^{1}and David E. Manolopoulos

^{1,a)}

### Abstract

We further develop the ring polymermolecular dynamics (RPMD) method for calculating chemical reaction rates [I. R. Craig and D. E. Manolopoulos, J. Chem. Phys.122, 084106 (2005)]. We begin by showing how the rate coefficient we obtained before can be calculated in a more efficient way by considering the side functions of the ring-polymer centroids, rather than averaging over the side functions of the individual ring-polymer beads. This has two distinct advantages. First, the statistics of the phase-space average over the ring-polymer coordinates and momenta are greatly improved. Second, the resulting flux-side correlation function converges to its long-time limit much more rapidly. Indeed the short-time limit of this flux-side correlation function already provides a “quantum transition state theory” approximation to the final rate coefficient. In cases where transition state recrossing effects are negligible, and the transition state dividing surface is put in the right place, the RPMD rate is therefore obtained almost instantly. We then go on to show that the long-time limit of the new flux-side correlation function, and hence the fully converged RPMD reaction rate, is rigorously independent of the choice of the transition state dividing surface. This is especially significant because the optimum dividing surface can often be very difficult to determine for reactions in complex chemical systems.

We would like to thank David Chandler for suggesting Eq. (22) and Bill Miller for some stimulating discussions. This work was supported by the U.S. Office of Naval Research under Contract No. N000140510460.

I. INTRODUCTION

II. THEORY AND RESULTS

A. Reaction rate theory

B. Ring polymermolecular dynamics

C. Centroid correlation functions

D. A symmetric Eckart barrier

E. The short-time limit

F. The long-time limit

G. An asymmetric Eckart barrier

H. Multidimensional generalization

III. SUMMARY

### Key Topics

- Correlation functions
- 27.0
- Surface reactions
- 23.0
- Surface states
- 20.0
- Transition state theory
- 20.0
- Polymers
- 17.0

## Figures

Schematic illustration of the original RPMD method in Eq. (20) for calculating a reaction rate. At time zero, the first bead of the ring-polymer necklace is pinned to the transition state dividing surface and contributes a velocity factor of to the flux-side correlation function . The polymer then evolves under the classical equations of motion in Eqs. (17) and (18) and contributes a side factor of to the correlation function at time .

Schematic illustration of the original RPMD method in Eq. (20) for calculating a reaction rate. At time zero, the first bead of the ring-polymer necklace is pinned to the transition state dividing surface and contributes a velocity factor of to the flux-side correlation function . The polymer then evolves under the classical equations of motion in Eqs. (17) and (18) and contributes a side factor of to the correlation function at time .

Schematic illustration of the new RPMD method in Eq. (24) for calculating a reaction rate. At time zero, the centroid of the ring-polymer necklace is pinned to the transition state dividing surface and contributes a velocity factor of to the flux-side correlation function . The polymer then evolves under the classical equations of motion in Eqs. (17) and (18) and contributes a side factor of [ or 1] to the correlation function at time .

Schematic illustration of the new RPMD method in Eq. (24) for calculating a reaction rate. At time zero, the centroid of the ring-polymer necklace is pinned to the transition state dividing surface and contributes a velocity factor of to the flux-side correlation function . The polymer then evolves under the classical equations of motion in Eqs. (17) and (18) and contributes a side factor of [ or 1] to the correlation function at time .

Comparison of the flux-side correlation function in Eq. (24) (solid line) with that in Eq. (20) (dashed line) for the symmetric Eckart barrier at 300 and 1000 K.

Comparison of the flux-side correlation function in Eq. (24) (solid line) with that in Eq. (20) (dashed line) for the symmetric Eckart barrier at 300 and 1000 K.

Histograms of the factors in Eq. (27) and in Eq. (28) for the symmetric Eckart barrier at 300 K.

Histograms of the factors in Eq. (27) and in Eq. (28) for the symmetric Eckart barrier at 300 K.

(a) An Arrhenius plot of the rate coefficient for the symmetric Eckart barrier. The solid line is the RPMD result obtained from Eq. (24) 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).

(a) An Arrhenius plot of the rate coefficient for the symmetric Eckart barrier. The solid line is the RPMD result obtained from Eq. (24) 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).

Top panel: the location of five different dividing surfaces on the symmetric Eckart barrier. Middle panel: the computed RPMD flux-side correlation functions at 300 K for each of these dividing surfaces. (Note that the dividing surfaces at and give the same correlation functions by symmetry.) Bottom panel: dependence of the QTST and RPMD rate coefficients on the location of the dividing surface.

Top panel: the location of five different dividing surfaces on the symmetric Eckart barrier. Middle panel: the computed RPMD flux-side correlation functions at 300 K for each of these dividing surfaces. (Note that the dividing surfaces at and give the same correlation functions by symmetry.) Bottom panel: dependence of the QTST and RPMD rate coefficients on the location of the dividing surface.

Computed QTST (dashed line) and RPMD (solid line) transmission coefficients for the asymmetric Eckart barrier at three different temperatures as a function of the location of the dividing surface .

Computed QTST (dashed line) and RPMD (solid line) transmission coefficients for the asymmetric Eckart barrier at three different temperatures as a function of the location of the dividing surface .

Time-dependent RPMD transmission coefficients for the asymmetric Eckart barrier at the three different temperatures considered in Fig. 7.

Time-dependent RPMD transmission coefficients for the asymmetric Eckart barrier at the three different temperatures considered in Fig. 7.

## Tables

Transmission coefficients for the asymmetric Eckart barrier.

Transmission coefficients for the asymmetric Eckart barrier.

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