^{1}, Alexander V. Soudackov

^{1}and Sharon Hammes-Schiffer

^{1,a)}

### Abstract

A nonadiabatic rate expression for hydrogen tunnelingreactions in the condensed phase is derived for a model system described by a modified spin-boson Hamiltonian with a tunneling matrix element exponentially dependent on the hydrogen donor-acceptor distance. In this model, the two-level system representing the localized hydrogen vibrational states is linearly coupled to the donor-acceptor vibrational mode and the harmonic bath. The Hamiltonian also includes bilinear coupling between the donor-acceptor mode and the bath oscillators. This coupling provides a mechanism for energy exchange between the two-level system and the bath through the donor-acceptor mode, thereby facilitating convergence of the time integral of the probability flux correlation function for the case of weak coupling between the two-level system and the bath. The dependence of the rate constant on the model parameters and the temperature is analyzed in various regimes. Anomalous behavior of the rate constant is observed in the weak solvation regime for model systems that lack an effective mechanism for energy exchange between the two-level system and the bath. This theoretical formulation is applicable to a wide range of chemical and biological processes, including neutral hydrogen transfer reactions with small solvent reorganization energies.

This work was supported by NSF Grant No. CHE-05-01260 and NIH Grant No. GM56207. One of the authors (Y.O.) is grateful for the support by the Japanese Society for the Promotion of Science Postdoctoral Fellowships for Research Abroad.

I. INTRODUCTION

II. THEORETICAL FORMULATION

A. General model system

B. Spin-boson Hamiltonian

C. Nonadiabaticrate constant

1. Diagonalization of the unperturbed Hamiltonian

2. Derivation of rate constant for nonadiabatic quantum transitions

3. Probability flux correlation function for Ohmic dissipation

III. MODEL CALCULATIONS

A. Strong solvation regime

B. Weak solvation regime

IV. CONCLUSIONS

### Key Topics

- Reaction rate constants
- 52.0
- Correlation functions
- 33.0
- Tunneling
- 30.0
- Oscillators
- 23.0
- Non adiabatic couplings
- 22.0

## Figures

Schematic picture of a general hydrogen tunneling system. The double well potential energy curves are functions of the hydrogen coordinate , and the lowest two adiabatic hydrogen vibrational states are depicted for each hydrogen potential energy curve. Reorganization of the bath environment alters the relative energies of the wells of the hydrogen potential energy curves. Hydrogen tunneling is allowed for the symmetric double well potential. Increasing the donor-acceptor distance increases the barrier height and width, thereby decreasing the tunneling splitting between the adiabatic hydrogen vibrational states. In the diabatic representation, the bath reorganization leads to degeneracy of the reactant and product diabatic states, and the nonadiabatic coupling between these states decreases as the donor-acceptor distance increases.

Schematic picture of a general hydrogen tunneling system. The double well potential energy curves are functions of the hydrogen coordinate , and the lowest two adiabatic hydrogen vibrational states are depicted for each hydrogen potential energy curve. Reorganization of the bath environment alters the relative energies of the wells of the hydrogen potential energy curves. Hydrogen tunneling is allowed for the symmetric double well potential. Increasing the donor-acceptor distance increases the barrier height and width, thereby decreasing the tunneling splitting between the adiabatic hydrogen vibrational states. In the diabatic representation, the bath reorganization leads to degeneracy of the reactant and product diabatic states, and the nonadiabatic coupling between these states decreases as the donor-acceptor distance increases.

Contour plots of the reactant and product potential energy surfaces as functions of the donor-acceptor coordinate and a bath mode coordinate . The surfaces are shown for systems with (a) uncoupled donor-acceptor and bath coordinates and (b) coupled donor-acceptor and bath coordinates. Note that the minima of the potential energy surfaces are not affected by the coupling between the coordinates.

Contour plots of the reactant and product potential energy surfaces as functions of the donor-acceptor coordinate and a bath mode coordinate . The surfaces are shown for systems with (a) uncoupled donor-acceptor and bath coordinates and (b) coupled donor-acceptor and bath coordinates. Note that the minima of the potential energy surfaces are not affected by the coupling between the coordinates.

Schematic diagram of the model for hydrogen tunneling. The three subsystems are the two-level system (TLS) with splitting , the donor-acceptor mode with mass and frequency , and the bath consisting of modes with masses and frequencies . The subsystems are coupled through the coupling constants , , and .

Schematic diagram of the model for hydrogen tunneling. The three subsystems are the two-level system (TLS) with splitting , the donor-acceptor mode with mass and frequency , and the bath consisting of modes with masses and frequencies . The subsystems are coupled through the coupling constants , , and .

Probability flux correlation function in the strong solvation regime with and . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (dashed line) and (solid line). The donor-acceptor mode frequency is and the temperature is .

Probability flux correlation function in the strong solvation regime with and . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (dashed line) and (solid line). The donor-acceptor mode frequency is and the temperature is .

Dependence of the rate on in the strong solvation regime with and . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (dashed line) and (solid line). The temperature is and the donor-acceptor mode frequency is varied.

Dependence of the rate on in the strong solvation regime with and . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (dashed line) and (solid line). The temperature is and the donor-acceptor mode frequency is varied.

Dependence of the rate constant on the energy bias in the strong solvation regime with , , and . The dashed, dot-dashed, and solid lines correspond to , , and , respectively. The thin vertical line separates the inverted regime (left) and the normal regime (right) for .

Dependence of the rate constant on the energy bias in the strong solvation regime with , , and . The dashed, dot-dashed, and solid lines correspond to , , and , respectively. The thin vertical line separates the inverted regime (left) and the normal regime (right) for .

Dependence of the rate on in the strong solvation regime with , , and . The dashed and solid lines correspond to (inverted regime) and (normal regime), respectively.

Dependence of the rate on in the strong solvation regime with , , and . The dashed and solid lines correspond to (inverted regime) and (normal regime), respectively.

Probability flux correlation function in the weak solvation regime with and . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (a) and (b) . The donor-acceptor mode frequency is and the temperature is .

Probability flux correlation function in the weak solvation regime with and . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (a) and (b) . The donor-acceptor mode frequency is and the temperature is .

Dependence of the rate constant on the energy bias in the weak solvation regime with . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (oscillatory thin line) and (smooth thick line). The donor-acceptor mode frequency is and the temperature is .

Dependence of the rate constant on the energy bias in the weak solvation regime with . The friction constants corresponding to the coupling between the donor-acceptor mode and the bath are (oscillatory thin line) and (smooth thick line). The donor-acceptor mode frequency is and the temperature is .

Temperature dependence of the rate constant in the weak solvation regime for the case of a low donor-acceptor mode frequency . The parameters for this model system are , , , , and .

Temperature dependence of the rate constant in the weak solvation regime for the case of a low donor-acceptor mode frequency . The parameters for this model system are , , , , and .

Temperature dependence of the rate constant in the weak solvation regime for the case of a high donor-acceptor mode frequency . The parameters that are the same for both model systems are , , and . The other parameters are (a) , , , , and and (b) , , , , and .

Temperature dependence of the rate constant in the weak solvation regime for the case of a high donor-acceptor mode frequency . The parameters that are the same for both model systems are , , and . The other parameters are (a) , , , , and and (b) , , , , and .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content