^{1,a)}, Fermín Huarte-Larrañaga

^{2}and Uwe Manthe

^{1,b)}

### Abstract

Rigorous quantum dynamics calculations of reaction rates and initial state-selected reaction probabilities of polyatomic reactions can be efficiently performed within the quantum transition state concept employing flux correlation functions and wave packet propagation utilizing the multi-configurational time-dependent Hartree approach. Here, analytical formulas and a numerical scheme extending this approach to the calculation of state-to-state reaction probabilities are presented. The formulas derived facilitate the use of three different dividing surfaces: two dividing surfaces located in the product and reactant asymptotic region facilitate full state resolution while a third dividing surface placed in the transition state region can be used to define an additional flux operator. The eigenstates of the corresponding thermal flux operator then correspond to vibrational states of the activated complex. Transforming these states to reactant and product coordinates and propagating them into the respective asymptotic region, the full scattering matrix can be obtained. To illustrate the new approach, test calculations study the D + H_{2}(ν, *j*) → HD(ν′, *j*′) + H reaction for *J* = 0.

Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. F.H.L. acknowledges Project Nos. CTQ2009-12215 (Spanish MICINN) and SGR2009-17 (Generalitat de Catalunya).

I. INTRODUCTION

II. THEORY

A. Quantum transition state concept

B. Flux correlation functions and state-to-state observables

C. Scattering matrix and state-to-state reaction probabilities

D. Asymptotic state analysis

III. QUANTUM DYNAMICS

A. MCTDH approach

B. Coordinate transformation

IV. SYSTEM AND NUMERICAL DETAILS

A. Coordinate systems

B. Thermal flux eigenstates

C. Real time propagation

D. ABC calculations

V. RESULTS AND DISCUSSION

A. Cumulative reaction probability

B. Convergence with respect to flux eigenstates

C. State-to-state reaction probabilities

VI. CONCLUDING REMARKS

### Key Topics

- Wave functions
- 26.0
- Hydrogen reactions
- 22.0
- Surface states
- 15.0
- Atom reactions
- 13.0
- Chemical reaction cross sections
- 13.0

## Figures

Cumulative reaction probability of D + H_{2} → DH + H(*J* = 0) calculated in the two different Jacobi coordinate systems with the second dividing surface located at *R* _{ d } = 7.5 bohr or bohr and by summing all state-to-state reaction probabilities. The ABC results are given as symbols.

Cumulative reaction probability of D + H_{2} → DH + H(*J* = 0) calculated in the two different Jacobi coordinate systems with the second dividing surface located at *R* _{ d } = 7.5 bohr or bohr and by summing all state-to-state reaction probabilities. The ABC results are given as symbols.

State-to-state reaction probability for D + H_{2}(ν = 0, *j* = 0) → H + DH(ν′ = 0, *j*′ = 0) calculated using different numbers of eigenstates of the thermal flux operator. The ABC results are given as symbols.

State-to-state reaction probability for D + H_{2}(ν = 0, *j* = 0) → H + DH(ν′ = 0, *j*′ = 0) calculated using different numbers of eigenstates of the thermal flux operator. The ABC results are given as symbols.

Partial summed state-to-state reaction probabilities for D + H_{2}(ν = 0, *j* = 0) → H + DH(ν′ = *all*, *j*′). The ABC results are given as symbols.

Partial summed state-to-state reaction probabilities for D + H_{2}(ν = 0, *j* = 0) → H + DH(ν′ = *all*, *j*′). The ABC results are given as symbols.

Partial summed state-to-state reaction probabilities for D + H_{2}(ν = 0, 1, *j* = 0) → H + DH(ν′, *j*′ = *all*). The ABC results are given as symbols.

Partial summed state-to-state reaction probabilities for D + H_{2}(ν = 0, 1, *j* = 0) → H + DH(ν′, *j*′ = *all*). The ABC results are given as symbols.

State-to-state reaction probabilities for D + H_{2}(ν = 0, *j* = 0) → H + *DH*(ν′ = 0, *j*′). The ABC results are given as symbols.

State-to-state reaction probabilities for D + H_{2}(ν = 0, *j* = 0) → H + *DH*(ν′ = 0, *j*′). The ABC results are given as symbols.

State-to-state reaction probabilities for D + H_{2}(ν = 0, 1, *j* = 0) → H + DH(ν′, *j*′ = 0). The ABC results are given as symbols.

State-to-state reaction probabilities for D + H_{2}(ν = 0, 1, *j* = 0) → H + DH(ν′, *j*′ = 0). The ABC results are given as symbols.

State-to-state reaction probabilities for D + H_{2}(ν = 0, *j* = 3) → H + DH(ν′ = 0, *j*′). The ABC results are given as symbols.

State-to-state reaction probabilities for D + H_{2}(ν = 0, *j* = 3) → H + DH(ν′ = 0, *j*′). The ABC results are given as symbols.

## Tables

Number of grid points (*N*) and SPFs (*n*) used in the flux eigenstate calculation.

Number of grid points (*N*) and SPFs (*n*) used in the flux eigenstate calculation.

Eigenvalues of the thermal flux operator.

Eigenvalues of the thermal flux operator.

Number of grid points (*N*) and SPFs (*n*) used in the propagation to the reactant and product asymptotic region.

Number of grid points (*N*) and SPFs (*n*) used in the propagation to the reactant and product asymptotic region.

Parameters used in the ABC calculation.

Parameters used in the ABC calculation.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content