^{1,2}, Soo-Y. Lee

^{2}, Hua Guo

^{3,a)}and Dong H. Zhang

^{1,a)}

### Abstract

A detailed comparison of the time-dependent wave packet method using the split operator propagator and recently introduced Chebyshev real wave packet approach for calculating reactive scattering processes is reported. As examples, the state-to-state differential cross sections of the reaction, the state-to-state reaction probabilities of the reaction, the reaction, and the reaction are calculated, using an efficient reactant-coordinate-based method on an L-shape grid which allows the extraction of the state-to-state information of the two product channels simultaneously. These four reactions have quite different dynamic characteristics and thus provide a comprehensive picture of the relative advantages of these two propagation methods for describing reactive scattering dynamics. The results indicate that the Chebyshev real wave packet method is typically more accurate, particularly for reactions dominated by long-lived resonances. However, the split operator approach is often more cost effective, making it a method of choice for fast reactions. In addition, our results demonstrate accuracy of the reactant-coordinate-based method for extracting state-to-state information.

Z.G.S. and S.Y.L. were supported by Research Grant No. T207B1222 from the Ministry of Education, Singapore. Z.G.S. and D.H.Z. were supported by the Knowledge Innovation Program of the Chinese Academy of Science (Grant Nos. DICP R200402 and Y200601) and by the National Natural Science Foundation of China (Grant Nos. 20688301 and 20773127). H.G. was supported by the U.S. Department of Energy (Grant No. DE-FG02-05ER15694).

I. INTRODUCTION

II. THEORY

A. Hamiltonian and basis functions in reactant coordinates

B. Construction of the initial wave packet

C. Wave packet propagation using the SO method and the CRWP approach

D. Extraction of state-to-state cross sections

E. Absorbing potential

III. RESULTS AND DISCUSSION

A. reaction

B. exchange reaction

C. reaction

D. reaction

IV. CONCLUSIONS

### Key Topics

- Hydrogen reactions
- 42.0
- Wave functions
- 13.0
- Chemical reaction cross sections
- 10.0
- Angular momentum
- 7.0
- Collision induced chemical reactions
- 7.0

## Figures

Total reaction probabilities from the initial state with and of the reaction calculated by the CRWP approach and the time-dependent wave packet method using the SO propagator with time steps , 15, and 25 a.u. These results agree with each other very well which indicate the SO propagator converges well even with time step for this reaction. The corresponding ABC results are not given for clarity because they agree very well with the results obtained by the wave packet methods.

Total reaction probabilities from the initial state with and of the reaction calculated by the CRWP approach and the time-dependent wave packet method using the SO propagator with time steps , 15, and 25 a.u. These results agree with each other very well which indicate the SO propagator converges well even with time step for this reaction. The corresponding ABC results are not given for clarity because they agree very well with the results obtained by the wave packet methods.

State-to-state DCSs at collision energies of 0.4 and 0.8 eV of the reaction (left column) and of the reaction (right column), calculated by the ABC code, the CRWP approach and the time-dependent wave packet method using the SO propagator. It is seen that with a time step , the SO method can still give highly converged DCSs for this reaction.

State-to-state DCSs at collision energies of 0.4 and 0.8 eV of the reaction (left column) and of the reaction (right column), calculated by the ABC code, the CRWP approach and the time-dependent wave packet method using the SO propagator. It is seen that with a time step , the SO method can still give highly converged DCSs for this reaction.

State-to-state DCSs at collision energies of 0.4 and 0.8 eV of the reaction, calculated by the ABC code, the CRWP approach, and the time-dependent wave packet method using the SO propagator. It is seen that none of the used time steps (, 15.0, and 25.0 a.u.) in time propagation gives satisfactory results for this reaction.

State-to-state DCSs at collision energies of 0.4 and 0.8 eV of the reaction, calculated by the ABC code, the CRWP approach, and the time-dependent wave packet method using the SO propagator. It is seen that none of the used time steps (, 15.0, and 25.0 a.u.) in time propagation gives satisfactory results for this reaction.

Total reaction probabilities of the reaction [(a) and (b)] and the reaction [(c) and (d)] calculated by the CRWP approach and the wave packet method using the SO propagator with time steps , 10.0, and 15.0 a.u. The results in panels (b) and (d) are the enlarged parts of those in panels (a) and (c). The plots indicate that the SO propagator only with a time step is able to give well converged results for this reaction.

Total reaction probabilities of the reaction [(a) and (b)] and the reaction [(c) and (d)] calculated by the CRWP approach and the wave packet method using the SO propagator with time steps , 10.0, and 15.0 a.u. The results in panels (b) and (d) are the enlarged parts of those in panels (a) and (c). The plots indicate that the SO propagator only with a time step is able to give well converged results for this reaction.

State-to-state reaction probabilities of the reaction (, [(a) and (b)] and , [(c) and (d)]) calculated by the CRWP approach and the SO method with time steps of 5.0, 10.0, and 15.0 a.u. Only with a time step , the SO method gives well converged numerical results for this reaction.

State-to-state reaction probabilities of the reaction (, [(a) and (b)] and , [(c) and (d)]) calculated by the CRWP approach and the SO method with time steps of 5.0, 10.0, and 15.0 a.u. Only with a time step , the SO method gives well converged numerical results for this reaction.

Total reaction probabilities of the reaction calculated by the CRWP approach and the wave packet method using the SO propagator with time steps , 10.0, and 15.0 a.u. The results in right panel are enlarged part of those in the left panel. The plots indicate that the SO propagator with a time step as small as 5.0 a.u. cannot give well converged results for this reaction.

Total reaction probabilities of the reaction calculated by the CRWP approach and the wave packet method using the SO propagator with time steps , 10.0, and 15.0 a.u. The results in right panel are enlarged part of those in the left panel. The plots indicate that the SO propagator with a time step as small as 5.0 a.u. cannot give well converged results for this reaction.

State-to-state reaction probabilities for the reaction (, [(a) and (b)] and , [(c) and (d)]) calculated by the CRWP approach and the wave packet method using the SO propagator with time steps , 10.0, and 15.0 a.u. Even with a time step of 5.0 a.u., the SO propagator cannot give well converged results for this reaction.

State-to-state reaction probabilities for the reaction (, [(a) and (b)] and , [(c) and (d)]) calculated by the CRWP approach and the wave packet method using the SO propagator with time steps , 10.0, and 15.0 a.u. Even with a time step of 5.0 a.u., the SO propagator cannot give well converged results for this reaction.

Total reaction probabilities of the reaction calculated by the CRWP approach (a) and the wave packet method using the SO propagator with time steps of 5.0 a.u. (b) and 10.0 a.u. (c), as compared to that obtained by the ABC code. The total reaction probabilities obtained by summing the product state-resolved probabilities using RCB method are also given. It is seen that an unsuitable time step used for the SO propagator not only results in errors in total reaction probabilities calculated by a flux method, but also results in errors in final boundary conditions matching for obtaining the state-to-state information.

Total reaction probabilities of the reaction calculated by the CRWP approach (a) and the wave packet method using the SO propagator with time steps of 5.0 a.u. (b) and 10.0 a.u. (c), as compared to that obtained by the ABC code. The total reaction probabilities obtained by summing the product state-resolved probabilities using RCB method are also given. It is seen that an unsuitable time step used for the SO propagator not only results in errors in total reaction probabilities calculated by a flux method, but also results in errors in final boundary conditions matching for obtaining the state-to-state information.

## Tables

Parameters used in the numerical calculations (atomic units are used if not otherwise stated).

Parameters used in the numerical calculations (atomic units are used if not otherwise stated).

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