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Quantum interference in H + HD → H2 + D between direct abstraction and roaming insertion pathways

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Science  15 May 2020:
Vol. 368, Issue 6492, pp. 767-771
DOI: 10.1126/science.abb1564

Intriguing interference mechanism

Quantum interference (QI) effects play a fundamental role in the dynamics of chemical reactions. Xie et al. detected unusual QI oscillations in the differential cross section measured in the recoil scattering direction of the prototypical elementary reaction H + HD → H2 + D (see the Perspective by Aoiz). Topological analysis showed that this pattern originates from the QI between a direct abstraction and previously unknown rebounding insertion pathways, which are affected by the geometric phase at energies far below the conical intersection. The QI observed between these two distinctive pathways in the three-atom system is a clear example of the quantum nature of chemical reactivity.

Science, this issue p. 767; see also p. 706

Abstract

Understanding quantum interferences is essential to the study of chemical reaction dynamics. Here, we provide an interesting case of quantum interference between two topologically distinct pathways in the H + HD → H2 + D reaction in the collision energy range between 1.94 and 2.21 eV, manifested as oscillations in the energy dependence of the differential cross section for the H2 (v′ = 2, j′ = 3) product (where v′ is the vibrational quantum number and j′ is the rotational quantum number) in the backward scattering direction. The notable oscillation patterns observed are attributed to the strong quantum interference between the direct abstraction pathway and an unusual roaming insertion pathway. More interestingly, the observed interference pattern also provides a sensitive probe of the geometric phase effect at an energy far below the conical intersection in this reaction, which resembles the Aharonov–Bohm effect in physics, clearly demonstrating the quantum nature of chemical reactivity.

Quantum effects are ubiquitous in chemical reactions, which are, in essence, atomic and molecular collision processes. Thus, understanding quantum effects such as reaction resonance, quantum tunneling, and quantum interference (QI) in chemical reactions is crucial in the study of chemical reaction dynamics (1). Experimental and theoretical studies of reaction resonances in elementary chemical reactions, such as F + H2 and Cl + H2, have provided deep insights into the quantum nature of transition states in chemical reactions (25). Quantum tunneling was also found to be essential in bimolecular reactions (6, 7) as well as in biochemical reactions at low temperatures (8, 9). A more general quantum phenomenon in atomic and molecular collisions is QI, and often the study of QI is vital to understanding the dynamics of inelastic bimolecular collisions (1013), chemical reactions (1416), and photochemical reactions (17, 18). In a typical elementary chemical reaction, normally there is a single reaction pathway that goes through only one transition state. QI in such a reaction arises between partial waves with different total angular momenta (J), which belong to the same topological reaction path (19) and are usually manifested in the differential cross section (DCS) or the scattering angular distribution, as in the case of the H + D2 reaction (15, 16). However, a QI phenomenon in a chemical reaction is not easy to observe, because the averaging over a range of scattering angles and over many partial waves usually blurs any pattern expected to be associated with interferences (11, 15).

For molecular dynamics processes with more than one pathway, especially more than one topologically distinct pathway, QI could occur between different pathways, and a few of them have been identified (15, 1720). For example, QI between different conical intersection (CI) pathways was observed in the unimolecular dissociation of the water molecule, manifested as oscillations in product rotational state distribution (17). A similar QI phenomenon was also probed in the HCO photodissociation (18). In a bimolecular reaction with CIs between the ground and excited electronic states, the reaction could occur through two topologically different pathways around the CI (fig. S1), and QI occurs strongly between them (19). That is the case for the H + H2 → H2 + H reaction and its isotopologs, which have a well-characterized CI with a D3h symmetry at 2.75 eV in total energy (21).

The H + H2 → H2 + H reaction and its isotopologs have been the most important benchmark systems for studies of interesting quantum dynamics in chemical reactions (2226). The first collinear quantum dynamics calculations were carried out on this system in the early 1970s by Schatz and Kuppermann (27). Full-dimensional quantum dynamics calculations were then conducted in 1988 (28). Among all the interesting quantum dynamics effects in this system, an intriguing effect involving its CI has been the geometric phase (GP) effect (29). There has been great effort to search for the GP effect in this system, both experimentally and theoretically. Kuppermann and co-workers (30, 31) investigated the GP effect on the H + H2 reaction using the multivalued basis approach, which was revealed to be erroneous by later theoretical studies (3234) and experimental results (23, 35). Quantum dynamics calculations by the Kendrick and Althorpe groups showed that the GP effect is negligible at a total energy below 1.6 eV (34, 3640), whereas the GP effect could be considerable at high collision energies. However, how to effectively detect the elusive GP effect experimentally remained a big challenge.

Searching experimentally for the GP effect in the H + H2 reaction continued in the past few decades (16, 2226), with no evidence detected until recently. In 2018, a high-resolution crossed-beam imaging study on the H + HD → H2 + D reaction at a collision energy of 2.77 eV (41), which is 0.24 eV above the CI, observed the GP effect by measuring H2 product state-resolved angular distributions. Through accurate theoretical analysis, a previously uncovered reaction channel other than the direct abstraction channel was inferred at this high collision energy. This distinctive channel at this collision energy is topologically different from the direct abstraction channel. More interesting questions thus follow: Can we directly detect strong QI between the two topologically distinct reaction pathways in the H + HD → H2 + D reaction, and what is the exact mechanism of this distinctive reaction pathway?

A preliminary theoretical study shows that for certain specific rovibrationally excited H2 products from the H + HD → H2 + D reaction, strong oscillations are present in the energy dependence of the DCS in the backward scattering direction. These strong oscillations are intriguing, and the dynamics origin of these DCS oscillations in the backward scattering direction is not explicit. We thus initiated a combined experimental and theoretical study of this interesting phenomenon.

The experiment was carried out using an improved crossed-beam apparatus based on the D atom Rydberg tagging method shown in fig. S2 in the supplementary materials (SM) (42). A molecular beam of pure HI was generated by supersonic expansion with a stagnation pressure of 1.7 bar from a pulsed valve. A deep ultraviolet laser at 213 nm with a pulse energy of about 10 mJ, which was produced by the fifth harmonic of a yttrium-aluminum-garnet–Nd laser (YAG-Nd) laser, was focused to a spot of ~1 mm in diameter and intersected the HI beam about 5 mm downstream from the nozzle to dissociate the HI molecule. The fast H atoms with a velocity of 22,945 m/s, corresponding to the ground-state iodine atom product, were selected to cross the HD beam, which was generated by a supersonic expansion through a pulsed valve cooled to liquid nitrogen temperature. The cold expansion ensured that nearly all HD molecules in the beam populated the lowest rovibrational state (v = 0, j = 0) (where v is the vibrational quantum number and j is the rotational quantum number). The velocity of the HD molecules was determined to be 1240 m/s, with a speed ratio better than 20. The HD beam was rotatable around the center of the crossing region, so it was possible to vary the collision energy in the range of 1.94 to 2.21 eV by changing the crossing angle from 55° to 130°, with an energy interval of about 0.02 eV. The product D atom was detected by the D atom Rydberg tagging technique (22). Time-of-flight (TOF) spectra at different collision energies were accumulated at the corresponding backward scattering direction. To reduce the experimental errors, TOF spectra were accumulated by scanning the collision energies back and forth more than 100 times.

Figure 1A shows the D atom TOF spectra at five different collision energies. The TOF spectra measured at various collision energies show many sharp peaks, which can be clearly assigned to rovibrational states of the H2 product from the reaction of H + HD (v = 0, j = 0). Some of the peaks, such as the H2 (v′ = 0, j′ = 5) product, are rather well resolved, whereas some are partially resolved, such as the H2 (v′ = 2, j′ = 3) product, which partially overlaps with H2 (v′ = 1, j′ = 9) and H2 (v′ = 2, j′ = 4) (Fig. 1B). From the partially resolved TOF spectral features of the D atom, it was possible to extract the accurate peak height of the corresponding H2 (v′ = 2, j′ = 3) product for all collision energies in the backward scattering direction by fitting the experimental spectra at all collision energies using three Gaussian peaks for the corresponding quantum states (Fig. 1B). The obtained peak height was used to compute the DCS for the H2 (v′ = 2, j′ = 3) product in the backward scattering direction.

Fig. 1 TOF spectra of the D atom product from the H + HD → H2 + D reaction.

(A) The D atom TOF spectra in the backward scattering direction (180° ± ~5°) of the reaction of H + HD → H2 + D at five selected collision energies: 1.958 (black), 1.990 (red), 2.050 (green), 2.162 (blue), and 2.211 eV (magenta). The peaks marked by asterisks are contributed from H2 (v′ = 2, j′ = 3), H2 (v′ = 1, j′ = 9), and H2 (v′ = 2, j′ = 4). arb. unit, arbitrary unit. (B) The Gaussian fit of the peak marked by the asterisk at the collision energy of 1.958 eV. The black squares are the experimental data. The red, blue, and green solid lines are the Gaussian fits for H2 (v′ = 2, j′ = 3), H2 (v′ = 1, j′ = 9), and H2 (v′ = 2, j′ = 4), respectively. The magenta solid line is the sum of the three Gaussian peaks.

Figure 2 shows the DCS for the H2 (v′ = 2, j′ = 3) product in the backward scattering direction in the collision energy range between 1.94 and 2.21 eV. The error bars in this figure were estimated by analyzing the signal fluctuation in these scans and should be taken as ±1 SD of uncertainty. As the collision energy increased, the DCS for H2 (v′ = 2, j′ = 3) in the backward scattering direction clearly underwent strong oscillations (up-down-up-down). This oscillation can also be roughly seen from the corresponding energy dependence of the peak height in the TOF spectra shown in Fig. 1A, as it becomes higher (1.990 eV) and lower (2.050 eV), and then becomes higher (2.162 eV) and lower (2.211 eV) again with increasing energy.

Fig. 2 Comparison of the backward scattering for product H2 (v′ = 2, j′ = 3) from theory and experiment.

Experimental (black dots) and theoretical (lines) DCS for the backward-scattering H2 (v′ = 2, j′ = 3) product (with uncertainty about ±5°) over the collision energy range between 1.94 and 2.21 eV. The red and blue lines are the NGP and GP theoretical results, respectively, where the experimental angular broadening has been convoluted. The error bars were estimated by analyzing the signal fluctuations and taken as ±1 SD of uncertainty.

In an effort to understand the origin of these oscillations in the collision energy dependence of the DCS for the H2 (v′ = 2, j′ = 3) product, we carried out quantum dynamics calculations and analysis on the refined Boothroyd–Keogh–Martin–Peterson (BKMP2) potential energy surface (PES) (43) using the time-dependent quantum wave-packet approach (44, 45). The corresponding theoretical results shown in Fig. 2 are convoluted by including the overall angular broadening of the crossed-beam experiment, which is about 10.0° in the center-of-mass frame. For more details on the theoretical simulation of the DCS, see the SM. We first carried out quantum dynamics calculations using the adiabatic PES (without the upper cone of the CI) and including no GP (NGP) effects. There were clear oscillations in the DCS from the NGP calculations for H2 (v′ = 2, j′ = 3) in the backward scattering direction (Fig. 2). Surprisingly, the oscillations calculated without GP effects were almost completely out of phase with the experimentally observed oscillations, suggesting the possibility that the GP has a large effect on the observed oscillations.

We then performed time-dependent quantum dynamics calculations using the same adiabatic BKMP2 PES but including the GP as a vector potential, as has been done previously (34, 37, 41). The calculated DCS for the H2 (v′ = 2, j′ = 3) product by including the GP, shown in Fig. 2, is now in notable agreement with the experimental result. This clearly demonstrates that the GP could have a profound effect on the dynamics of the H + HD → H2 + D reaction, even at energies well below the CI. To verify this, we performed quantum dynamics calculations based on the diabatic PES, including the lowest two electronic states of the H3 system, which are strongly coupled through the CI at a total energy of about 2.75 eV. The diabatic results agree well with the GP results (fig. S4). This verifies that the adiabatic results that include the GP should be accurate enough for this reaction.

Observation of the GP effect on the energy dependence of the DCS is intriguing, especially at an energy substantially below the CI. In an effort to understand the origin of the oscillations in the energy dependence of the DCS in the backward direction for the H2 (v′ = 2, j′ = 3) product, theoretical results in the NGP and GP calculations are presented in an expanded collision energy range (Fig. 3A). Notable oscillations of the DCS as a function of collision energy were observed in the two sets of calculations, with the oscillations of the NGP results completely out of phase with those of the GP ones. Such oscillations were also observed for many other rovibrational states of the H2 product (fig. S5).

Fig. 3 Backward scatterings and their phases for product H2 (v′ = 2, j′ = 3) from theory in an expanded energy range.

(A) The collision energy dependence of the DCS in the backward scattering direction for the H2 (v′ = 2, j′ = 3) product from the GP and NGP calculations and the backward scattering for product H2 (v′ = 2, j′ = 3) generated through path 1 and path 2 in a range of collision energies. a.u., arbitrary units. (B) The phases Φ1(E) and Φ2(E) calculated from scattering amplitudes f1(θ = 180°, E) and f2(θ = 180°, E) for path 1 and path 2. rad., radians.

In our previous work, we applied the topological theory proposed by Althorpe et al. (19, 34, 37) to show that there are two possible reaction paths in the H + HD reaction: One goes clockwise around the CI through a single transition state (path 1), and the other passes through two transition states in an opposite direction around the CI (path 2). These two paths interfere with each other, which leads to the angular oscillation pattern change in the forward scattering sphere at a particular collision energy.

In the present work, we similarly analyze the energy-dependent oscillations in the DCS in the backward scattering direction observed using the topological theory. According to this theory, the scattering nuclear wave functions for path 1 and path 2 can be expressed explicitly by ψ1=(ψNGP+ψGP)/2 and ψ2=(ψNGPψGP)/2, respectively, where ψNGP and ψGP are the calculated scattering wave functions without and with the GP, respectively. The scattering amplitudes from path 1 and path 2 in the backward scattering direction can thus be calculated using the following expressions:

f1(E)=[fNGP(E) + fGP(E)]/2(1)f2(E)=[fNGP(E)  fGP(E)]/2(2)

Because f1(E) and f2(E) are both complex functions, they can be written as f1(E) = |f1(E)|*exp[1(E)] and f2(E) = |f2(E)|*exp[2(E)], where Φ1(E) and Φ2(E) are the phases of the scattering amplitudes of the two paths. The square moduli of f1(E) and f2(E) give the product angular distribution, that is, the DCS at collision energy E in the backward scattering direction,σNGP(E)=|fNGP(E)|2=|f1(E) + f2(E)|22={|f1(E)|2+|f2(E)|2+2|f1(E)|*|f2(E)|*cos[Φ1(E)Φ2(E)]}/2 and

σGP(E)= |fGP(E)|2 =|f1(E)f2(E)|22=(|f1(E)|2+|f2(E)|2+2|f1(E)|*|f2(E)|*cos{π+[Φ1(E)Φ2(E)]})/2

Using Eqs. 1 and 2, it is possible to obtain |f1(E)|, |f2(E)|, and Φ1(E) and Φ2(E) in the backward scattering direction, and they are shown in Fig. 3. Surprisingly, the |f1(E)| and |f2(E)| functions were both rather smooth curves, and the two phases were roughly linearly dependent on the collision energy above 1.3 eV. More interestingly, Φ1(E) decreased with the collision energy, and Φ2(E) increased with the collision energy. The relative phase between the two paths, |Φ1(E)Φ2(E)|, thus increased rapidly and caused fast oscillations in the energy dependence of the DCS in the backward scattering direction for the H2 (v′ = 2, j′ = 3) product.

From the above analysis, it is clear that the oscillations in both σNGP(E) and σGP(E) were caused by the crossing terms between the two pathways, with a phase difference of π, which is exactly the relative GP change between the two reaction paths. Thus, we concluded that the observed oscillations in the energy dependence of the DCS of the H2 (v′ = 2, j′ = 3) product can only be attributed to the QI between the two reaction pathways. To the best of our knowledge, such a QI pattern in the form of energy-dependent DCS oscillations between two topologically distinct reaction pathways in a chemical reaction has not been observed and understood previously.

It is also quite notable that it was possible to detect the QI, or the GP effect, for the H2 (v′ = 2, j′ = 3) product channel in the backward scattering direction at a collision energy as low as 1.94 eV, because the reactivity of path 2 was only 0.28% of path 1 at this energy. Theoretically, one could also see the oscillations at a collision energy as low as 1.5 eV, which means that the GP effect in this system could theoretically be seen at a collision energy down to 1.5 eV, much lower in energy than its CI. This clearly shows that the oscillations in the energy dependence of the state-resolved DCS provide a distinctive way to investigate QI and the GP effect at a low collision energy in this benchmark system. Further analysis shows that the DCS contribution to the H2 (v′ = 2, j′ = 3) product from path 2 at this energy was predominantly backward scattered (fig. S6), which thus made the oscillations in the energy dependence of the DCS more easily observable in the backward scattering direction. The dynamics mechanism of this backward scattered reaction path (path 2) must be different from previous findings about path 2 in the H + H2 reaction by Althorpe and co-workers, where they focused on the forward scatterings and a direct insertion mechanism was found (34, 37).

To understand the reaction mechanism of path 2 in the classical dynamics picture, quasi-classical trajectory (QCT) calculations were carried out on the adiabatic BKMP2 PES. Although the QCT theory breaks down when there are strong quantum effects, it could nevertheless provide an intuitive picture of the reaction mechanism (19). The QCT methodology is described in the SM. A total of 50 million trajectories were propagated at a 2.01-eV collision energy, giving an overall statistical error of 0.046% in the total reactive cross section. Of these, 4.03 million trajectories were found to be reactive to produce the H2 product, with ~99.77% reacted through path 1 and only ~0.23% (9256 trajectories) through path 2. This ratio is quite consistent with the quantum scattering calculations.

The reaction mechanisms for the two paths are different. Path 1 is the well-known direct abstraction mechanism, and the mechanism of path 2 is interesting. The snapshots of a representative QCT trajectory of the path 2 mechanism for backward scattered H2 (v′ = 2, j′ = 3) product are shown in a movie sequence (Fig. 4, A to F). These snapshots show that the incoming H atom initially approaches the HD molecule via the CI region toward the D atom end, then roams around the D atom in HD. When the incoming H atom approaches the CI region, the HD bond starts to stretch, making it possible for the roaming H atom to insert into the stretched HD molecule. The incoming H atom then forms a new chemical bond with the H atom in HD to scatter mostly into the backward sphere. To the best of our knowledge, such a distinctive insertion reaction pathway with predominantly backward scattering has not been observed previously in any other elementary direct chemical reactions. The roaming insertion mechanism accounts for nearly all trajectories of product that was backward scattered through path 2 around this collision energy. However, previously in the H + H2 reaction, the direct insertion mechanism for path 2 at a much higher energy was found, which was forward scattered (37). The movies of the two representative trajectories of these two reaction mechanisms are also provided in the SM for reference (movies S1 and S2).

Fig. 4 Representative classical trajectories for the H + HD → H2 + D reaction by the roaming mechanism.

(A to F) A representative trajectory of the H + HD → H2 (v′ = 2, j′ = 3) + D reaction in the backward scattering direction (θ = 173°) via the roaming mechanism (path 2) moving with time in Cartesian coordinates. The black curves represent the trajectories of the incoming H atom. The positions of the atoms are plotted at a series of time intervals (23, 33, 47, 62, 74, and 83 fs) in frames (A) to (F) on top of the potential energy surface in a space-fixed frame located at the center of the mass of reactant HD molecule. The color map shows the potential energy surface captured at times 23, 33, 47, 62, 74, and 83 fs for frames (A) to (F), respectively. In the map, blue color indicates the lowest energy, red color indicates the highest energy, and green color indicates the intermediate energy. The crosses indicate the locations of the CIs.

From the above study, we concluded that the interesting oscillations in the energy dependence of the DCS observed for the H2 (v′ = 2, j′ = 3) product in the backward scattering direction were caused by the QI between the two topologically distinct reaction paths. To stress the difference between these two kinds of trajectories, typical trajectories calculated by the QCT theory, which correspond to the snapshots in Fig. 4 and movies S1 and S2, were plotted in the hyperspherical coordinates (46) with a fixed hyperspherical radius (ρ) in Fig. 5. Reaction by path 1 proceeds in clockwise direction around the CI by passing one transition state with the usual direct abstraction mechanism, whereas reaction by path 2 occurs through a topologically different pathway in a counterclockwise direction around the CI by passing two transition states, with a distinctive roaming insertion mechanism to produce mainly backward scattering products. This distinctive reaction path allows us to sensitively probe the QI between the two paths at the backward scattering direction. According to the topological argument proposed by Althorpe and co-workers (19), the two components of the nuclear wave function encircling the CI in different directions belong to two different branching spaces or two different homotopy classes, which interfere quantum mechanically with each other in the product channel. The inclusion of the GP altering their relative phases of the scattering amplitudes from these two pathways thus changes the interference pattern. The interference pattern observed in this work can thus be used to sensitively probe the GP effect at an energy far below the CI. The picture of QI between the two topologically distinct pathways in a chemical reaction presented here also resembles the Aharonov–Bohm experiment, (47) providing an excellent example of quantum effect in chemical reactions.

Fig. 5 Representative direct and roaming trajectories in hyperspherical coordinates.

(A and B) Typical trajectories calculated by the QCT theory for the direct abstraction reaction path (A) and the roaming insertion path (B) in the hyperspherical coordinates with a fixed ρ. The pink crosses indicate the locations of the CIs, and the blue lines, which indicate transition states (T), separate three different atom-diatom channels. The trajectory in (A) passes over only one transition state, but the trajectory in (B) passes over two transition states. These two trajectories are the same as those used to generate movies S1 and S2.

Supplementary Materials

science.sciencemag.org/content/368/6492/767/suppl/DC1

Materials and Methods

Figs. S1 to S6

Tables S1 and S2

References

Movies S1 and S2

Data S1

References and Notes

Acknowledgments: Funding: This work was supported by the National Natural Science Foundation of China (nos. 21688102, 21590800, 21825303, and 21822305), the Chinese Academy of Sciences (grant no. XDB 17010000), and the Ministry of Science and Technology. Author contributions: Y.X., Y.W., T.W., C.X., and X.Y. performed the crossed-beam experiments and data analysis. H.Z., X.X., Z.S., and D.H.Z. performed the quantum dynamics calculations and data analysis. Y.H. and Z.S. performed the QCT analysis. C.X., Z.S., D.H.Z., and X.Y. designed the research. C.X., Z.S., D.H.Z., and X.Y. wrote the manuscript. Competing interests: The authors declare no competing interests. Data and materials availability: All data are available in the supplementary data file.

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