Chemical physics

A delayed reaction

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The simple hydrogen exchange reaction was well understood until an unexpected effect emerged in detailed experimental measurements of the process. An explanation for this effect has now been found.

The scattering of a hydrogen atom from a hydrogen molecule (H + H2 → H2 + H) is the simplest possible chemical reaction, as it involves only three nuclei and three electrons. This reaction has served as a benchmark system for understanding chemical reaction dynamics for nearly a century, and quantum-mechanical calculations of the electronic aspects of the process have been made with unrivalled accuracy. As a result, quantitative agreement between theory and experiment for the related reaction involving a deuterium molecule, H + D2 → HD + D, was achieved in the middle of the last decade1.

But more refined experiments and improved theoretical methods have since raised an intriguing question about the dynamics of this reaction at the quantum level2,3. Although the reaction is dominated by a direct recoil mechanism — the incident hydrogen atom recoils along its original path after abstracting a deuterium atom to form the HD product molecule — some of the more subtle aspects of the reaction dynamics revealed by recent experiments have been traced to a second, slower mechanism that occurs with a time delay of around 25 femtoseconds2. The question concerns the physical origin of this time delay, which has become the subject of some debate3,4. On page 281 of this issue, Harich et al.5 provide an explanation.

In the H + D2 → HD + D reaction, effects beyond the direct recoil mechanism are expected to become apparent only when the quantum states of the reactant and product molecules are directly interrogated. A useful, measurable quantity is the 'state-to-state differential cross-section' — the probability for the reaction to proceed from reactants in a specific initial quantum state to products in a specific final quantum state as a function of the scattering angle between them. This quantity was first measured for the H + D2 reaction at a single collision energy by Welge and co-workers1 in the 1990s.

Following the suggestion of Miller and Zhang6, Althorpe et al.2 have since measured the state-to-state differential cross-section over a range of collision energies, in the hope of gaining greater insight into the reaction dynamics of the H + D2 → HD + D process. By varying the wavelength of the photolysis laser used to photodissociate HBr precursor molecules (and hence generate the reagent H atoms), they were able to measure the differential cross-section for collision energies from 1.39 to 1.85 eV. Their reward was the observation, at a collision energy of around 1.64 eV, of an intriguing forward-scattering peak for the HD product molecule in a quantum state with a vibrational quantum number of three and a rotational angular momentum quantum number of zero. A theoretical analysis by the same authors2 of this forward scattering showed that it was delayed relative to the backward-scattering recoil mechanism by some 25 femtoseconds, but their analysis did not produce a physical explanation.

In fact, there are two conceivable explanations for such a time delay3. The more obvious is a simple 'threshold' effect: as the reactants approach a potential-energy barrier they lose classical kinetic energy and slow down. This situation is depicted in Fig.1a, which shows the symmetric scattering wavefunction for a one-dimensional potential-energy barrier when the reactant energy is equal to that of the barrier maximum. The slowing down of the reactants in classical mechanics is reflected by an increase in the amplitude of the quantum-mechanical wavefunction Ψ, and hence an increase in the probability |Ψ|2 of finding the system in the region of the barrier maximum. But because the reactants simply slow down at the top of the barrier rather than becoming trapped there, the wavefunction associated with this reaction threshold effect still has significant amplitude away from the barrier maximum.

Figure 1: Wavefunctions and barriers.
figure1

There are two possible explanations for the time delay seen in forward scattering in the reaction H + HD → H2 + D. a, The scattering energy (dashed black line) coincides with the top of a potential-energy barrier (solid blue line); crossing this 'threshold' causes the reactants to slow down and leads to the increased amplitude of the quantum-mechanical wavefunction (solid red line) at the barrier maximum. b, A 'quasi-bound' quantum state exists in the well between the double maxima of a potential-energy barrier (solid blue line) before decaying into reaction products. The highly localized wavefunction (solid red line) indicates a 'resonance' effect. Harich and co-workers' analysis5 of wavefunctions in the H + HD → H2 + D reaction suggests that the time delay is due to the threshold effect, not the resonance process.

The second explanation does not have a classical analogue, and is therefore a little more exotic. It is possible for the reactants to become trapped on the potential-energy surface, forming a 'quasi-bound' quantum state. This situation also leads to a time delay, owing to the time it takes for the quasi-bound state to decay into reaction products, and it is known as a quantum reactive scattering resonance. The simplest example of such a resonance is illustrated in Fig. 1b, which shows the computed scattering wavefunction at the energy of a quasi-bound state supported by the well between the two potential-energy maxima of a double barrier.

Although these two explanations are not unrelated, because one situation can be deformed into the other by changing the potential-energy surface7, they are mathematically different8,9, and the physical implications of this mathematical difference are too significant to regard the two situations as manifestations of the same phenomenon. Furthermore, it is clear from Fig. 1 that the scattering wavefunction behaves very differently in the threshold and resonance situations.

This last distinction has been exploited by Harich et al.5 to elucidate the origin of the time delay associated with forward scattering in the hydrogen exchange reaction. Rather than study the H + D2 version of the reaction, they used a crossed molecular beam apparatus to study the related H + HD → H2 + D reaction, at a collision energy of 1.2 eV. State-to-state differential cross-sections were measured by Rydberg tagging the product deuterium atom (and thereby inferring the quantum state of the hydrogen product molecule), a powerful and highly sensitive technique that was first used for the H + D2 reaction by Welge and co-workers1. Although the experiment was only done for a single collision energy, a distinct forward-scattering peak was seen in the differential cross-section of the H2 product with a vibrational quantum number of zero and a rotational angular momentum quantum number of one.

This forward peak in the H + HD reaction has similar characteristics to that seen in the H + D2 reaction by Althorpe et al.2. Both peaks occur for product states with low rotational quantum numbers, and the theoretical analysis of Harich et al.5 shows that the forward scattering in the H + HD reaction is again associated with a time delay, in this case of around 20 femtoseconds. But this analysis goes further and extracts the quantum-mechanical wavefunction that underlies the time-delayed reaction mechanism. The wavefunction turns out to be like the one shown in Fig. 1a, so Harich et al. conclude that the time delay in forward scattering in the H + HD reaction is caused by a reaction threshold effect rather than a reactive scattering resonance. The same is likely to be true in the case of the H + D2 reaction.

This is, in fact, the best possible outcome for the field. A genuine quantum reactive scattering resonance has recently been identified through a theoretical analysis of integral and differential cross-section measurements on the F + HD → HF + D reaction10. Hence, we now have concrete examples of both phenomena — reactive thresholds and reactive resonances — and the effects they have on experimental observations of the dynamics of chemical reactions.

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Correspondence to David E. Manolopoulos.

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Manolopoulos, D. A delayed reaction. Nature 419, 266–267 (2002) doi:10.1038/419266a

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