^{1,2}, Hua Guo

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

^{1,a)}

### Abstract

The -matrix for a scattering system provides the most detailed information about the dynamics. In this work, we discuss the calculation of -matrix elements for the , type reaction. Two methods for extracting -matrix elements from a single wave packet in reactant Jacobi coordinates are reviewed and compared. Both methods are capable of extracting the state-to-state attributes for both product channels from a single wave packet propagation. It is shown through the examples of , , and reactions that such reactant coordinate based methods are easy to implement, numerically efficient, and accurate. Additional efficiency can be gained by the use of a -shaped grid with two-dimensional fast Fourier transform.

Z.G.S. and D.H.Z. were supported by the Knowledge Innovation Program of the Chinese Academy of Science (DICP Grant Nos. 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 discretization

B. Initial wave packet

C. Real Chebyshev wave packet on -shaped grid

D. Scattering boundary conditions

E. Extraction of S-matrix elements and state-to-state cross sections

F. Final state projection in reactant coordinates

1. Collocation RCB method

2. Coordinate transform RCB method

III. RESULTS AND DISCUSSION

A. Demonstration of methods: reaction

B. Advantages of -shaped grid: reaction

C. Position of projection plane: reaction

D. Choice between the PCB and RCB methods

IV. CONCLUSIONS

### Key Topics

- Hydrogen reactions
- 42.0
- Wave functions
- 19.0
- Chemical reaction cross sections
- 12.0
- Chemical reaction theory
- 11.0
- Exchange reactions
- 11.0

## Figures

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 using the collocation and coordinate transform RCB methods. It is seen that the two methods give identical and converged DCSs for this reaction with parameters listed in Table I.

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 using the collocation and coordinate transform RCB methods. It is seen that the two methods give identical and converged DCSs for this reaction with parameters listed in Table I.

2D contours of the potential energy surface of the reaction along and degrees of freedom on the -shaped grid with the energy in the angular coordinate optimized. The parameters in Table I are defined in the 2D plot.

2D contours of the potential energy surface of the reaction along and degrees of freedom on the -shaped grid with the energy in the angular coordinate optimized. The parameters in Table I are defined in the 2D plot.

Total reaction probabilities obtained from initial wave packet placed at different positions , 12.0, and 16.0 a.u. The results indicate that the entire van der Waals well in the reactant channel should be included for an accurate calculation of the total reaction probabilities above the collision energy of 0.45 eV.

Total reaction probabilities obtained from initial wave packet placed at different positions , 12.0, and 16.0 a.u. The results indicate that the entire van der Waals well in the reactant channel should be included for an accurate calculation of the total reaction probabilities above the collision energy of 0.45 eV.

Total reaction probabilities for the reaction obtained by the state-to-state reaction probabilities for , 20, 30, 40, 50, and 60 of the abstraction reaction (a) and exchange reaction (e).

Total reaction probabilities for the reaction obtained by the state-to-state reaction probabilities for , 20, 30, 40, 50, and 60 of the abstraction reaction (a) and exchange reaction (e).

Comparison between the total reaction probabilities for the reaction obtained by summing the state-to-state reaction probabilities with those calculated directly by a flux method for , 20, and 40. The upper panel is for the total reaction while the lower panel is for the abstraction reaction only.

Comparison between the total reaction probabilities for the reaction obtained by summing the state-to-state reaction probabilities with those calculated directly by a flux method for , 20, and 40. The upper panel is for the total reaction while the lower panel is for the abstraction reaction only.

A comparison for a state-to-state ICSs for the reaction, at selected collision energies of 0.4, 0.8, and 1.2 eV for the abstraction reaction (left panels) and 0.8, 1.0, and 1.2 eV for the exchange reaction (right panel). The product analysis planes are placed at , 9.0, and 10.0 a.u. for the abstraction reaction and at , 10.0, and 11.0 a.u. for the exchange reaction. The positions of the product state analysis plane has little influence on the results, even they are not at the true asymptote.

A comparison for a state-to-state ICSs for the reaction, at selected collision energies of 0.4, 0.8, and 1.2 eV for the abstraction reaction (left panels) and 0.8, 1.0, and 1.2 eV for the exchange reaction (right panel). The product analysis planes are placed at , 9.0, and 10.0 a.u. for the abstraction reaction and at , 10.0, and 11.0 a.u. for the exchange reaction. The positions of the product state analysis plane has little influence on the results, even they are not at the true asymptote.

Same as in Fig. 6, but for a comparison for a state-to-state ICSs as a function of collision energy.

Same as in Fig. 6, but for a comparison for a state-to-state ICSs as a function of collision energy.

Same as in Fig. 7, but for a comparison of the total DCSs.

Same as in Fig. 7, but for a comparison of the total DCSs.

Same as in Fig. 8, but for a comparison at a state-to-state level for product states as , (0,5) and (0,8).

Same as in Fig. 8, but for a comparison at a state-to-state level for product states as , (0,5) and (0,8).

## 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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