^{1}and David E. Manolopoulos

^{1}

### Abstract

We show how the ring-polymer molecular dynamics method can be adapted to calculate approximate Kubo-transformed flux-side correlation functions, and hence rate coefficients for condensed phase reactions. An application of the method to the standard model for a chemical reaction in solution—a quartic double-well potential linearly coupled to a bath of harmonic oscillators—is found to give results of comparable accuracy to those of the classical Wigner model and the centroid molecular dynamics method. However, since the present method does not require that one evaluate the Wigner transform of a thermal flux operator or that one perform a separate path integral calculation for each molecular dynamics time step, we believe it will prove easier to apply to more general problems than either of these alternative techniques. We also present a (logarithmic) discretization scheme for the Ohmic bath in the system-bath model that gives converged results with just nine bath modes—a surprisingly small number for a model of a condensed phase reaction. Finally, we present some calculations of the transmission through an Eckart barrier which show that the present method provides a satisfactory (although not perfect) description of the deep quantum tunneling regime. Part of the reason for the success of the method is that it gives the exact quantum-mechanical rate constant for the transmission through a parabolic barrier, as we demonstrate analytically in the Appendix.

The authors would like to thank Bill Miller for some helpful suggestions, David Chandler for his continued encouragement, and David Logan for a discussion that led them to the bath discretization scheme presented in Sec. IV.

I. INTRODUCTION

II. ALTERNATIVE TECHNIQUES

A. The classical Wigner model

B. Forward-backward semiclassical dynamics

C. Centroid molecular dynamics

D. Summary

III. RING-POLYMER MOLECULAR DYNAMICS

IV. THE SYSTEM-BATH MODEL

V. TRANSITION STATE SAMPLING

VI. RESULTS AND DISCUSSION

A. Computational details

B. The system-bath model

C. The Eckart barrier

D. Discussion

VII. CONCLUSIONS AND FUTURE WORK

### Key Topics

- Molecular dynamics
- 25.0
- Correlation functions
- 24.0
- Normal modes
- 19.0
- Transition state theory
- 14.0
- Chemical reactions
- 13.0

## Figures

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 .

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.

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.

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).

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).

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).

(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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