^{1}and Dmitri Babikov

^{1,a)}

### Abstract

A mixed quantum-classical approach to the description of collisional energy transfer is proposed in which the vibrational motion of an energized molecule is treated quantum mechanically using wave packets, while the collisional motion of the molecule and quencher and the rotational motion of the molecule are treated using classical trajectories. This accounts rigorously for quantization of vibrational states, zero-point energy, scattering resonances, and permutation symmetry of identical atoms, while advantage is taken of the classical scattering regime. Energy is exchanged between vibrational, rotational, and translational degrees of freedom while the total energy is conserved. Application of this method to stabilization of the van der Waals states in ozone is presented. Examples of mixed quantum-classical trajectories are discussed, including an interesting example of supercollision. When combined with an efficient grid mapping procedure and the reduced dimensionality approximation, the method becomes very affordable computationally.

This research was supported by the National Science Foundation (NSF) Atmospheric Chemistry Program, Division of Atmospheric Sciences, grant number 0842530. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the (U.S.) Department of Energy (DOE) under Contract No. DE-AC02–05CH11231.

I. INTRODUCTION

II. THEORETICAL FRAMEWORK

A. Collisional degrees of freedom

B. Vibrational degrees of freedom

C. Rotational degrees of freedom

III. PRACTICAL IMPLEMENTATION AND NUMERICAL METHODS

A. Potential energy surface

B. Adiabatic Bending Approximation

C. Mapping and masking of the 2D-grid

D. Transformation to the laboratory frame

E. Time propagation

IV. RESULTS AND DISCUSSION

A. Test of the adiabatic bending correction

B. Energy and angular momentum conservation

C. Insight into the mechanism of ozone stabilization

V. CONCLUSIONS

### Key Topics

- Ozone
- 138.0
- Energy transfer
- 30.0
- Wave functions
- 28.0
- Potential energy surfaces
- 21.0
- Angular momentum
- 17.0

## Figures

Potential energy surface of O_{3} in two dimensions. The main well and two van der Waals wells are clearly seen. The insert shows an expanded view of the vdW well and reef. Contour lines of the main well are given from –1 eV to 0 (dissociation threshold) with a step of 0.2 eV. Contour lines of the vdW well are given from −200 cm^{−1} to 0 with a step of 50 cm^{−1}. The cross-shaped part of the grid (white) is used in calculations; grey area of the configuration spase is neglected.

Potential energy surface of O_{3} in two dimensions. The main well and two van der Waals wells are clearly seen. The insert shows an expanded view of the vdW well and reef. Contour lines of the main well are given from –1 eV to 0 (dissociation threshold) with a step of 0.2 eV. Contour lines of the vdW well are given from −200 cm^{−1} to 0 with a step of 50 cm^{−1}. The cross-shaped part of the grid (white) is used in calculations; grey area of the configuration spase is neglected.

Points of the working grid transferred from the internal coordinates into the laboratory reference frame (Cartesian). Equilibrium geometry of O_{3} is shown. Geometry of the vdW minimum is indicated by arrows. Note that molecular geometries are distorted in the figure due to different ranges (and scales) for *X* and *Y*.

Points of the working grid transferred from the internal coordinates into the laboratory reference frame (Cartesian). Equilibrium geometry of O_{3} is shown. Geometry of the vdW minimum is indicated by arrows. Note that molecular geometries are distorted in the figure due to different ranges (and scales) for *X* and *Y*.

Wave functions of several states near the dissociation threshold of ^{16}O^{18}O^{16}O with *J* = 19 (*K* _{a} = 4, *K* _{b} = 12). Energies and assignments are given in Table II. One contour line is given at ± 0.01.

Wave functions of several states near the dissociation threshold of ^{16}O^{18}O^{16}O with *J* = 19 (*K* _{a} = 4, *K* _{b} = 12). Energies and assignments are given in Table II. One contour line is given at ± 0.01.

Trajectory of Ar + O_{3} collision showing large transfer of the angular momentum. Different frames of the figure give time evolution of the (a) distance between Ar and O_{3}; (b) mean interaction potential; (c) internal energy of O_{3}; (d) angular momentum of O_{3}; (e) rotational energy of O_{3}. Frame (f) shows geometry of this collision.

Trajectory of Ar + O_{3} collision showing large transfer of the angular momentum. Different frames of the figure give time evolution of the (a) distance between Ar and O_{3}; (b) mean interaction potential; (c) internal energy of O_{3}; (d) angular momentum of O_{3}; (e) rotational energy of O_{3}. Frame (f) shows geometry of this collision.

Same as Fig. 4, but for the case when the rotational energy transfer is small, while the vibrational energy transfer is significant. Frame (e) gives time evolution of projection of angular momentum vector. Geometry of collision is discussed in the text.

Same as Fig. 4, but for the case when the rotational energy transfer is small, while the vibrational energy transfer is significant. Frame (e) gives time evolution of projection of angular momentum vector. Geometry of collision is discussed in the text.

Same as Figs. 4 and 5, but for the case of supercollision. Two sequential Ar-O encounters are easily recognized in frame (b). The high-*J* plateau is seen in frame (c). Parameters of collision are discussed in the text.

Same as Figs. 4 and 5, but for the case of supercollision. Two sequential Ar-O encounters are easily recognized in frame (b). The high-*J* plateau is seen in frame (c). Parameters of collision are discussed in the text.

## Tables

Fitting coefficients of the bending energy correction function of Eq. (70) for ^{16}O^{18}O^{16}O.

Fitting coefficients of the bending energy correction function of Eq. (70) for ^{16}O^{18}O^{16}O.

Spectrum of ^{16}O^{18}O^{16}O for *J* = 19 (*K* _{a} = 4, *K* _{b} = 12) near the dissociation threshold.

Spectrum of ^{16}O^{18}O^{16}O for *J* = 19 (*K* _{a} = 4, *K* _{b} = 12) near the dissociation threshold.

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