Schematic illustration of the RPMD method for calculating a reaction rate. At time zero, the first bead of the ring-polymer necklace is pinned to the transition state dividing surface (the vertical line) by the function in Eq. (25), and this bead contributes a velocity factor to the flux-side correlation function . The polymer then evolves under the classical equations of motion in Eqs. (19) and (20), and contributes a side factor to the correlation function at time .
Convergence of the classical transmission coefficient of the system-bath model with respect to the number of bath modes retained in the discretization of Eqs. (31) and (32), as a function of the reduced coupling strength at 200 and 300 K. The long-dashed, dashed, dotted, and solid lines correspond to 3, 6, 9, and 12 bath modes, respectively. Note that the dotted line is almost entirely obscured by the solid line at both temperatures.
Transmission coefficients for the system-bath model as a function of the reduced coupling strength at 300 K. The filled circles are the exact real-time path integral results of Topaler and Makri (Ref. 26). The long-dashed, dashed, dotted, and solid lines are the RPMD results with 1, 2, 4, and 16 beads, respectively. The first of these lines is the purely classical result shown in Fig. 2, and the last is fully converged with respect to the number of beads.
As in Fig. 3, but at a temperature of 200 K and with 32 beads in the converged RPMD calculation (solid line).
Time-dependent transmission coefficients from the RPMD calculation at 300 K in Fig. 3 for two different system-bath coupling strengths: (a) (high friction); (b) (low friction).
(a) An Arrhenius plot of the rate coefficient for the Eckart barrier. The long-dashed line is the classical result, the solid line the converged RPMD result, and the filled circles indicate the exact quantum-mechanical rate. (b) Percentage error in the RPMD result over the same temperature range as in (a).
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