Schematic illustrating the use of the Im F model to describe a reaction rate in the deep-tunneling regime. The potential energy (black curve) is distorted (red curve) to give a well that accommodates a series of long-lived resonances (blue lines). The decay of the resonances through the barrier gives an approximation to the rate.
RP geometries at the instanton saddle point for (a) the asymmetric Eckart barrier at and (b) the symmetric Eckart barrier at . The green circles are the positions of the centroids. The arrows represent the dominant contributions to the unstable normal mode, which is a concerted centroid-shift and overall stretch in (a) and just a centroid-shift in (b). Panel (c) shows the RP geometries at various points along the minimum energy path for the asymmetric Eckart barrier at
Approximate form of the optimal free-energy dividing surface for a (one-dimensional) asymmetric barrier. The surface depends on the free-RP normal modes and and is cone-shaped, owing to the cyclic permutation symmetry of the beads within the polymer. Also shown are the directions of the normal modes and at one particular instanton geometry .
(a) Variation with of the coefficient for the asymmetric (dashed green line) and symmetric (blue solid line) Eckart barriers. (b) Variation with of the unstable frequency for the asymmetric (dashed green line) and symmetric (blue solid line) barriers. Also shown is the variation with of the centroid-component of for the asymmetric barrier (dotted red line).
Diagram summarizing the relations between the various RPMD and instanton rates. All abbreviations and formulae are defined in the text. Each arrow represents an approximation [e.g., the ORP-TST rate is obtained from the -inst rate by making the approximation ]. Note that this diagram applies only to rates calculated in the deep-tunneling regime.
Tunneling correction factors for the asymmetric Eckart barrier. The ORP-TST results were obtained using different approximations to the optimal dividing surface , in which was allowed to depend on (a) just the centroid coordinate , (b) and as in Eq. (33), and (c) all degrees of freedom. The -ORP-TST results were obtained by multiplying the ORP-TST results in column (b) by the factor to give the free-energy instanton rate [see Eq. (27)].
As Table I, for the symmetric Eckart barrier. The temperature is also given in kelvin to facilitate comparison with previous work (Refs. 10, 35, and 36).
Parameters characterizing the optimal dividing surface for the asymmetric Eckart barrier. is the shift along the axis that maximizes the free energy, and is the resulting percentage change in the tunneling correction factor . is the pitch of the cone (see Fig. 3).
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