Classical dynamics of chemical reactions in a quantum spirit

Abstract

Molecular beam experiments provide fascinating data on how atoms move in the course of chemical reactions. In order to theoretically reproduce these data at relatively low computational cost and to interpret them, nuclei are often treated as classical particles, even though we have known for about a century that, in the range of energies usually available to chemical systems, their motion is best described by the laws of quantum mechanics. Nevertheless, over the last decade this approximation has been shown to work unexpectedly well, provided that a few constraints are introduced into the calculations in order to take into account the quantisation of product internal motions. Why this is so and the nature of the previous constraints are the central issues of this review article.

Keywords:

• Reaction dynamics
• QCT method
• semiclassical approaches
• Bohr quantisation
• Standard and Gaussian binnings
• backward description.

Acknowledgements

The author is especially grateful to Professor Jean-Claude Rayez for about 20 years of stimulating and fruitful discussions and for creating a space of freedom as well as a rich scientific environment in his laboratory. The author is also particularly indebted to Dr Pascal Larrégaray, Dr Cédric Crespos and Dr Philippe Halvick for their close collaboration, and to Dr Aurélie Perrier, Dr Maykel Léonardo González Martínez and Wilmer Arbelo González for their impressive work during their PhD theses. The author also wishes to thank all the colleagues who contributed in different and complementary ways to improving his understanding of molecular collisions and chemical reactivity, in particular, Pr. Vincenzo Aquilanti, Pr. Simonetta Cavalli, Pr. William H. Miller, Pr. Robert Littlejohn, Pr. Toshio Kasai, Pr. Alberto Beswick, Pr. Maurice Monnerville, Pr. Joaquín Espinosa García, Pr. Miguel González, Dr Bruno Lepetit and Dr Foudhil Bouakline. Last but not least, deep appreciation is expressed to Pr. José Gayoso for transmitting his passion for science and poetry.

Notes

1. We will not go into detail regarding the theory of vector observables in this paper. On the one hand, the theory is more complex than for scalar observables. On the other hand, quantisation of product internal motions seems to have more effect on scalar than on vector properties.

2. In Reference [84], q_n , q_j , and q_l are denoted by q_j , α_j and α_l .

3. In Reference [82], GB was stated to be a by-product of CSMT within the random phase approximation. However, the developments of Section 2.2 show that CSMT plus an efficient quenching of interference effects by summing over non-measured quantum numbers are the actual justification of GB.

4. In Reference [84], the sum in Equation (12) should run from to (instead of ). MC in Section 3 is for Monte Carlo. In the first sentence below Figure 6, the word denominator should replace numerator. In addition to that, it is worth emphasising that the reasoning in that sentence is valid because the flux towards one of the two product channels and the flux towards the reagents are roughly identical. The developments in step 1 of Appendix A are originally given by Miller in Reference [10] of [92].

5. In all rigour, Equation (2) must be slightly modified for a photodissociation. The initial quantum numbers in have indeed to be replaced by those specifying the state of the molecule before the optical excitation.

6. The quantum cross section presented here is a smooth fit of the exact results given in Reference [164].

7. On the contrary, we have found that for the reaction D⁺ + H₂ (n ₁ = 0, j ₁ = 0), this domain is negligible. The mass combination for this process is indeed more favourable to energy transfer from translation to vibration motion than for the H⁺ + H₂ (n ₁ = 0, j ₁ = 0) and H⁺ + D₂ (n ₁ = 0, j ₁ = 0) reactions, leading therefore to capture and complex formation with a larger probability.

8. Although the state-correlated translational energy distributions published in [227] have never been theoretically reproduced to date, the three vibrational states available to the products are nicely predicted in Reference [170], as seen at the end of Section 2.9.

9. Equations (A.1) and (A.2) correspond to Equations (A.34) and (A.35) of Reference [84] (the factor in Equation (A.2) is however lacking in Equation (A.35) of Reference [84]). The list of references given in this paper is intended to be representative of the past and present research activity related to the topics of the review. It is of course not exhaustive.