Arrhenius plot of the rate constant for a 1D Eckart barrier (with parameters correspondingly approximately to the reaction). The solid line is the exact quantum value and the dashed line that given by classical mechanics. The circles are values given by the full SC-IVR and the squares its linearized approximation (LSC-IVR).
Transmission coefficient, , for the isomerization rate in a double well potential coupled to a harmonic bath ( is the rate constant and the 1D classical rate), as a function of coupling to the bath. The solid lines are the results of the linearized approximation to the SC-IVR (LSC-IVR), and the solid points the accurate quantum values of Topaler and Makri, Ref. 61 (a) ; (b) .
Probability distribution of the vibrational coordinate of (modeled as a 1D Morse oscillator) at time ( vibrational periods after excitation). The dashed line and solid line (almost indistinguishable) are the exact quantum and forward-backward SC-IVR (FB-IVR) results, respectively, and the dash-dot line the results of linearized approximation to the SC-IVR (LSC-IVR).
Same quantity as Fig. 3, but with the addition of a harmonic bath that is coupled to the Morse oscillator ; is the temperature of the harmonic bath. The solid line is the result of the FB-IVR, and the dash-dot line that of the LSC-IVR.
Schematic depiction of the forward and backward trajectories in the “exact” forward-backward (EFB-IVR) approach described in Sec. III C. The trajectory begins at with initial conditions and is evolved to time , where its coordinates and momenta are ; here the momentum is changed to and the trajectory evolved backward in time to , where its values are .
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