Time-dependent wave packet averaged vibrational frequencies from femtosecond stimulated Raman spectra

Femtosecond stimulated Raman spectroscopy (FSRS) on the Stokes side arises from a third order polarization, P(3)(t), which is given by an overlap of a first order wave packet, |Ψ2(1)(pu,t)>, prepared by a narrow band (ps) Raman pump pulse, Epu(t), on the upper electronic e2 potential energy surface (PES), with a second order wave packet, <Ψ1(2)(pr(∗),pu,t)|, that is prepared on the lower electronic e1 PES by a broadband (fs) probe pulse, Epr(t), acting on the first-order wave packet. In off-resonant |FSRS, Ψ2(1)(pu,t)> resembles the zeroth order wave packet |Ψ1(0)(t)> on the lower PES spatially, but with a force on |Ψ2(1)(pu,t)> along the coordinates of the reporter modes due to displacements in the equilibrium position, so that <Ψ1(2)(pr(∗),pu,t)| will oscillate along those coordinates thus giving rise to similar oscillations in P(3)(t) with the frequencies of the reporter modes. So, by recovering P(3)(t) from the FSRS spectrum, we are able to deduce information on the time-dependent quantum-mechanical wave packet averaged frequencies, ω̄j(t), of the reporter modes j along the trajectory of |Ψ1 (0)(t)>. The observable FSRS Raman gain is related to the imaginary part of P(3)(ω). The imaginary and real parts of P(3)(ω) are related by the Kramers-Kronig relation. Hence, from the FSRS Raman gain, we can obtain the complex P(3)(ω), whose Fourier transform then gives us the complex P(3)(t) to analyze for ω̄j(t). We apply the theory, first, to a two-dimensional model system with one conformational mode of low frequency and one reporter vibrational mode of higher frequency with good results, and then we apply it to the time-resolved FSRS spectra of the cis-trans isomerization of retinal in rhodopsin [P. Kukura et al., Science 310, 1006 (2005)]. We obtain the vibrational frequency up-shift time constants for the C12-H wagging mode at 216 fs and for the C10-H wagging mode at 161 fs which are larger than for the C11-H wagging mode at 127 fs, i.e., the C11-H wagging mode arrives at its final frequency while the C12-H and C10-H wagging modes are still up-shifting to their final values, agreeing with the findings of Yan et al. [Biochemistry 43, 10867 (2004)].