)-(11) it may be interesting to sort out the influence of the higher-order effects. 

Finally, as emphasized below Eq. (1), the present re- sults are for ℏ ω off resonance, in contrast to those of e.g. Refs. [11] and [14]. In particular, Smith and Cina considered pulses whose central frequencies are near res- onance with an electronic transition, in a model with two electronic levels and one vibrational degree of free- dom. They found that the momentum increment (for the nuclei) falls off less rapidly with increasing offset (from resonance) than does the population loss (from the elec- tronic ground state), and they presented a rather com- plex strategy for optimizing the various parameters for these “preresonant” pulses. The work of this group and others is thus complementary to that of the present pa- per. 

In summary, we find remarkable agreement between the general model of Eqs. (2)-(3) and the detailed DFT- based simulations based on Eqs. (8)-(11) for C$_{60}$ and a carbon nanotube. At fixed pulse intensity, both of these 

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