^{1}and Sharon Hammes-Schiffer

^{1,a)}

### Abstract

A hybrid quantum/classical molecular dynamics approach is applied to a proton transferreaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theoryrate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.

We are grateful to Nancy Makri for providing the data from her calculations and for helpful discussions. This work was supported by NIH Grant No. GM56207, AFOSR Grant No. FA9550-04-1-0062, and NSF Grant No. CHE-05-01260.

I. INTRODUCTION

II. THEORY AND METHODS

A. The model

B. Reactive flux approach for rare events

C. Classical treatment of the proton

1. Classical transition state theoryrate constant

2. Classical transmission coefficient

D. Quantum treatment of the proton

1. Hybrid quantum/classical transition state theoryrate constant

2. Hybrid quantum/classical transmission coefficient

E. Simulation details

1. Classical rate constant

2. Hybrid quantum/classical rate constant

III. RESULTS AND DISCUSSION

A. Classical approach

B. Hybrid quantum/classical approach

IV. CONCLUSIONS

### Key Topics

- Friction
- 67.0
- Protons
- 58.0
- Transition state theory
- 46.0
- Transmission coefficient
- 43.0
- Reaction rate constants
- 29.0

## Figures

The overall classical transmission coefficient as a function of the reduced friction for the model proton transfer system coupled to a dissipative bath at . The results were obtained with the plateau reactive flux method (triangles) and Keck’s reactive flux method (squares). The results obtained previously with the plateau reactive flux method by Topaler and Makri (Ref. 34) are shown as circles. The Kramers-Grote-Hynes transmission coefficient (Refs. 96 and 97) (dashed line) and the Pollak-Grabert-Hänggi turnover theory (Ref. 98) (solid line) are also shown. The equations for these theories and the corresponding data were obtained from Ref. 34.

The overall classical transmission coefficient as a function of the reduced friction for the model proton transfer system coupled to a dissipative bath at . The results were obtained with the plateau reactive flux method (triangles) and Keck’s reactive flux method (squares). The results obtained previously with the plateau reactive flux method by Topaler and Makri (Ref. 34) are shown as circles. The Kramers-Grote-Hynes transmission coefficient (Refs. 96 and 97) (dashed line) and the Pollak-Grabert-Hänggi turnover theory (Ref. 98) (solid line) are also shown. The equations for these theories and the corresponding data were obtained from Ref. 34.

The proton potential energy curves averaged over (a) reactant bath configurations and (b) transition state bath configurations. The results are shown for the following values of the reduced friction : 1.0 (solid), 1.5 (dashed), 3.0 (dotted), and 4.5 (dot-dashed).

The proton potential energy curves averaged over (a) reactant bath configurations and (b) transition state bath configurations. The results are shown for the following values of the reduced friction : 1.0 (solid), 1.5 (dashed), 3.0 (dotted), and 4.5 (dot-dashed).

The average proton potential energy curves shown in Fig. 2 and the corresponding proton vibrational states for (left) and (right). The proton potentials averaged over reactant bath configurations are shown on the top, and the proton potentials averaged over transition state bath configurations are shown on the bottom.

The average proton potential energy curves shown in Fig. 2 and the corresponding proton vibrational states for (left) and (right). The proton potentials averaged over reactant bath configurations are shown on the top, and the proton potentials averaged over transition state bath configurations are shown on the bottom.

The potential of mean force , where , for the following values of the reduced friction : 1.0 (solid), 1.5 (dashed), 3.0 (dotted), and 4.5 (dot-dashed).

The potential of mean force , where , for the following values of the reduced friction : 1.0 (solid), 1.5 (dashed), 3.0 (dotted), and 4.5 (dot-dashed).

Schematic illustration of a proton transfer reaction in the condensed phase. The proton potential energy curves and the corresponding ground and excited vibrational states are depicted. The solid and dashed lines represent the occupied and unoccupied adiabatic vibrational states, respectively. Initially the reactant well is stabilized, and the proton is in the ground state localized in the reactant well. Fluctuations of the environment lead to a symmetric double well potential, and the proton vibrational wave functions are delocalized over the reactant and product wells. Further fluctuations stabilize the product well. In the adiabatic pathway, the system remains in the ground state and the proton becomes localized in the product well. In the nonadiabatic pathway, the system switches to the excited vibrational state and the proton remains localized in the reactant well.

Schematic illustration of a proton transfer reaction in the condensed phase. The proton potential energy curves and the corresponding ground and excited vibrational states are depicted. The solid and dashed lines represent the occupied and unoccupied adiabatic vibrational states, respectively. Initially the reactant well is stabilized, and the proton is in the ground state localized in the reactant well. Fluctuations of the environment lead to a symmetric double well potential, and the proton vibrational wave functions are delocalized over the reactant and product wells. Further fluctuations stabilize the product well. In the adiabatic pathway, the system remains in the ground state and the proton becomes localized in the product well. In the nonadiabatic pathway, the system switches to the excited vibrational state and the proton remains localized in the reactant well.

Analysis of a representative nonadiabatic MDQT trajectory. The reaction coordinate , the quantum probabilities and the vibrational state energies for (solid) and (dashed), and the nonadiabatic coupling between the lowest two vibrational states are depicted as functions of time. For clarity, only the portion of the trajectory near the dividing surface (i.e., near time ) is shown. This trajectory exhibits two nonadiabatic transitions from the ground state to the first excited state at and and two nonadiabatic transitions from the first excited state down to the ground state at and . These nonadiabatic transitions lead to a recrossing that prevents the reaction.

Analysis of a representative nonadiabatic MDQT trajectory. The reaction coordinate , the quantum probabilities and the vibrational state energies for (solid) and (dashed), and the nonadiabatic coupling between the lowest two vibrational states are depicted as functions of time. For clarity, only the portion of the trajectory near the dividing surface (i.e., near time ) is shown. This trajectory exhibits two nonadiabatic transitions from the ground state to the first excited state at and and two nonadiabatic transitions from the first excited state down to the ground state at and . These nonadiabatic transitions lead to a recrossing that prevents the reaction.

The overall transmission coefficient as a function of the reduced friction for the model proton transfer system coupled to a dissipative bath at . The hybrid quantum/classical reactive flux approach results (filled circles) and the numerically exact results obtained by Topaler and Makri (Ref. 34) (open circles) are shown. The Wolynes transmission factor (Ref. 99) from quantum Grote-Hynes theory (Ref. 96) (dotted line) and the quantum turnover theories due to Hänggi *et al.* (Refs. 35 and 100) (solid line) and to Rips and Pollak (Ref. 101) (dashed line) are also shown. The equations for these theories and the corresponding data were obtained from Ref. 34.

The overall transmission coefficient as a function of the reduced friction for the model proton transfer system coupled to a dissipative bath at . The hybrid quantum/classical reactive flux approach results (filled circles) and the numerically exact results obtained by Topaler and Makri (Ref. 34) (open circles) are shown. The Wolynes transmission factor (Ref. 99) from quantum Grote-Hynes theory (Ref. 96) (dotted line) and the quantum turnover theories due to Hänggi *et al.* (Refs. 35 and 100) (solid line) and to Rips and Pollak (Ref. 101) (dashed line) are also shown. The equations for these theories and the corresponding data were obtained from Ref. 34.

## Tables

The values for the TST rate constant , the two factors comprising the TST rate constant, the transmission coefficient , and the overall transmission coefficient for four different friction values. The first factor comprising the TST rate constant is and the second factor is . The TST rate constant is given in Eq. (25), the transmission coefficient is given in Eq. (24), and the overall transmission coefficient is defined in Eq. (9).

The values for the TST rate constant , the two factors comprising the TST rate constant, the transmission coefficient , and the overall transmission coefficient for four different friction values. The first factor comprising the TST rate constant is and the second factor is . The TST rate constant is given in Eq. (25), the transmission coefficient is given in Eq. (24), and the overall transmission coefficient is defined in Eq. (9).

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