ALBERT

Description: 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, Ψ 2 ( 1 ) ( p u , 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, Ψ 1 ( 2 ) ( p r ∗ , p u , t ) , that is prepared on the lower electronic e 1 PES by a broadband (fs) probe pulse, E pr ( t ), acting on the first-order wave packet. In off-resonant FSRS, Ψ 2 ( 1 ) ( p u , t ) resembles the zeroth order wave packet Ψ 1 ( 0 ) ( t ) on the lower PES spatially, but with a force on Ψ 2 ( 1 ) ( p u , t ) along the coordinates of the reporter modes due to displacements in the equilibrium position, so that Ψ 1 ( 2 ) ( p r ∗ , p u , 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, ω ̄ j ( t ) , of the reporter modes j along the trajectory of Ψ 1 ( 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 ω ̄ j ( 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. [Biochemistry 43 , 10867 (2004)].