For molecules and materials responding to femtosecond-scale optical laser pulses, we predict maximum relative excitation of a Raman-active vibrational mode with period T when the pulse has a full-width-at-halfmaximum duration Ϸ 0.42T. This result follows from a general analytical model, and is precisely confirmed by detailed density-functional-based dynamical simulations for C 60 and a carbon nanotube, which include anharmonicity, nonlinearity, no assumptions about the polarizability tensor, and no averaging over rapid oscillations within the pulse. The mode specificity is, of course, best at low temperature and for pulses that are electronically off-resonance, and the energy deposited in any mode is proportional to the fourth power of the electric field. For a quarter century there has been considerable interest in optimizing the vibrational response of molecules and materials to ultrafast laser pulses. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] This problem is directly relevant to the broader issue of coherent control in physical, chemical, [16][17][18][19] and biological 20-23 systems.Here we consider excitation via impulsive stimulated Raman scattering and related techniques using femtosecondscale optical pulses. We find that the optimum full-width-athalf-maximum ͑FWHM͒ pulse duration for exciting a specific vibrational mode with angular frequency 2 / T is given by Ϸ 0.42T. Our prediction results from a general analytical model, and is precisely confirmed by completely independent density-functional-based simulations for C 60 24-28 and a small carbon nanotube. [29][30][31] Unlike the model, these simulations include anharmonic effects in the vibrations, nonlinear effects in the response to the applied field and no simplifying assumptions about the electronic polarizability tensor.Our general model consists of the following: ͑1͒ the electric field has the formwith ӷ / and ប off resonance. Since the oscillations of sin 2 ͑ t + ␦͒ will average out to 1/2 over a period that is short compared to the response time of the vibrating nuclei, we will actually replace the square of Eq. ͑1͒ by the envelope functionwith Ē 0 2 = E 0 2 / 2. This form has the following nice features: 32 ͑i͒ the duration is finite and need not be truncated. ͑ii͒ The FWHM duration is exactly half the full duration 2 . ͑iii͒ A plot reveals that it closely resembles a Gaussian. ͑iv͒ The slope is zero at beginning and end. ͑2͒ The initial conditions Q k ͑0͒ = Q k ͑0͒ = 0 are imposed on the normal-mode coordinates Q k ͑t͒. This approximation is valid for the expectation value below Eq. ͑4͒ at low temperature, or an ensemble average at higher temperature, in the linearized Eq. ͑3͒. ͑3͒ The equation of motion is given by the standard ͑Placzek͒ model for Raman-active modes 8,21,33where the anharmonic and damping terms ͑in the normalmode coordinates Q k ͒, nonlinear terms ͓in the electric field E͑t͔͒, and off-diagonal terms ͑in the polarizability tensor ␣͒ have been neglected, with the electric dipole moment given by tot = + ind , ind = ␣ · ...