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, vertical bar Psi((1))(2) (pu, t)>, prepared by a narrow band (ps) Raman pump pulse, E-pu(t), on the upper electronic e(2) potential energy surface (PES), with a second order wave packet, . resembles the zeroth order wave packet vertical bar Psi((0))(1)(t)>. on the lower PES spatially, but with a force on vertical bar Psi((1))(2) (pu, t)>. along the coordinates of the reporter modes due to displacements in the equilibrium position, so that . The observable FSRS Raman gain is related to the imaginary part of P-(3)(omega). The imaginary and real parts of P-(3)(omega) are related by the Kramers-Kronig relation. Hence, from the FSRS Raman gain, we can obtain the complex P-(3)(omega), whose Fourier transform then gives us the complex P-(3)(t) to analyze for (omega) over barj(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 C-12-H wagging mode at 216 fs and for the C-10-H wagging mode at 161 fs which are larger than for the C-11-H wagging mode at 127 fs, i.e., the C-11-H wagging mode arrives at its final frequency while the C-12-H and C-10-H wagging modes are still up-shifting to their final values, agreeing with the findings of Yan et al. (C) 2016 AIP Publishing LLC.