_{2}-H

_{2}collisions: Internal energy and rotational angular momentum conservation effects

^{1,a)}, N. Balakrishnan

^{1}, S. Lepp

^{2}, G. Quéméner

^{3}, R. C. Forrey

^{4}, R. J. Hinde

^{5}and P. C. Stancil

^{6}

### Abstract

We present a full dimensional quantum mechanical treatment of collisions between two H_{2} molecules over a wide range of energies.Elastic and state-to-state inelastic cross sections for ortho-H_{2} + para-H_{2} and ortho-H_{2} + ortho-H_{2}collisions have been computed for different initial rovibrational levels of the molecules. For rovibrationally excited molecules, it has been found that state-to-state transitions are highly specific. Inelastic collisions that conserve the total rotational angular momentum of the diatoms and that involve small changes in the internal energy are found to be highly efficient. The effectiveness of these quasiresonant processes increases with decreasing collisionenergy and they become highly state-selective at ultracold temperatures. They are found to be more dominant for rotational energy exchange than for vibrational transitions. For non-reactive collisions between ortho- and para-H_{2} molecules for which rotational energy exchange is forbidden, the quasiresonant mechanism involves a purely vibrational energy transfer albeit with less efficiency. When inelastic collisions are dominated by a quasiresonant transition calculations using a reduced basis set involving only the quasiresonant channels yield nearly identical results as the full basis set calculation leading to dramatic savings in computational cost.

We thank Roman Krems for many useful discussions and initial collaboration in this project. This work was supported by NSF Grant Nos. PHY-0855470 (N.B.), ATM-0635715 (N.B.), PHY-0854838 (R.C.F.), and AST-0607733 (P.C.S.).

I. INTRODUCTION

II. THEORY

III. RESULTS

A. Computational details

B. Ortho-ortho calculations

1. Vibrational relaxation rate coefficients

C. Ortho-para calculations

IV. CONCLUSION

### Key Topics

- Angular momentum
- 23.0
- Elasticity
- 11.0
- Band gap
- 10.0
- Wave functions
- 10.0
- Elastic collisions
- 8.0

## Figures

Jacobi coordinates for the tetratomic system.

Jacobi coordinates for the tetratomic system.

Elastic cross sections for initial combined molecular states 1001, 1101, 1103, 1201, and 1301 as functions of the incident kinetic energy.

Elastic cross sections for initial combined molecular states 1001, 1101, 1103, 1201, and 1301 as functions of the incident kinetic energy.

Total IN and state-to-state cross sections as functions of the incident kinetic energy for H_{2}(*v* = 1, *j* = 1) + H_{2}(*v* = 0, *j* = 1) collisions.

Total IN and state-to-state cross sections as functions of the incident kinetic energy for H_{2}(*v* = 1, *j* = 1) + H_{2}(*v* = 0, *j* = 1) collisions.

EL, total IN, and state-to-state cross sections as functions of the incident kinetic energy for 1103 collisions. Upper panel: Elastic and total inelastic cross section; middle panel: Total inelastic and dominant inelastic cross sections along with a few less prominent state-to-state cross sections; lower panel: State-to-state cross sections that make negligible contributions to the total inelastic cross section.

EL, total IN, and state-to-state cross sections as functions of the incident kinetic energy for 1103 collisions. Upper panel: Elastic and total inelastic cross section; middle panel: Total inelastic and dominant inelastic cross sections along with a few less prominent state-to-state cross sections; lower panel: State-to-state cross sections that make negligible contributions to the total inelastic cross section.

Total IN and state-to-state cross sections as functions of the incident kinetic energy for H_{2}(*v* = 1, *j* = 3) + H_{2}(*v* = 0, *j* = 1) collisions. The upper panel shows the total inelastic and some dominant state-to-state cross sections while the lower panel shows the negligible contribution from many inelastic channels.

Total IN and state-to-state cross sections as functions of the incident kinetic energy for H_{2}(*v* = 1, *j* = 3) + H_{2}(*v* = 0, *j* = 1) collisions. The upper panel shows the total inelastic and some dominant state-to-state cross sections while the lower panel shows the negligible contribution from many inelastic channels.

Temperature dependence of calculated vibrational relaxation rate constants (solid lines) for ortho-ortho H_{2} collisions. The upper panel shows the initial state resolved rate coefficients for 1101, 1103, and 1301 together with the available results of Pogrebnya *et al.* ^{41} for the 1101 initial state (triangles). The lower panel displays vibrational relaxation rate coefficients averaged over a thermal population of rotational levels in *v* = 0 and *v* = 1 evaluated using Eq. (9). Experimental results of Audibert *et al.* (shown in both panels)^{63} and theoretical results of Flower^{31} are also included for comparison. The rate coefficients for the 1101, 1103, and 1301 initial states in the upper panel, and the averaged vibrational relaxation rate coefficient in the lower panel are provided as supplementary material, see Ref. 62.

Temperature dependence of calculated vibrational relaxation rate constants (solid lines) for ortho-ortho H_{2} collisions. The upper panel shows the initial state resolved rate coefficients for 1101, 1103, and 1301 together with the available results of Pogrebnya *et al.* ^{41} for the 1101 initial state (triangles). The lower panel displays vibrational relaxation rate coefficients averaged over a thermal population of rotational levels in *v* = 0 and *v* = 1 evaluated using Eq. (9). Experimental results of Audibert *et al.* (shown in both panels)^{63} and theoretical results of Flower^{31} are also included for comparison. The rate coefficients for the 1101, 1103, and 1301 initial states in the upper panel, and the averaged vibrational relaxation rate coefficient in the lower panel are provided as supplementary material, see Ref. 62.

Inelastic cross sections as functions of the incident kinetic energy for the 1201 collisions. Upper panel: total inelastic cross section and leading state-to-state cross sections; Middle panel: Cross sections for all other inelastic channels (the legend follows the order of the cross section values at 100 K); Bottom panel: Partial cross sections for different values of *J* that contribute to the resonance peaks at 0.07 K, 1.2 K, and 18.3 K. The various partial cross sections are labeled by *J*ε_{ I }.

Inelastic cross sections as functions of the incident kinetic energy for the 1201 collisions. Upper panel: total inelastic cross section and leading state-to-state cross sections; Middle panel: Cross sections for all other inelastic channels (the legend follows the order of the cross section values at 100 K); Bottom panel: Partial cross sections for different values of *J* that contribute to the resonance peaks at 0.07 K, 1.2 K, and 18.3 K. The various partial cross sections are labeled by *J*ε_{ I }.

Inelastic cross sections as functions of the incident kinetic energy for 1001 collisions. The legend follows the decreasing order of the cross section values at 100 K.

Inelastic cross sections as functions of the incident kinetic energy for 1001 collisions. The legend follows the decreasing order of the cross section values at 100 K.

Comparison between cross sections obtained using the full basis set and a reduced basis set for 1001 collisions. The reduced basis set included only the quasiresonant channels. Upper panel: comparison between EL and total IN cross sections; Lower panel: comparison between the dominant inelastic cross sections.

Comparison between cross sections obtained using the full basis set and a reduced basis set for 1001 collisions. The reduced basis set included only the quasiresonant channels. Upper panel: comparison between EL and total IN cross sections; Lower panel: comparison between the dominant inelastic cross sections.

## Tables

Rovibrational energies of various combined molecular states of the two ortho-H_{2} molecules included in the calculations. Energies are relative to the bottom of the H_{2} potential and the zero-point energy of the ortho-ortho complex in this case is 4595.263 cm^{−1}. Some of the channels that are excluded in the calculations by imposing a cut-off energy are also shown.

Rovibrational energies of various combined molecular states of the two ortho-H_{2} molecules included in the calculations. Energies are relative to the bottom of the H_{2} potential and the zero-point energy of the ortho-ortho complex in this case is 4595.263 cm^{−1}. Some of the channels that are excluded in the calculations by imposing a cut-off energy are also shown.

Percentage contributions of selected final states to the total inelastic cross section for the 1101 initial state from calculations assuming the molecules to be distinguishable (upper panel) and indistinguishable (lower panel).

Percentage contributions of selected final states to the total inelastic cross section for the 1101 initial state from calculations assuming the molecules to be distinguishable (upper panel) and indistinguishable (lower panel).

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