^{1}and Stuart C. Althorpe

^{1}

### Abstract

We demonstrate that the ring-polymer molecular dynamics (RPMD) method is equivalent to an automated and approximate implementation of the “Im F” version of semiclassical instanton theory when used to calculate reaction rates in the deep-tunneling regime. This explains why the RPMD method is often reliable in this regime and also shows how it can be systematically improved. The geometry of the beads at the transition state on the ring-polymer potential surface describes a finite-difference approximation to the “instanton” trajectory (a periodic orbit in imaginary time on the inverted potential surface). The deep-tunneling RPMD rate is an approximation to the rate obtained by applying classical transition-state theory(TST) in ring-polymer phase-space using the optimal dividing surface; this TST rate is in turn an approximation to a free-energy version of the Im F instanton rate. The optimal dividing surface is in general a function of several modes of the ring polymer, which explains why centroid-based quantum-TSTs break down at low temperatures for asymmetric reaction barriers. Numerical tests on one-dimensional models show that the RPMD rate tends to overestimate deep-tunneling rates for asymmetric barriers and underestimate them for symmetric barriers, and we explain that this is likely to be a general trend. The ability of the RPMD method to give a dividing-surface-independent rate in the deep-tunneling regime is shown to be a consequence of setting the bead-masses equal to the physical mass.

It is a pleasure to thank David E. Manolopoulos, William H. Miller, David Chandler, and Thomas Stecher for reading through and commenting on the article. J.O.R. acknowledges a Doctoral Training Account Ph.D. studentship from the UK Engineering and Physical Sciences Research Council.

I. INTRODUCTION

II. RPMD RATE-THEORY

III. CONNECTION WITH SEMICLASSICAL INSTANTON THEORY

A. The Im F approximation

B. Saddle points on the RP potential surface

C. RP form of the Im F instanton rate

D. Connection with the harmonic RP-TST rate

E. Connection with the RP-TST rate

F. Connection with the full RPMD rate

IV. NUMERICAL COMPARISONS OF RPMD, INSTANTON, AND QUANTUM RATES

A. Locating the optimal dividing surface using

B. More approximate expressions for

C. Computation of the coefficient

D. Predicting the magnitude of

V. MULTIDIMENSIONAL GENERALIZATION

VI. CONCLUSIONS

### Key Topics

- Polymers
- 36.0
- Free energy
- 23.0
- Normal modes
- 15.0
- Transition state theory
- 15.0
- Semiclassical theories
- 12.0

## Figures

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.

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

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 .

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).

(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.

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.

## Tables

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)].

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).

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).

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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