Sideways scattering oscillations reveal geometric phase effect and isotope effect in the H + D_2 → HD + D reaction

Domain: physics
Published: 2025-12-16
Editorial outcome: accept

Abstract
Quantum interference between reaction pathways plays a crucial role in understanding the microscopic mechanisms of chemical reactions. The hydrogen exchange reaction, involving a well-characterized conical intersection, exhibits rich dynamics arising from interference between multiple reactive pathways. Here, we report high-resolution scattering measurements at a collision energy of 2.38 eV, which reveal pronounced sideways angular oscillations in the HD product distribution. Together with exact quantum theoretical calculations, these results provide unambiguous evidence of geometric phase-induced quantum interference in the H + D2 reaction. Detailed analysis reveals that the observed sideways scattering oscillations, hallmarks of quantum interference, are strongly influenced by transient D–D bond elongation in D2 reactants. This behavior, in contrast to the reaction with HD, highlights a pronounced isotopic effect and reflects the unique dynamic competition along the roaming insertion pathway. Collectively, these findings underscore the critical role of pathway interference and provide new mechanistic insights into how the isotopic effect modulated the geometric phase in an elementary chemical reaction.

Introduction
Understanding chemical reaction pathways, the detailed routes through which reactants transform into products, is fundamental to the study of chemistry1–5. Investigating these pathways not only clarifies the final products of a reaction, but also provides essential insights into the underlying mechanisms that govern chemical transformations. Such insights enable the potential control of reaction rates, product yields, and branching ratios6–10, and are particularly vital in fields like atmospheric chemistry, interstellar chemistry, combustion science, and chemical engineering2–10.

In a molecular reaction system involving a conical intersection (CI), when nuclear motion traces a closed loop around a CI, the change in the electronic wave function must be compensated by a phase change in the nuclear wave function to maintain the single-valuedness of the total wave function. This topological effect, known as the geometric phase (GP), plays a central role in quantum reaction dynamics11–14. According to topological theory, adiabatic and non-adiabatic wave functions can be recombined to extract two types of distinct reaction pathways, corresponding to odd and even numbers of crossings around the CI15.

The H + H2 → H2 + H reaction and its isotopic variants, featuring a D3h-symmetric CI at a total energy of 2.75 eV, have served as benchmark systems for studying the GP effect and associated quantum interference phenomena12,16,17. Notably, in the H3 reaction, pathways that loop around the CI more than once are effectively nonexistent, simplifying the dynamics to two fundamental pathways: path 1 (with 0 CI crossings, i.e., direct pathway) and path 2 (with 1 CI crossing, i.e., looping pathway). When two such distinct pathways, each represented by a different wave function, lead to the same final state, quantum interference arises. This interference could manifest as measurable oscillatory patterns in the reaction observables18–23. These distinctive interference patterns in reactive scattering offer fundamental insights into the quantum dynamics of elementary chemical reactions2–10,24,25.

Over the past few years, by combining high-resolution reactive scattering experiments with highly accurate theoretical calculations, researchers have been able to directly capture key information about the two fundamental reaction pathways in hydrogen exchange reactions. In particular, the reaction H + HD → H2 + D has been well studied, providing compelling evidence for these quantum effects16,21,22,26. Notably, the GP effect has been experimentally observed in the H + HD → H2 + D reaction21,22,26, where it demonstrably modulates the reaction dynamics via pathway interference. The experimental identification of quantum interference between the direct abstraction pathway and the roaming insertion pathway represents a significant breakthrough in understanding reaction mechanisms at the quantum level.

While the GP effect has been well identified in H + HD reactions16,20,21,27, experimental studies on the isotopic variant H + D2 system, despite extensive investigation at collision energies up to 3.26 eV28–32, have so far been unable to discover any GP signatures. This is particularly surprising given that the H + D2 reaction yields only a single product channel (HD + D). The challenge lies in disentangling the complex quantum-mechanical behavior of the system, particularly the subtle interference patterns encoded in the nuclear wave functions. Therefore, a detailed quantum-level investigation of the reaction pathways in this system is indispensable.

In this work, we probe the quantum dynamics of the H + D2 reaction by performing high-resolution scattering experiments on the H + o-D2 → HD + D reaction at a collision energy of 2.38 eV. The measurements reveal clear angular oscillatory structures in the quantum state specific differential cross sections (DCSs) for HD products, with pronounced sideways scattering as a characteristic feature. By combining these observations with accurate quantum theoretical calculations, we unambiguously identify the GP effect in the H + D2 reaction. Further analysis shows that the interaction between the incoming H atom and the D2 molecule elongates the D–D bond, promoting product formation via Path 2, and demonstrates the role of the potential barrier height in modulating the GP effect. In contrast to the H + HD reaction, the results reflect that the isotopic effect plays a key role in relative contributions of Path 2 in the two systems. These findings provide the definitive evidence for the GP effect in the H + D2 reaction, highlighting the crucial role of the isotopic effect in modulating reactive quantum interference and the GP effect. Our results offer a new platform for understanding fundamental quantum effects in chemical reactivity and underscore the importance of coherent pathway analysis in elementary chemical reactions.

Results
A rotational state-resolved experimental image was obtained using the crossed molecular beams method combined with the velocity map ion imaging technique. The D-atom products were probed using an near-threshold laser ionization scheme. (for details, see Methods). The H-atom beam was generated by photolyzing HI at 213 nm (5th harmonic of an Nd:YAG laser, Continuum Powerlite DLS9020; produced via sum frequency mixing of 1064 nm fundamental and 266 nm 4th harmonic using a BBO crystal). The faster H-atom component (22.89 km/s) was selected. The D2 beam (1030 m/s) had ~92% in (v=0, j=0). The H and D2 beams crossed at 160°, with the D-atom product velocity map ion image shown in Fig. 1 (collision energy 2.38 eV).

In Fig. 1, the concentric rings in the image of D-atom products correspond to the ro-vibrational states of the co-product HD molecules. The red arrows in Fig. 1 indicate the specific HD product states: (v’=3, j’=12), (v’=4, j’=8) and (v’=1, j’=18), as well as (v’=2, j’=14), (v’=4, j’=5) and (v’=1, j’=17). Notably, the fine angular oscillations of these states in the sideways scattering direction, indicated by ring 1 and ring 2, were experimentally observed. Based on the total kinetic energy release spectrum (TKERs) and rotational state-specific DCSs of the products can be obtained (please see data analysis in Supplementary Note 1).

To better illustrate the fine angular structure of products at specific rotational states, the experimental angular distributions of these states (indicated by ring 1 and ring 2) in the angular range of 0–70° are shown alongside the theoretical results in Fig. 2a and Fig. 2b, respectively.

The hollow points in Fig. 2 represent the experimental DCS results, which are compared with the outcomes of two different theoretical calculations. The light blue lines correspond to the DCS calculated using theoretical models that incorporate the GP, while the light red lines represent theoretical results obtained without considering the GP. The experimental angular oscillatory structure and its fine details are in good agreement with the theoretical calculations that include the GP effect, clearly deviating from the results that omit the GP. As a result, the GP effect in the H + D2 reaction has been unambiguously identified. Notably, the pronounced sideways oscillatory pattern serves as a distinctive signature, confirming the presence of the GP effect in the H + D2 → HD + D reaction.

Theoretical analysis reveals that the (v’=3, j’=12) state primarily contributes to the angular distribution shown in Fig. 2a, with the observed oscillatory pattern largely arising from the fine structure of this state (see Fig. 2c). Therefore, in the following, we focus on investigating the oscillations associated with the (v’=3, j’=12) state. Using a straightforward topological approach proposed by Althorpe and coworkers12, the adiabatic and diabatic scattering amplitude results were recombined into two reaction pathways around the CI21. The scattering amplitudes for these pathways are expressed as:

f_path1 = 1/sqrt(2) * (f_NGP + f_GP) (1)

f_path2 = 1/sqrt(2) * (f_NGP - f_GP) (2)

As previously mentioned, pathways encircling the CI more than once are negligible. Therefore, the reconstructed paths represent a direct abstraction (path 1) and a looping path around the CI (path 2).

Figure 2d illustrates the contributions of these two paths to the DCSs for the product (v’=3, j’=12). Notably, path 1 overwhelmingly dominates, producing ~100 times more products than path 2. Additionally, products from path 1 are primarily distributed in the forward-sideways direction, while those from path 2 exhibit a more uniform distribution.

To evaluate the contribution of each partial wave to the scattering, Fig. 3a, c presents the QM generalized deflection functions (GDFs) together with the QCT results. The quantum GDF represents the quantum correlation between the total angular momentum, J, and the scattering angle, and can be considered as a joint quasi-probability distribution function of J and θ that includes all coherences between different partial waves. The summation of this function over all partial waves recovers the exact differential cross section33. For a more direct comparison with the trajectory analysis, the functions are presented with the collision angle θ. atom–diatom reactions can often be interpreted within the framework of the billiard ball model32. In this model, back-scattering products primarily originate from head-on collisions involving low partial waves, whereas sideways-scattering products arise from glancing collisions involving high partial waves. Figure 3a shows a scattering pattern for path 1 that is consistent with these qualitative predictions. In sharp contrast, the scattering pattern for path 2 displays an unexpected trend: products from low-partial-wave collisions exhibit forward scattering, whereas high-partial-wave collisions predominantly yield backward scattering. Moreover, Fig. 3 reveals a distinctive feature of path 2. The peaks of the DCSs shift progressively from the forward to the backward direction as the impact parameter increases, challenging conventional understanding of scattering mechanisms.

Notably, the partial waves from path 2 that contribute to the observed sideways (30°–60°) angular oscillations are concentrated in two distinct ranges: J=0–15 and J=16–30. To assess their contributions, we calculated the product amplitudes from path 2 separately for J=0–15 and J=16–40, and combined them with those from path 1 to obtain the corresponding differential cross sections (DCSs). As shown in Fig. 4, there is no significant interference between the products from path 1 and those from path 2 at higher partial waves (J>15). This suggests that the oscillations in the sideways and forward scattering angles primarily originate from interference between path 1 and the low-J (J≤15) contributions of path 2. To further support this, we have plotted the reaction probability P(J) for path 2 of the H + D2 → D + HD (v’=3, j’=12) reaction as a function of the total angular momentum J. As shown in Supplementary Fig. S5, P(J) decreases significantly for J>15.

A partial wave analysis was also performed for the dominant product channels shown in Fig. 2b, (v′=2, j′=14 and v′=1, j′=17; see Supplementary Fig. S6), with the results shown in Supplementary Figs. S7, S8. For path 2 in these states, the DCS peaks progressively shift from the forward to the backward direction as the impact parameter increases, consistent with the trend observed in Fig. 3b, d. The partial-wave decomposition further reveals that, as in the v’=3, j’=12 case, the oscillatory features in the 30°–60° range arise mainly from interference between path 1 and the low-J (J≤15) component of path 2. The consistency of these features across both cases shown in Fig. 2 indicates that the unconventional roaming-insertion mechanism associated with path 2 is a general aspect of the reaction dynamics, persisting even when the HD product populates different final vibrational states.

The looping pathway (path 2) was considered to be a roaming-like pathway20,21. Roaming describes a process in which a reactant deviates from the minimum energy path, with a fragment of the molecule orbiting the remaining part until it encounters a reactive site to form products. This mechanism was previously assumed to be rare in the reaction system with minimal electronic complexity5,34–38. The H + D2 reaction is a system with only three electrons yields only one product channel (HD + D). The fact that a roaming insertion pathway arises even in this more constrained and electronically minimal system highlights its generality and makes its appearance here particularly remarkable.

To gain insight into the underlying dynamics, we performed a quasi-classical trajectory (QCT) study. In this study, we ran a total of 20 million trajectories, of which about 13.65% exhibited reactivity, resulting in a total reaction cross section of about 1.08 Å2. The majority of these reactive trajectories followed the direct abstraction mechanism (path 1). However, 0.42% of the reactive trajectories occurred via path 2, which involves a two-transition-state mechanism. We focused our analysis on a specific product state with vibrational and rotational quantum numbers v’=3 and j’=12. For trajectories following path 2, both QCT and quantum dynamics results revealed a pronounced angular dependence in the scattering outcomes. Trajectories with higher angular momentum, corresponding to larger partial waves, tended to produce backward scattering. In contrast, those with lower angular momentum favored forward scattering. This trend differs markedly from the dynamics observed for path 1.

In Fig. 5b, d, the green curve represents the variation of the system’s potential energy with time. For the ball-and-stick models included therein, “H” stands for a hydrogen atom, “a” stands for Da, and “b” stands for Db. The bond lengths between molecules are represented by the straight lines and curves between them.

To further examine the physical origin of this behavior, we analyzed the effect of the impact parameter on path 2 trajectories. Collisions with larger impact parameters that led to backward scattering displayed features similar to the roaming insertion mechanism previously reported for the H + HD → H2 + D reaction21. In contrast, for smaller impact parameters in the H + D2 system, we observed a distinct roaming behavior characterized by a different scattering direction. Specifically, the incoming H atom circled one of the D atoms (Da), rotated by approximately 180 degrees, and then inserted into the Da–Db bond to form a transient D–H–D configuration. This intermediate subsequently evolved into the final product.

To clearly illustrate the roaming process in the H + D2 reaction, we analyzed the time-dependent changes in bond lengths during the reaction (Fig. 5a, c). The critical step in the roaming process, which occurs during the dynamically competing interactions at 30–45 femtoseconds, was determined. Interestingly, the initial step of this phenomenon closely resembles the inelastic collision results reported by Greaves et al. in 2008 for the H + D2 system, which involved a vibrational excitation mechanism commonly observed in neutral particles with strong mutual attractions39. In the present study, during the dynamic competing interactions, the H–Da and Da–Db bonds soften and elongate when the interatomic forces are approximately balanced. This enables the H atom to orbit around the Da atom. Simultaneously, the interaction between the H atom and Da competes with the Da–Db bond, stretching it and reducing steric hindrance to facilitate the insertion process. The H atom then inserts between the Da–Db bond and approaches this site, ultimately forming the reaction products. Notably, the roaming insertion pathway is more efficient and feasible than the direct insertion pathway. As a result, most path 2 reactions process via the roaming pathway regardless of impact parameters, leading to the unconventional scattering direction of path 2 products.

It is instructive to compare the present findings for the H + D2 reaction with the H + HD reaction. To enable a direct comparison, we performed a detailed QCT analysis for H + HD under the same initial state and collision energy as for H + D2. The comparison clarifies the subtle mechanistic differences in the roaming-like insertion pathway between the two reactions. The contributions of the two pathways to the DCSs, and the classical trajectories for the forward and backward scattering products are illustrated and presented in the new Supplementary Figs. S9–S11 of the revised Supplementary Information. A side-by-side comparison of the time evolution of bond length and potential energy along the roaming-like insertion pathway (path 2) was conducted and presented in the new Supplementary Figs. S12–S15 of the revised Supplementary Information. As mentioned above, the critical step in the roaming process, which occurs during the dynamically competing interactions at 30–45 femtoseconds, was determined. To compare the differences in the reaction mechanisms of the two isotope reactions though path 2, theoretical calculations were conducted on the product quantum states (v’=0, j’=0) and (v’=3, j’=12), systematically investigating the bond length stretching characteristics of the two reactant molecules within the 30–45 femtosecond timer range (as shown in Supplementary Fig. S13). As shown in the Supplementary Fig. S13, for the product with the same ro-vibrational state, the stretching rate of the D2 bond in the H + D2 reaction is slower than that of the HD bond in the H + HD reaction; this result indicates that the reaction rate of H + D2 via path 2 is slower than that of H + HD via path 2. It’s commonly accepted that the reaction rate strongly relies on the height of the potential barrier along the reaction pathway. The evolution of the forward and backward potential energies of the H + D2 → D + HD (v’, j’) reaction via path 2 is further analyzed. The results for the H + D2 → D + HD (v’=3, j’=12) reaction ranging from 20 to 75 fs are presented in Fig. 5b, d. The potential energy evolution along path 2 can be viewed as an effective barrier that reactants must overcome to form products. In order to explore the differences in barrier height in the isotopic reaction system and its impact on the path 2 reaction. Further comparison of the time evolution of the total potential energy for product in the (v’=0, j’=0) and (v’=3, j’=12) states between the H + HD → D + H2 and the H + D2 → D + HD reaction, as shown in Supplementary Figs. S14, 15. In Supplementary Fig. S14, for high-impact-parameter trajectories, the effective barrier along path 2 is lower than the one with low impact parameter in both systems, which explains the predominance of backward scattering and the GP effect observed for H + HD at a collision energy of 1.72 eV21. In contrast, for low-impact-parameter trajectories, the path 2 barrier in H + D2 is about 0.2 eV higher than in H + HD, leading to a sideways path 2 yield in H + D2 that is approximately half of that in H + HD in our QCT calculations as shown in Supplementary Fig. S13a, b. Furthermore, as shown in Supplementary Fig. S15, the path 2 barrier for the reaction H + D2 → D + HD (v’=0, j’=0) is ~0.36 eV higher than that for H + HD → D + H2 (v’=0, j’=0). This comparison clarifies both the common features and the subtle mechanistic differences in the roaming-like insertion pathway between the two reactions. In summary, the QCT simulations highlight the dynamical significance of the newly identified roaming-like pathway. They reveal the emergence of a roaming-like path 2 involving transient D–D bond stretching, which provides a clear explanation for why the GP effect is much more difficult to probe in H + D2 than in its isotopic variants.

In the H + DaDb reaction at collision energy 2.38 eV, dynamic competing interactions lead to the softening and extension of H–Da and Da–Db bonds, resulting in the vibrational excitation (v≥3) of D2, similar to what was reported in previous work about inelastic collisions in H + D2 39. As shown in Fig. 6, this excitation, via the roaming pathway, results in the formation of HD products. This also gives insight into the reactive collision between the highly vibrationally excited D2 molecule, which was prepared via collisions. While this type of molecular vibrational excitation is not an actively induced process, it provides a valuable research strategy and experimental basis for studying vibrationally excited reaction dynamics.

Through high-resolution scattering measurements, we have revealed significant sideways angular oscillations in the HD product distribution of the H + D2 → HD + D reaction. Combined with accurate quantum mechanical calculations, these results provide unambiguous evidence of GP-induced quantum interference. The observed sideways scattering is strongly modulated by dynamic competition along the roaming insertion pathway, driven by transient elongation of the D–D bond. Unlike the reaction with HD, this behavior highlights a pronounced isotopic effect while reflecting the unique dynamic competition within the roaming insertion pathway. Collectively, these findings not only underscore the critical role of pathway interference but also provide novel mechanistic insights into how the isotopic effect modulates the geometric phase in an elementary chemical reaction.

Methods
Experimental setup
The H + D2 → HD + D reaction was investigated using a velocity map imaging (VMI)-crossed molecular beam (CMB) apparatus. The apparatus consists of three differentially pumped chambers: the scattering chamber, the fixed source chamber, and the rotating source chamber. In this experiment, a liquid N2-cooled Even Lavie valve was mounted in the rotating source chamber to generate the pulsed molecular beam of ortho-D2 (o-D2). The o-D2 sample was obtained by catalytic conversion of normal D2 (n-D2) at low temperatures. To catalytically convert the n-D2 into o-D2, n-D2 was frozen within a spiral copper pipe containing the catalyst, which was mounted to the second stage of a closed cycle cryocooler maintained at a temperature below 20 K. Subsequently, the catalytically converted D2 was collected with an aluminum gas cylinder to prevent back conversion. The backing pressure of the D2 molecular beam was 13 bar. The most probable velocity of the D2 molecules was 1030 m/s. The pulsed o-D2 atom beam was collimated by two skimmers (Beam Dynamics, Model 50.8, size 2 mm and Model 1, size 1.5 mm). Approximately 90% of the D2 molecules in the molecular beam are populated in the ground rovibrational state, (v, j)=(0, 0). The rotational state distribution of the D2 beam is measured using the (3+1) resonant enhanced multiphoton ionization (REMPI) method at around 280.22 nm via the D2 (C1Πu, v=5 ← X1Σ+, v=0), R(0), and R(1)) transitions.

In the study of the H + D2 → HD + D reaction at 2.38 eV, pure HI molecules are supersonically expanded by a pulsed Parker valve (Series 9, vertically mounted in the fixed source chamber) at the backing pressure of 1 bar. After the expansion, the HI molecules are dissociated by a linearly polarized laser at 213 nm (30 mJ/pulse) about 10 mm above the nozzle of the pulsed valve. The photodissociation laser at 213 nm is generated from the 5th harmonic of an Nd:YAG laser (Continuum, Powerlite DLS9020). The 5th harmonic is realized by the sum frequency of the fundamental output (1064 nm) and the 4th harmonic (266 nm) of the Nd:YAG using a BBO crystal. At this wavelength, H+I(2P3/2) and H+I(2P1/2) dissociation channels of the HI molecule are both energetically allowed, which give rise to two sharp and separated peaks in the velocity distribution of H atom products, exhibiting either faster or slower velocities. Due to the different angular distributions of the H atom in the two channels, in this experiment, when the photolysis laser is set to be vertically polarized, the faster H atom at a speed of 22.89 km/s from the H+I(2P3/2) channel is selectively collimated to the scattering chamber using a skimmer, intersecting with D2. The crossing angle of H and D2 beams is set to be 160°, giving a collision energy of 1.72 eV. Other experimental conditions are consistent with the aforementioned.

The D-atom products generated from the H + D2 reaction was ionized using a near-threshold resonance-enhanced multiphoton ionization (REMPI) approach with a 1+1’ scheme, employing laser wavelengths of 121.6 nm and 364.5 nm. The D atoms were excited from n=1 to n=2 state by the VUV laser beam at 121.6 nm, and subsequently pumped by a UV laser beam at 364.5 nm to an energy which is just above the ionization limit, and thus ionized. The ionized species were then directed toward a microchannel plate (MCP) detector via ion optical components, while the resulting ion images were captured using a charge-coupled device (CCD) camera.

Theoretical method
We performed adiabatic quantum dynamics calculations on the BKMP2 PES and extracted state-to-state dynamical information by using the product-coordinate-based wave packet method40,41. The numerical parameters provided in the supplementary materials facilitate the convergence of DCSs for collision energies up to 2.4 eV. To further explore the underlying dynamics of H + D2 and H + HD reaction, we carried out a quasi-classical trajectory (QCT) dynamics study. All the calculations were performed on the BKMP2 PES42.

Quantum scattering calculations
The state-to-state quantum dynamics information were extracted using the product-coordinate-based time-dependent wave packet method. We prepared an initial wave function in reactant coordinates. A coordinate transformation was then carried out to transfer the wave packet from the reactant coordinates to the D+HD product coordinates. After the transformation, we propagated the wave packet for additional 800 iterations with Δt=10 in the product coordinates. We applied a total of 161 sine basis functions (including 127 for the interaction region) in the R range of 0.1–15.0 bohr, a total of 110 potential optimized discrete variable representation basis functions (including 10 for the asymptotic region) in the r range of 0.4–14.0 bohr, and 110 rotational basis functions to converge calculation results. The range of the total angular momentum was taken as 0≤J≤45, and all helicity channels were included. Damping functions were employed to prevent the wave packet from reflecting back from the boundaries in R=11.0 bohr and r=12.0 bohr.

Quasi-classical trajectory calculations
To confirm the reaction mechanism in the classical dynamics picture, quasi-classical trajectory (QCT) calculations were performed on the adiabatic BKMP2 PES. Although these simulations are unable to predict quantum effects such as the GP, they are expected to accurately capture the underlying dynamics, which are essentially semiclassical. In our calculations, the maximum impact parameter bmax is set to be 3 bohr and the time step is equal to 3 in atom unit. Similar to the calculations of H + D2, for H + HD reaction, we ran a total of 20 million trajectories, of which about 8.98% exhibited reactivity, resulting in a total reaction cross section of about 0.71 Å2. As illustrated in Supplementary Figs. S10 and S11, the position of the transition state can be identified in hyperspherical coordinates, and the number of trajectories passing through the transition state allow the assignment of Path 1 and Path 2. The majority of these reactive trajectories followed the direct abstraction mechanism (path 1). However, 0.44% of the reactive trajectories occurred via path 2. The snapshots of a representative QCT trajectory of the path 2 mechanism for forward and backward scattered HD (v’=3, j’=12) product are shown in Supplementary Figs. S3 and S4. In the trajectory, the incoming H atom approaches D2 molecule and passes the first H-Da-Db linear transition state. As the Da-Db molecular stretches, H atom inserts between Da-Db bond by experiencing the second Da-H-Db transition state. For comparison, the snapshots of a representative H + HD QCT trajectory of the path 2 mechanism for forward and backward scattered H2 (v’=3, j’=12) product are also given in Supplementary Figs. S10 and S11.

Data availability
Data supporting the findings of this study are available from the corresponding authors upon request. Additionally, source data are provided with this paper and can be accessed at https://doi.org/10.6084/m9.figshare.30417382.

Code availability
The code that supports the theoretical calculations in this paper is available from the corresponding authors upon request.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Shihao Li and coworkers present a new study on the H+D2 reaction at a collision energy of 2.38 eV. With a careful study of the DCS and supported by exact quantum calculations, they were able to infer the role of the conical intersection on the quantum interference between 2 different pathway leading to products, thus providing interesting insights on reaction mechanism.\n\nThe work is interesting and very well written. The only issue in my point of view is the lack of description of the theoretical calculations. I would have expected some details on the calculations in the \"method\" section, both for QM and QCT calculations, and also the associated references, in particular for the PES that have been employed to perform the calculations.\n\nAfter those details are included, I recommend publication of this very nice work.", "reviewer_2": "See attached report.", "reviewer_3": "Through combined theoretical and experimental investigations Li et al. study the quantum dynamics of the three-atom reactive scattering H + D2 --> HD + D at a collision energy of 2.38 eV that lies below the conical interaction of the ground and excited state of the three atom (H3) potential. They analyze the high resolution scattering angular distribution of state specific (v’, j’) product HD using rigorous quantum dynamical and quasi classical trajectory calculations. The experimentally observed oscillations in the angular distribution that arises from the interference between the direct and the looping pathways around the conical intersection of the three atom potential tallies with theory only when the topological phase effect is included in the calculation. A clear phase shift of the oscillatory structure of the measured angular distribution relative to that calculated excluding the GP effect clearly illustrates how the geometrical phase controls this reactive scattering.\n\nTo gain deeper understanding they focus on the specific product state HD (v’=3, j’=12) that has strong contribution to the observed sideways scattering. Detailed partial wave analysis reveal that oscillations observed in the forward and sideways scattering distribution are mainly due to the interference effect between the direct path (path 1) of the higher partial waves and the looping path (path 2) of the lower partial waves (J<15). The back scattering from path 2 contributed mainly by the higher order partial waves is in stark contrast to the conventional scattering theory, and thus, provides a strong signature of the quantum phase effect of this reaction pathway that goes circling around the conical intersection. The partial wave analysis analysis further reveals the detailed dynamics of H atom insertion via the two transition states of path 2. It exhibits roaming like behavior of the H-atom around the D-atoms which is facilitated by the bond stretching of the D2 molecule. This raises further interesting question, whether prior vibrational excitation of the reactant will favor the looping path, thus enhancing the topological phase effect.\n\nThe combined experimental and theoretical studies of this particular H + D2 reaction have been carried out before [Refs 27-31], however, these investigations failed to detect the geometrical phase effect. The present work clearly demonstrates the topological phase effect in this specific three atom reaction, which is new. However, the GP effects from the interference of the direct and the looping reaction paths in an isotopic variant reaction H + HD --> H2 +D has been demonstrated multiple times by the same group of authors [Refs 16, 20, 21], and these quantum dynamical studies are closely parallel to the current work. In this sense the current study is more of an extension of their previous works. The strength of the present work lies in their rigorous theoretical analysis of the experimental scattering angular distribution via discrimination of the partial wave contributions. The partial wave analysis manifests the quantum nature of the looping path. It reveals the roaming like behavior of path 2 and shows how the insertion dynamics proceeds via transient vibrational excitation of the D2 molecule.\n\nAlthough these results could be published in a suitable journal, in my mind, it lacks sufficient novelty to justify publication in Nature Communication.", "reviewer_4": "This manuscript presents a joint experimental and theoretical study on the hydrogen exchange reaction, H + D2 → HD + D, focusing on geometric phase-induced quantum interference effects. These are attributed to interference between the direct abstraction and roaming insertion pathways.\n\nThe roaming insertion pathway shows an interesting effect: low-partial-wave collisions preferentially lead to forward scattering, whereas high-partial-wave collisions mostly lead to backward scattering. The opposite is generally observed, and is also the case for the direct abstraction pathway in this manuscript.\n\nThe roaming insertion pathway has been investigated in more detail using quasi-classical trajectory calculations, offering valuable insight in the underlying reaction mechanisms.\n\nIn previous beautiful studies on H + HD, an isotopic variant of the same reaction, geometric phase effects have been reported and analysed (refs 16, 20, 21, 26 and Yuan et al., Nat. Comm. 11, 3640 (2020), which to our surprise has not been cited in this work). In these cases, they showed up as angular oscillations in either the forward or backward scattered directions. As mentioned by the authors on page 4, numerous experimental studies have investigated the H + D2 system already, but until now, geometric phase effects have not been observed.\n\nThe experimental image recorded for this work, in a similar way as in previous work by the same research groups, is beautiful and of extremely high quality. Oscillations are clearly visible in the angular scattering distributions for angles between 0-70 degrees (sideways scattering). Fig. 2 very convincingly shows that these oscillations are due to geometric phase effects. This manuscript therefore presents the first observation of the geometric phase effect for H + D2.\n\nUsing QCT calculations, the origin of these oscillations has been investigated, and it was found that a roaming insertion pathway is involved, as was also seen for H + HD (ref. 21 for instance). This pathway has been investigated in a more detailed way, and the underlying mechanism has been clarified.\n\nThis manuscript presents beautiful and high-quality work that in our opinion clearly has the potential to be published in Nature Communications. However, to strengthen the clarity of the manuscript and to more clearly highlight the novelty of the current findings – especially with respect to previous work on H + HD by the same groups – we ask the authors to address the following points. We believe this is crucial before the manuscript can be considered for acceptance in Nature Communications.\n\n- We recommend that the authors revise the last part of the manuscript, which discusses the underlying dynamics of the observed phenomenon (starting from page 11) to make it more accessible to a broader audience, including non-specialists.\n- While Fig. 2 presents oscillations for two different rings, the manuscript afterwards focusses exclusively on the mechanisms underlying the oscillatory structures shown in panel a, without addressing the oscillations shown in panel b. We think the authors should extend their analysis to include the oscillations shown in panel b. Does the newly identified roaming insertion mechanism also play a role in this case? If so, this would further strengthen the findings. This is particularly important as the final vibrational states of the HD product is different in panel b compared to panel a.\n- We ask the authors to more clearly show and highlight the difference between the roaming insertion mechanisms found for H + HD and H + D2. On page 13, the authors write that the roaming mechanism for H + D2 relies on a symmetric stretch mode, which is absent in the H + HD system. This is a particularly compelling conclusion that would clearly distinguish the present study from previous work. We suggest that the authors clearly show that this is the case, for instance by demonstrating that the roaming mechanism responsible for oscillations in the backward direction for H + HD does not play a role for H + D2. Analogously, the authors can show that the newly identified roaming mechanism is really absent in H + HD (also at sidescattered angles where the theoretical DCS in H+HD shows clear oscillations as well. Are these sidescattered oscillations in H+HD indeed fundamentally different than the sidescattered oscillations found in this work for H + D2?).\nIdeally, one would like to see plots like Figure 3,4,5,S6 and S7 for the H + D2 and H + HD systems next to each other to clearly illustrate the differences in interpretation of the roaming mechanisms involved.\n\nSmaller comments, questions and suggestions:\n- On page 5, the authors refer to Supplementary Note 1 for experimental details regarding the ionization of the D-atom products. However, there is no information on the ionization scheme in this note, nor in the Methods section. This, and additional information about how the image in Fig. 1 is measured, should be included.\n- In the last paragraph on page 5, which continues on page 6, many experimental details are given that are also described in both the Methods section as well as in Supplementary Note 1. We advise the authors to shorten this section.\n- All information that is given in Supplementary Note 1 is also given in the Methods section. We therefore advise the authors to remove Supplementary Note 1.\n- On page 7, the authors write “rotation-state resolved image”. We advise the authors to rephrase this, since several rovibrational states contribute to these rings. Now, it sounds like every ring corresponds to a single rotational state.\n- Fig. 2 shows theoretical calculations with and without inclusion of the geometric phase (which are not mentioned in the caption). However, the manuscript provides no details at all on how these calculations were performed. This information should be added.\n- On page 8, the authors discuss a topological approach proposed by Althorpe and coworkers, and refer to ref. 21. We suggest the authors to here refer to ref. 12, and at the end of the same sentence (…pathways around the CI) refer to ref. 21 (their previous work on H + HD, in which they also applied the approach by Althorpe and coworkers).\n- Page 9: ‘Figs. 3a and 3c show that the DCSs…’. In this sentence, 3c should be 3b.\n- Page 9: ‘Figs. 3a and 3c show that the DCSs…’. The authors write that partial waves with J = 25 and 35 dominate in the sideways direction. However, we don’t see this so clearly from Fig. 3. We recommend the authors to elaborate on this.\n- Fig. 2 shows the angular scattering distributions from 0-70 degrees, whereas Fig. 3 only starts at 30 degrees. Is there a specific reason for this? If so, we recommend the authors to mention this in the manuscript.\n- Page 9: ‘Previously, the collision dynamics of…’ It seems like a reference is missing in this sentence.\n- Page 10: we recommend the authors to move the sentence ‘Interestingly, Fig. 3 also highlights…’ and place it before the sentence ‘Notably, the partial waves…’.\n- The authors often use the word ‘collision parameters’. We advise the authors to use partial waves or impact parameters instead.\n- Page 10: ‘Notably, the partial waves…’. We recommend the authors to reformulate this sentence for clarity. Moreover, we would say that the ranges of J=15-25 and J=5-10 should be a bit larger, when inspecting Fig. 3.\n- Page 10: ‘The looping pathway…’. It seems like references are missing here.\n- Page 10, last paragraph: it is discussed that the roaming insertion pathway in this reaction system is unusual. However, it is also already discussed and used in other papers on H + HD (ref. 21 for instance). So we advise the authors to update this sentence.\n- Page 12: ‘On the other hand … as shown in Figs. S6 and S7…’. This should probably only be Fig. S6. Fig. S7 is for backward scattering and is useful for the previous sentence (‘By comparing the trajectories, …’).\n- Fig. 4: we advise the authors to improve the caption, especially regarding panels c and d.\n- Page 12-14: It is not always clear whether the authors refer to the forward scattering (low impact parameter) roaming pathway or to the backward scattering (high impact parameter) roaming pathway. Often, the authors just use Path 2 to indicate both of them or maybe only one of them. It would help to clarify this.\n- Page 13: ‘Simultaneously, the competing interaction…’. Please clarify which interaction is meant here.\n- Page 13: ‘Moreover, the roaming mechanism (path 2) in the H + D2 reaction relies on a symmetric stretch…’ we assume the authors base this on Fig. 4a and c, for times between 30-45 fs, where the H-Da and Da-Db distances are similar. If so, please clarify this in the text.\n- Regarding Fig. 5 and the corresponding interpretation: the authors mention that D2 is excited to v=3. Does the exact vibrational level of D2 (so v=3) really follow from the calculations, is v=3 used since the mechanism was found for HD(v=3), or is this speculation (maybe based on ref. 37)? We recommend the authors to clarify this in the text.\n- Methods: ‘The experimental setup has been described in detail in a previous article…’ A reference could be added here.\n- Supplementary Note 2: ‘In Fig. S1, the peaks of v’=0, j’=1-8 can be well resolved.’ We advise the authors to here use j’=2-8, since j’=1 and 0 seem to overlap.\n- Fig. S3, in the caption: ‘bule’ should be ‘blue’\n- Supplementary Note 4: It would be useful to provide some more detail on how the QCT trajectory calculations were performed (or to add references if the methods are similar to what has been used for H + HD for instance). A reference to the BKMP2 PES should be included as well. Moreover, the abbreviation PES is not introduced.\n- In Supplementary Note 4, the authors use as notation H’ and D. We advise the authors to here use similar notation as in the main text (H, Da and Db).\n- Fig. S6 and S7: it is not described what the red arrows in the left panels are (probably the D-atom movement). Moreover, labels a-e are missing in the figure itself (the caption refers to these labels). From the caption, it seems like the conical intersection (CI) is indicated by crosses in all panels. However, this is only for panel e.\n- Fig. S7, in the caption: ‘The potential energy surface contours… for a-e.’ Here, a-e should be a-d.\n- In some sentences, the English (grammar) can be improved.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Responses to the reviewers’ comments\nWe would like to express our sincere gratitude to all the reviewers for their\ninsightful and thoughtful comments, which are very important for further improving\nthe quality and clarity of the manuscript. In the following, we provide detailed, point-\nby-point responses to each comment.\nWe are grateful to the reviewer for the thoughtful and valuable comments, which have\nbeen very helpful in improving the manuscript.\nReviewer:\nShihao Li and coworkers present a new study on the H+D reaction at a collision\n2\nenergy of 2.38 eV. With a careful study of the DCS and supported by exact quantum\ncalculations, they were able to infer the role of the conical intersection on the\nquantum interference between 2 different pathways leading to products, thus providing\ninteresting insights on reaction mechanism.\nThe work is interesting and very well written. The only issue in my point of view is\nthe lack of description of the theoretical calculations. I would have expected some\ndetails on the calculations in the \"method\" section, both for QM and QCT\ncalculations, and also the associated references, in particular for the PES that have been employed to\nperform the calculations.\nAfter those details are included, I recommend publication of this very nice work. I\nwould have expected some details on the calculations in the \"method\" section, both for\nQM and QCT calculations, and also the associated references, in particular for the PES\nthat have been employed to perform the calculations.\nReply:\nWe would like to thank the reviewer for the positive comment. We fully agree with\nthe reviewer and have now added more details regarding the theoretical calculations in\nthe Methods section. In particular, we provide expanded descriptions of both the QM\nand QCT calculations, together with the relevant references, especially for the PES\nemployed in this work.\nRevised text in the Methods section“: We performed adiabatic quantum dynamics\ncalculations on the BKMP2 PES and extracted state-to-state dynamical information by\nusing the product-coordinate-based wave packet method40,41. The numerical\nparameters provided in the supplementary materials facilitate the convergence of DCSs\nfor collision energies up to 2.4 eV. To further explore the underlying dynamics of H+D\n2\nand H +HD reaction, we carried out a quasi-classical trajectory (QCT) dynamics study.\nQuantum scattering calculations\nThe state-to-state quantum dynamics information were extracted using the\nproduct-coordinate-based time-dependent wave packet method. We prepared an initial\nwave function in reactant coordinates. A coordinate transformation was then carried\nout to transfer the wave packet from the reactant coordinates to the D + HD product\ncoordinates. After the transformation, we propagated the wave packet for additional\n800 iterations with Δt = 10 in the product coordinates. We applied a total of 161 sine\nbasis functions (including 127 for the interaction region) in the R range of 0.1–15.0\nbohr, a total of 110 potential optimized discrete variable representation basis functions\n(including 10 for the asymptotic region) in the r range of 0.4–14.0 bohr, and 110\nrotational basis functions to converge calculation results. The range of the total angular\nmomentum was taken as 0 ≤ J ≤ 45, and all helicity channels were included. Damping\nfunctions were employed to prevent the wave packet from reflecting back from the\nboundaries in R = 11.0 bohr and r = 12.0 bohr.\na a\nQuasi-classical trajectory calculations\nTo confirm the reaction mechanism in the classical dynamics picture, quasi-\nclassical trajectory (QCT) calculations were performed on the adiabatic BKMP2 PES.\nAlthough these simulations are unable to predict quantum effects such as the GP, they\nare expected to accurately capture the underlying dynamics, which are essentially\nsemiclassical. In our calculations, the maximum impact parameter b is set to be 3\nmax\nbohr and the time step is equal to 3 in atom unit. Similar to the calculations of H + D ,\n2\nfor H+HD reaction, we ran a total of 20 million trajectories, of which about 8.98%\nexhibited reactivity, resulting in a total reaction cross section of about 0.71 Å2. The\nmajority of these reactive trajectories followed the direct abstraction mechanism (path\n1). However, 0.44% of the reactive trajectories occurred via path 2. The snapshots of a\nrepresentative QCT trajectory of the path 2 mechanism for forward and backward\nscattered HD (v’=3, j’=12) product are shown in Figs. S3 and S4. In the trajectory, the\nincoming H atom approaches D molecule and passes the first H-D -D linear\n2 a b\ntransition state. As the D -D molecular stretches, H atom inserts between D-D bond\na b\nby experiencing the second D -H-D transition state. For comparison, the snapshots of\na b\na representative H+HD QCT trajectory of the path 2 mechanism for forward and\nbackward scattered H (v’=3, j’=12) product are also given in Figs. S10 and S11.“\n2\nWe sincerely thank the reviewer for the insightful and thoughtful comments, which\nhave been very important in improving the manuscript.\nReviewer:\nThis manuscript presents a combined experimental and theoretical study of the H+o-\nD HD+D reaction exploring the effect of the geometric phase on the DCS at a\n2\ncollision energy of 2.38 eV, which is below the energy of the conical intersection. The\nexperiments are carried out in a crossed molecular beam machine using VMI for\nproduct’s detection. The H atoms produced by photolysis of HI at 213 nm and the D\nproduct were detected using 1+1’ near-threshold REMPI. These experiments\ndemonstrate superb state-to-state resolution enabling the determination of the state-\nresolved DCS of the coproduct HD from the angle-velocity D ion images. The main\nfinding of this work is the oscillations in sideways scattering for some HD states, which\ncan be accurately reproduced by quantum dynamics calculations requiring the inclusion\nof the geometric phase. Furthermore, the interpretation of these oscillations bas been\nshown to result from the interference of the wavefunction describing direct abstraction,\nwhich crossed only one transition state (path1), and the wavefunction describing motion\naround the conical intersection, which crosses two transition states (path2). Although\nthe contribution of the latter is 100 times smaller than that of direct abstraction, it has a\ndefinitive influence in the oscillation pattern experimentally observed that NGP\ncalculations cannot reproduce.\nThe same authors have found similar effects in the state-resolved DCSs of\nH+HDH +D(refs.16,21,20) in both backward and forward scattering and in\n2\noscillations as a function of collision energy at extreme backward scattering. This works\nreveals the effect of the GP in the H+D reaction at energies below the conical\n2\nintersection for the first time. Analysis of the results of the present work for path 2\nreveals that backward scattering can be associated with relatively high angular\nmomenta, whereas low J values are responsible for forward scattering, in contrast to\nwhat is observed in the direct case through path1.\nI have no reservation about recommending this article for its publication in Nat.\nComm. However, I have some criticisms and suggestions that I believe should consider\nfor the final version of the manuscript.\nWe express profound gratitude to the reviewer for the recognition and high praise\nof the current work. We highly value the reviewers' criticisms and suggestions, and have\nimplemented comprehensive revision to improve the manuscript.\nReviewer:\nSome of the figures presented in the Supplementary Material are shown and\ndiscussed in the main body of the paper: Figures S3, S4 and S5. To avoid readers having\nto jump between the article itself and the SM, I suggest that the authors include these\nfigures in the main article.\nReply:\nWe sincerely appreciate this valuable suggestion. Following the reviewer’s advice,\nand in order to make the manuscript easier to follow without requiring readers to switch\nbetween the main text and the Supplementary Information, we have incorporated\nFigures S3, S4, and S5 into the main article. In particular, Figures S3 and S4 have been\nmerged into Figure 2, and Figure S5 has been introduced as the new Figure 4 (with\nsubsequent figures renumbered accordingly). We fully agree with the reviewer that this\nadjustment improves the coherence and accessibility of the discussion, allowing readers\nto follow the relevant content more smoothly. We believe this revision further enhances\nthe readability and overall presentation of the manuscript.\nReviewer: I am having serious difficulties understanding Fig. 3asDCS(J)-DCS(J−1).\nIn some cases, it can be expected that the difference between the QM DCSs may lead\nto negative numbers (adding one J at a given scattering angle may decrease its value).\nIn the classical case, it would be more informative to plot a dot for each trajectory at its\nJ and θ values or to use another representation of the joint probability distribution of J\nand θ. In the quantum case, it would be much more informative to use the QM joint\nquasi-probability distribution proposed in Chem. Sci. 9, 4837 (2019), to which the\nauthors of that article refer to as the ‘generalised deflection function’. I strongly\nrecommend that the present authors consult that reference. In any case, plotting P(J)\nand the DCS or their evolution with the addition of J would be necessary.\nReply:\nrecommended approach (Chem. Sci. 2019, 9, 4837), which has now been cited in the\nrevised manuscript as ref. 33, we reanalyzed the results using the generalized deflection\nfunction (GDF) method. Accordingly, the original Figures 3a and 3b have been replaced\nwith the new GDF-based plots, while the previous Figures 3a and 3b (showing the\nvariation of the DCS with θ and J) have been removed. We have also added the plot of\nP(J) for the H + D₂ → D + HD (v’= 3, j’= 12) reaction (see Fig. S5) to illustrate how\nthe reaction probability changes with increasing J.\nReviewer: One interesting aspect is that there is not interference between the two paths\nabove J=15. A plot of the P(J) for path 2 would be due.\nReply:\nWe thank the reviewer for the helpful suggestion. We have calculated the plot of\nP(J) for path 2 and added it to the Supplementary Information (see Fig. S5). This result\nfurther confirms that no significant interference occurs between the two paths for J >\n15.\nReviewer:\nThe caption in Fig.S1 and S2 should clarify that red curve is the sum of all the QM\ncontributions (including GP) from all the v′ j′ states represented in the combs, and that\nthe open circles are the experimental measurements.\nReply:\nWe thank the reviewer for this helpful suggestion. The captions of Figures S1 and\nS2 have been revised to clarify the following: open circles correspond to the\nexperimental measurements; green peaks indicate the Gaussian fit of the experimental\nmeasurements for each HD product rotational state using accurate energy of the ro-\nvibrational levels of HD and the red curve represents the sum of all the fitted ro-\nvibrational states.\nReviewer:\nIn Fig S4 the caption should indicate that it is the DCS for v’=3, j’=12.\nReply:\nWe thank the reviewer for pointing this out. The caption of Figure S4 has been\nrevised to clarify that it shows the DCS for 𝑣’=3, 𝑗’=12.\nwithin the circles representing each atom in the PES. The authors should clarify their\nmeaning.\nReply:\nWe thank the reviewer for this helpful comment. In Figures S6 and S7, the black\ndot denotes the H atom, while the black open circle denotes the D atom. The red curves\ntrace the trajectories of the incoming H atom, and the red arrow around the open circles\nindicates the direction of motion of the D atom during the roaming process. We have\nrevised the captions of the original Figures S6 and S7 (Figures S3 and S4 in the revised\nmanuscript) accordingly to clarify these representations.\nWe sincerely thank the reviewer for the thoughtful and helpful comments, which have\nbeen very important in improving the manuscript.\nReviewer #3 (Remarks to the Author):\nThrough combined theoretical and experimental investigations Li et al. study the\nquantum dynamics of the three-atom reactive scattering H + D --> HD + D at a\n2\ncollision energy of 2.38 eV that lies below the conical interaction of the ground and\nexcited state of the three atom (H ) potential. They analyze the high resolution\n3\nscattering angular distribution of state specific (v’, j’) product HD using rigorous\nquantum dynamical and quasi classical trajectory calculations. The experimentally\nobserved oscillations in the angular distribution that arises from the interference\nbetween the direct and the looping pathways around the conical intersection of the three\natom potential tallies with theory only when the topological phase effect is included in\nthe calculation. A clear phase shift of the oscillatory structure of the measured angular\ndistribution relative to that calculated excluding the GP effect clearly illustrates how\nthe geometrical phase controls this reactive scattering.\nTo gain deeper understanding they focus on the specific product state HD (v’=3,\nj’=12) that has strong contribution to the observed sideways scattering. Detailed partial\nwave analysis reveal that oscillations observed in the forward and sideways scattering\ndistribution are mainly due to the interference effect between the direct path (path 1) of\nthe higher partial waves and the looping path (path 2) of the lower partial waves (J<15).\nThe back scattering from path 2 contributed mainly by the higher order partial waves\nis in stark contrast to the conventional scattering theory, and thus, provides a strong\nsignature of the quantum phase effect of this reaction pathway that goes circling around\nthe conical intersection. The partial wave analysis analysis further reveals the detailed\ndynamics of H atom insertion via the two transition states of path 2. It exhibits roaming\nlike behavior of the H-atom around the D-atoms which is facilitated by the bond\nstretching of the D molecule. This raises further interesting question, whether prior\n2\nvibrational excitation of the reactant will favor the looping path, thus enhancing the\ntopological phase effect.\nThe combined experimental and theoretical studies of this particular H + D reaction\n2\nhave been carried out before [Refs 27-31], however, these investigations failed to detect\n8\nthe geometrical phase effect. The present work clearly demonstrates the topological\nphase effect in this specific three atom reaction, which is new. However, the GP effects\nfrom the interference of the direct and the looping reaction paths in an isotopic variant\nreaction H + HD --> H +D has been demonstrated multiple times by the same group\n2\nof authors [Refs 16, 20, 21], and these quantum dynamical studies are closely parallel\nto the current work. In this sense the current study is more of an extension of their\nprevious works. The strength of the present work lies in their rigorous theoretical\nanalysis of the experimental scattering angular distribution via discrimination of the\npartial wave contributions. The partial wave analysis manifests the quantum nature of\nthe looping path. It reveals the roaming like behavior of path 2 and shows how the\ninsertion dynamics proceeds via transient vibrational excitation of the D molecule.\nsufficient novelty to justify publication in Nature Communication.\nReply:\nWe thank the reviewer for the thoughtful and detailed evaluation of our work, and\nwe deeply appreciate the recognition of the rigorous theoretical analysis and the insight\nprovided by the partial wave decomposition.\nWe agree that the H + D reaction has been extensively investigated both\n2\nexperimentally and theoretically (Refs. 27–32). However, the GP effect has not\npreviously been experimentally identified in this benchmark system, despite its clear\ndemonstration in isotopic variants such as H + HD → H + D (Refs. 16, 20, 21, 26).\neffect in the sideways scattering of the H + D , confirming a phenomenon that had\n2\neluded direct observation despite long-standing theoretical predictions.\nIn the following, we will elucidate to the reviewer the similarity and, more\nsignificantly, the differences and scientific ramifications in the differential cross\nsections (DCSs) and mechanism of the roaming like behavior in H + D and H + HD\n2\nreactions.\nUnique features in DCSs. DCS is one of the most important measurements in the\nelementary reaction dynamics, encapsulating intricate details regarding the reaction\nmechanism. In this work, our high-resolution measurements of the DCS of the H + D\n2\nreaction clearly revealed pronounced sideways angular oscillations in the HD product\ndistribution. High-level quantum dynamical calculations performed with and without\nthe GP, reveal a distinct phase shift in the oscillations of the angular distribution. This\n9\ndirectly verifies the role of the GP effect in this reaction and, to our knowledge,\nconstitutes the first experimental confirmation of this effect in H + D . While in the\n2\nisotopic variant of the H + HD reaction, the angular distribution features, which include\nthe essential information regarding GP effect, appear in the forward or backward\nscattering directions of the DCS, rather than sideways in the H+ D reaction. The\n2\ndisparity in the unique angular distribution features related to the GP effect indicates\ndistinct reaction mechanisms in these two reactions.\nNew mechanistic insights into roaming-like insertion pathway. To gain insight\ninto the dynamical details, accurate quantum mechanical calculations were carried out\nfor the H + D reaction. Detailed partial wave analysis uncovers previously unobserved\n2\nquantum features of the reactive dynamics, including interference between higher-J\ndirect (path 1) and lower-J looping (path 2) pathways, as well as the role of the conical\nintersection in shaping the sideways scattering pattern. These newly identified features\nwere not anticipated in prior studies.\nHistorically, Althorpe and co-workers (2005) predicted that GP effects in the\nforward scattering direction of hydrogen exchange reactions arise from interference\nbetween high-impact-parameter direct abstraction (path 1) and low-impact-parameter\ninsertion (path 2). Our group confirmed these predictions experimentally for H + HD\n→ D + H (refs 16 and 26), and subsequently revealed roaming-like behavior in path\n2\n2 that contributes to backward scattering (refs 20 and 21). In the present work, for the\nfirst time, we demonstrate that low-impact-parameter roaming-like path 2 and high-\nimpact-parameter path 1 also produce GP effects in the sideways direction. This\nbehavior, in contrast to the reaction with HD, highlights a pronounced isotopic effect\nand reflects the unique dynamic competition along the roaming insertion pathway,\nproviding new mechanistic insights into how the geometric phase shapes isotopic\nvariants of an elementary chemical reaction.\nIsotope effect and challenges in GP effect probing. While the GP effect has been\nwell identified in H + HD reactions (refs16,20-21,27), experimental studies on the\nisotopic variant H + D system, despite extensive investigation at collision energies up\n2\nto 3.26 eV (refs 28-32), have been so far unable to discover any GP signatures.\nTherefore, a detailed quantum-level investigation of the reaction pathways in this\nisotopic system is need to detailed investigated.\nTo enable a direct comparison for the H + D reaction with the H + HD reaction,\n2\nas also suggested by reviewer 3, we performed detailed quasi-classical trajectory (QCT)\n10\ntheoretical study on the two reactions under the same initial state and collision energy.\nThe time evolution of the bond-length and potential energy underlying the roaming-\nlike insertion pathway are notably different between the H+HD and the H+D reactions.\n2 2\nthe HD bond in the H+HD reaction. This result indicates that the reaction rate of the H\n+ D\n2\nvia path 2 is slower than that of the H + HD via path 2. Secondly, the time evolution\nof the total potential energy for the two reactions shows that, for high-impact-parameter\ntrajectories, the effective barrier along path 2 is lower than the one with low impact\nparameter in both systems, which explains the predominance of backward scattering\nand the GP effect observed for H + HD at a collision energy of 1.72 eV (ref 21). In\ncontrast, for low-impact-parameter trajectories, the path 2 barrier in H + D is about 0.2\n2\neV higher than in H + HD, leading to a sideways path 2 yield in H + D that is\n2\napproximately half of that in H + HD in our QCT calculations as shown in Figs.S13a-\nb in the revised Supplementary Information. This provides a clear explanation for why\nthe GP effect is much more difficult to probe in H + D than in its isotopic variants.\nThe comparison clarifies both the common features and the subtle mechanistic differences\nin the roaming-like insertion pathway between the two reactions.\nTherefore, we believe that the experimental detection and theoretical elucidation\nof the GP effect in H + D represent a distinct and significant advance in reaction\n2\ndynamics.\nWe sincerely thank the reviewer for the thoughtful and insightful comments, which\nhave been invaluable in further strengthening the manuscript.\nReviewer:\nThis manuscript presents a joint experimental and theoretical study on the\nhydrogen exchange reaction, H + D → HD + D, focusing on geometric phase-induced\n2\nquantum interference effects. These are attributed to interference between the direct\nabstraction and roaming insertion pathways. The roaming insertion pathway shows an\ninteresting effect: low-partial-wave collisions preferentially lead to forward scattering,\nwhereas high-partial-wave collisions mostly lead to backward scattering. The opposite\nis generally observed, and is also the case for the direct abstraction pathway in this\nmanuscript. The roaming insertion pathway has been investigated in more detail using\nquasi-classical trajectory calculations, offering valuable insight in the underlying\nreaction mechanisms. In previous beautiful studies on H + HD, an isotopic variant of\nthe same reaction, geometric phase effects have been reported and analyzed (refs 16,\n20, 21, 26 and Yuan et al., Nat. Comm. 11, 3640 (2020), which to our surprise has not\nbeen cited in this work). In these cases, they showed up as angular oscillations in either\nthe forward or backward scattered directions. As mentioned by the authors on page 4,\nnumerous experimental studies have investigated the H + D system already, but until\n2\nnow, geometric phase effects have not been observed. The experimental image recorded\nfor this work, in a similar way as in previous work by the same research groups, is\nbeautiful and of extremely high quality. Oscillations are clearly visible in the angular\nscattering distributions for angles between 0-70 degrees (sideways scattering). Fig. 2\nvery convincingly shows that these oscillations are due to geometric phase effects. This\nmanuscript therefore presents the first observation of the geometric phase effect for H\n+ D . Using QCT calculations, the origin of these oscillations has been investigated,\n2\nand it was found that a roaming insertion pathway is involved, as was also seen for H\n+ HD (ref. 21 for instance). This pathway has been investigated in a more detailed way,\nand the underlying mechanism has been clarified. This manuscript presents beautiful\nand high-quality work that in our opinion clearly has the potential to be published in\nNature Communications. However, to strengthen the clarity of the manuscript and to\nmore clearly highlight the novelty of the current findings – especially with respect to\n12\nprevious work on H + HD by the same groups – we ask the authors to address the\nfollowing points. We believe this is crucial before the manuscript can be considered for\nacceptance in Nature Communications.\nReply:\nWe sincerely thank the reviewer for the positive comments and for highlighting\nthe important and relevant previous work on the H + HD reaction, particularly the study\nby Yuan et al. (Nat. Commun. 11, 3640 (2020)). We have now included a citation to this\nwork in the revised manuscript in page 5.\nIn addition, we are grateful to the reviewer for this helpful suggestion. We agree\nthat the manuscript would benefit from improved clarity. In the revised manuscript, we\nhave very carefully addressed all the points proposed by the reviewer, and revised the\ntext to enhance readability and clearly highlight the novelty of the current findings.\nReviewer:\nWe recommend that the authors revise the last part of the manuscript, which\ndiscusses the underlying dynamics of the observed phenomenon (starting from page 11)\nto make it more accessible to a broader audience, including non-specialists.\nReply:\nWe thank the reviewer for this thoughtful suggestion. In response, we have\nsubstantially revised the relevant section of the manuscript to improve clarity and\naccessibility for a broader readership, including those less familiar with reaction\ndynamics. Specifically, we have simplified the technical language, clarified the\ndescription of the angular dependence observed in the product state (𝑣’=3, 𝑗’=12), and\nprovided a more intuitive explanation of how the impact parameter influences the\nscattering behavior in path 2. We now emphasize the contrasting roles of high- and low-\nangular-momentum trajectories and their relation to backward and forward scattering\npatterns, respectively. In addition, starting from page 11, we have expanded the\ndiscussion of the roaming behavior observed in the H + D system, distinguishing it\n2\nfrom previously reported mechanisms in the H + HD → H + D reaction. This revision\n2\nincludes a clearer description of the unique pathway at smaller impact parameters, in\nwhich the incoming H atom rotates around one D atom and inserts into the D–D bond\nto form a transient D–H–D complex.\nWhile Fig. 2 presents oscillations for two different rings, the manuscript\nafterwards focusses exclusively on the mechanisms underlying the oscillatory\nstructures shown in panel a, without addressing the oscillations shown in panel b. We\nthink the authors should extend their analysis to include the oscillations shown in panel\nb. Does the newly identified roaming insertion mechanism also play a role in this case?\nIf so, this would further strengthen the findings. This is particularly important as the\nfinal vibrational states of the HD product is different in panel b compared to panel a.\nReply:\nWe thank the reviewer for this comment. In response, we have analyzed the\ndifferential cross section results for the three states shown in panel b of Fig. 2 and\npresented results in the new Fig. S6 of the Supplementary Information. The results show\nthat the 𝑣’=4, 𝑗’=5 state contributes weakly at 20°–70°. So, our analysis focused on\nthe 𝑣’=2, 𝑗’=14 and 𝑣’=1, 𝑗’=17 states. Topological methods were used to analyze the\ncontributions of different paths to DCSs for two ro-vibrational state, and the results\nwere shown in the new Figs.S7-S8. Calculations indicate that both states exhibit a\npronounced GP effect, reflecting interference between two topologically distinct\npathways. Further examination of path 2 for these states shows that the peaks of the\nDCSs progressively shift from the forward to the backward direction with increasing\nimpact parameter, demonstrating that path 2 follows the same mechanism as discussed\nin the main text for all the product states. The corresponding discussion has been\nincorporated into the revised manuscript (page 14).\nReviewer:\nWe ask the authors to more clearly show and highlight the difference between the\nroaming insertion mechanisms found for H + HD and H + D . On page 13, the authors\n2\nwrite that the roaming mechanism for H + D relies on a symmetric stretch mode, which\n2\nis absent in the H + HD system. This is a particularly compelling conclusion that would\nclearly distinguish the present study from previous work. We suggest that the authors\nclearly show that this is the case, for instance by demonstrating that the roaming\nmechanism responsible for oscillations in the backward direction for H + HD does not\nplay a role for H + D . Analogously, the authors can show that the newly identified\n2\nroaming mechanism is really absent in H + HD (also at side scattered angles where the\ntheoretical DCS in H + HD shows clear oscillations as well. Are these side scattered\n14\noscillations in H + HD indeed fundamentally different than the side scattered\noscillations found in this work for H + D ?). Ideally, one would like to see plots like\n2\nillustrate the differences in interpretation of the roaming mechanisms involved.\nReply:\nWe thank the reviewer for this valuable suggestion. We agree that it is instructive\nto compare the present findings for the H + D reaction with the H + HD reaction. To\n2\nenable a direct comparison, we performed a detailed QCT analysis for H + HD under\nthe same initial state and collision energy as for H + D . The comparison clarifies the\n2\nsubtle mechanistic differences in the roaming-like insertion pathway between the two\nreactions.\nThe contributions of the two pathways to the DCSs, and the classical trajectories\nfor the forward and backward scattering products are illustrated and presented in the\nnew Figs.S9 – S11 of the revised Supplementary Information. A side-by-side\ncomparison of the time evolution of bond length and potential energy along roaming-\nlike insertion pathway (path 2) were conducted and presented in the new Figs. S12–\nS15 of the revised Supplementary Information. The quasi-classical trajectories reveal\nthat the two reactions share some similarities on the previously unreported sideways\nroaming-like insertion mechanism. Detailed comparisons show that the time evolution\nof the bond-length and potential energy underlying the roaming-like insertion pathway\nare notably different between the H+HD and the H+D reactions. A detailed illustration\n2\nof the comparison is presented below.\nAs mentioned, in the main text, the critical step in the roaming process was\ndetermined to occur during the dynamically competing interactions at 30–45\nfemtoseconds. To compare the differences in the reaction mechanisms of the two\nisotope reactions though path2, theoretical calculations were conducted on the product\nquantum states (v’=0, j’=0) and (v’=3, j’=12), systematically investigating the bond\nlength stretching characteristics of the two reactant molecules within the 30–45\nfemtosecond time range (as shown in Fig. S13). The results in Fig. S13 show that, for\nthe product with the same ro-vibrational state, the stretching rate of the D bond in the\n2\nindicates that the reaction rate of H+D via path 2 is slower than that of H+HD via path\n2\n2. It’s commonly accepted that the reaction rate strongly relies on the height of the\npotential barrier along the reaction pathway, the evolution of the forward and backward\n15\npotential energies of the H + D → D + HD (v’, j’) reaction via path 2 is further\n2\nanalyzed. The results for the H + D → D + HD (v’=3, j’=12) reaction ranging from\n2\n20 to 75 fs are presented in Fig. 5b and 5d. The potential energy evolution along path 2\ncan be viewed as an effective barrier that reactants must overcome to form products.\nIn order to explore the differences in barrier height in the isotopic reaction system and its\nimpact on the path2 reaction. Further comparison of the time evolution of the total\npotential energy for product in the (v’=0, j’=0) and (v’=3, j’=12) states between the H\n+ HD → D + H and the H + D → D + HD reaction, as shown in Figs. S14-15. In\n2 2\nFig. S14，for high-impact-parameter trajectories, the effective barrier along path 2 is\nlower than the one with low impact parameter in both systems, which explains the\npredominance of backward scattering and the GP effect observed for H + HD at a\ncollision energy of 1.72 eV (Li, S. et al. Nature Communications 15, 1698, (2024)). In\ncontrast, for low-impact-parameter trajectories, the path 2 barrier in H + D is about 0.2\n2\neV higher than in H + HD, leading to a sideways path 2 yield in H + D that is\n2\napproximately half of that in H + HD in our QCT calculations as shown in Figs.S14a-\nb. Furthermore, as shown in Figure S15, the path 2 barrier for the reaction H + D → D\n2\n+ HD (v’=0, j’=0) is ~0.36 eV higher than that for H + HD→ D + H (v’=0, j’=0). This\n2\ncomparison clarifies both the common features and the subtle mechanistic differences\nin the roaming-like insertion pathway between the two reactions. In summary, the QCT\nsimulations highlight the dynamical significance of the newly identified roaming-like\npathway. They reveal the emergence of a roaming-like path 2 involving transient D–\nD bond stretching, which provides a clear explanation for why the GP effect is much\nmore difficult to probe in H + D than in its isotopic variants.(page19-20)\nOn page 5, the authors refer to Supplementary Note 1 for experimental details\nregarding the ionization of the D-atom products. However, there is no information on\nthe ionization scheme in this note, nor in the Methods section. This, and additional\ninformation about how the image in Fig. 1 is measured, should be included.\nReply:\nWe thank the reviewer for pointing out this important point. To address it, we have\nadded a detailed description of the D-atom ionization scheme in the Methods section\nand provided additional details on how the image in Fig. 1 is measured.\nIn the last paragraph on page 5, which continues on page 6, many experimental\ndetails are given that are also described in both the Methods section as well as in\nSupplementary Note 1. We advise the authors to shorten this section.\nReply:\nWe thank the reviewer for this helpful comment. In response, we have shortened\nthe experimental description on pages 6-7, retaining only the key steps (H-beam\ngeneration and D₂-beam preparation). A detailed experimental procedure has been\npresented in the Methods section. Furthermore, as suggested by the reviewer in the next\ncomment, to avoid repetition, we have removed the experimental details given in the\nSupplementary Note 1.\nReviewer: All information that is given in Supplementary Note 1 is also given in the\nMethods section. We therefore advise the authors to remove Supplementary Note 1.\nReply:\nThank you for the comment. We agree with the reviewer. We have therefore\nremoved the context in the original Supplementary Note 1.\nReviewer:\nOn page 7, the authors write “rotation-state resolved image”. We advise the authors\nto rephrase this, since several rovibrational states contribute to these rings. Now, it\nsounds like every ring corresponds to a single rotational state.\nReply:\nWe thank the reviewer for pointing out this ambiguity. We agree that the original\nphrase “rotation-state resolved image” could misleadingly suggest that each ring\ncorresponds to a single rotational state. Upon re-examining the data, we confirm that\nwhile some outer rings (e.g., HD 𝑣=0, 𝑗=2–8, in Fig. S1) can indeed be assigned to\nindividual rotational states, the inner rings result from overlapping contributions of two\nor three rotational states. To more accurately reflect this partial resolution, we have\nrevised the term to “partially rotational-state-resolved image” throughout the\nmanuscript. This revised phrasing clarifies that some rings correspond to individual\nstates, whereas others contain composite contributions, thereby addressing the\nreviewer’s concern.\nFig. 2 shows theoretical calculations with and without inclusion of the geometric\nphase (which are not mentioned in the caption). However, the manuscript provides no\ndetails at all on how these calculations were performed. This information should be\nadded.\nReply:\nWe thank the reviewer for pointing out this omission. In response, we have added\na description of the theoretical calculations to the Methods section, including details on\nhow the results with and without the geometric phase were obtained. This addition\nclarifies the computational approach underlying Fig. 2.\nReviewer:\nOn page 8, the authors discuss a topological approach proposed by Althorpe and\ncoworkers, and refer to ref. 21. We suggest the authors to here refer to ref. 12, and at\nthe end of the same sentence (…pathways around the CI) refer to ref. 21 (their previous\nwork on H + HD, in which they also applied the approach by Althorpe and coworkers).\nReply:\nWe sincerely thank the reviewer for this thoughtful suggestion. Following the\nrecommendation, we have updated the references in the revised manuscript accordingly,\nciting ref. 12 when introducing the topological approach proposed by Althorpe and co-\nworkers, and ref. 21 at the end of the sentence to acknowledge their previous application\nof this approach to the H + HD reaction. We believe this adjustment improves the\naccuracy and clarity of the referencing.\nReviewer:\nPage 9: ‘Figs. 3a and 3c show that the DCSs…’. In this sentence, 3c should be 3b.\nReply:\nWe sincerely appreciate the reviewer’s careful reading and attention to detail.\nFollowing reviewer 2’s suggestion, we have changed the result of Figs. 3a and 3c into\nthe QM generalized deflection functions (GDFs) results. Therefore, we have changed\nthe description as the sentence:\n“To evaluate the contribution of each partial wave to the scattering, Figs. 3a and\n3c present the QM generalized deflection functions (GDFs) together with the QCT\nresults. The GDF, which is additive over J and includes coherences among different J\n18\nvalues, was first introduced by Prof. Jambrina and Prof. Aoiz33. For a more direct\ncomparison with the trajectory analysis, the functions are presented without the usual\nfactor of sinθ.”\nReviewer:\nPage 9: ‘Figs. 3a and 3c show that the DCSs…’. The authors write that partial\nwaves with J = 25 and 35 dominate in the sideways direction. However, we don’t see\nthis so clearly from Fig. 3. We recommend the authors to elaborate on this.\nReply:\nWe thank the reviewer for this insightful comment. Upon re-examination, we also\nfound that the original presentation of the results did not clearly support the statement\nregarding the dominance of partial waves with J=25 and 35 in the sideways scattering\ndirection. In response to Reviewer 2’s earlier suggestion, we attempted to clarify this\npoint by replacing the original figure with one showing the quantum mechanical\ngeneralized deflection functions (GDFs). However, this revision made the results even\nless clear. Therefore, to avoid potential confusion and to maintain clarity, we have\ndecided to remove the sentence from the revised manuscript.\nReviewer:\nFig. 2 shows the angular scattering distributions from 0-70 degrees, whereas Fig.\n3 only starts at 30 degrees. Is there a specific reason for this? If so, we recommend the\nauthors to mention this in the manuscript.\nReply:\nWe sincerely thank the reviewer for raising this important question. The angular\nrange in Fig. 3 begins at 30° because the quasi-classical trajectory (QCT) method used\nfor these calculations is less reliable in the extreme forward direction (0–30°), where\nquantum effects play a dominant role. More importantly, the key interference features\ndiscussed in this work are most clearly manifested in the 30°–60° range. To avoid\nconfusion, we have added a clarifying statement in the caption of Fig. 3 to explain the\nchoice of angular range.\nReviewer:\nPage 9: ‘Previously, the collision dynamics of…’ It seems like a reference is\nmissing in this sentence.\nWe sincerely thank the reviewer for pointing out this omission. We have now\nadded the appropriate reference to support this statement in the revised manuscript.\nReviewer:\nPage 10: we recommend the authors to move the sentence ‘Interestingly, Fig. 3\nalso highlights…’ and place it before the sentence ‘Notably, the partial waves…’.\nReply:\nWe greatly appreciate the reviewer’s helpful suggestion. As suggested in the\nsecond comment by the reviewer for revising the last part of the manuscript (stating\nfrom page 11) to make the discussion more accessible to a broader audience, we have\nsubstantially revised the relevant section of the manuscript including the context\nmentioned in this comment.\nReviewer:\nThe authors often use the word ‘collision parameters. We advise the authors to use\npartial waves or impact parameters instead.\nReply:\nWe thank the reviewer’s thoughtful suggestion. To align with standard terminology,\nwe have revised the manuscript by replacing all instances of “collision parameters” with\neither “partial waves” (when referring to quantum-mechanical angular momentum\ncontributions) or “impact parameters” (when referring to classical trajectory dynamics).\nWe believe this change improves both the precision and clarity of the terminology used\nthroughout the manuscript.\nReviewer:\nPage 10: ‘Notably, the partial waves…’. We recommend the authors to reformulate\nthis sentence for clarity. Moreover, we would say that the ranges of J=16-25 and J=5-\n10 should be a bit larger, when inspecting Fig. 3.\nReply:\nWe are grateful to the reviewer for this helpful suggestion. In response, we have\nreformulated the sentence to improve clarity. We also agree that the partial wave ranges\nshould be slightly broadened based on Fig. 3, and have accordingly updated the text to\nspecify 𝐽=0–15 and 𝐽=16–30. We believe these revisions better reflect the data and\n20\nenhance the accuracy of the discussion.\nReviewer:\nPage 10: ‘The looping pathway…’. It seems like references are missing here.\nReply:\nWe appreciate the reviewer’s careful reading and for pointing this out. In the\nrevised manuscript, we have added the appropriate references to support the statement\nconcerning the looping pathway. We believe this addition strengthens the context and\naccuracy of the discussion.\nReviewer:\nPage 10, last paragraph: it is discussed that the roaming insertion pathway in this\nreaction system is unusual. However, it is also already discussed and used in other\npapers on H + HD (ref. 21 for instance). So, we advise the authors to update this\nsentence.\nReply:\nWe thank the reviewer for the helpful suggestion. While roaming insertion\npathways have indeed been observed in related systems such as H + HD (e.g., Ref. 21),\nwe emphasize that their appearance in the H + D reaction is particularly noteworthy.\nenergy underlying the roaming-like insertion pathway are notably different between the\nH+HD and the H+D reactions.\n2\nbond in the H+D reaction is slower than that of the HD bond in the H+HD reaction;\n2\nthis result indicates that the reaction rate of H+D via path 2 is slower than that of\nin H + D is about 0.2 eV higher than in H + HD, leading to a sideways path 2 yield in\n2\nin Figs.S14a-b. Furthermore, as shown in Figure S15, the path 2 barrier for the reaction\nH + D → D + HD (v’=0, j’=0) is ~0.36 eV higher than that for H + HD→ D + H\n2 2\n(v’=0, j’=0). Therefore, the QCT results clarified the subtle mechanistic differences in\nthe roaming-like insertion pathway between the two reactions.\nThe sentence has been updated to be as follows:\n“The H + D reaction is a system with only three electrons yields only one product\n2\n21\nchannel HD + D. The fact that a roaming insertion pathway arises even in this more\nconstrained and electronically minimal system highlights its generality and makes its\nappearance here particularly remarkable.”\nReviewer:\nPage 12: ‘On the other hand … as shown in Figs. S6 and S7…’. This should\nprobably only be Fig. S6. Fig. S7 is for backward scattering and is useful for the\nprevious sentence (‘By comparing the trajectories, …’).\nReply:\nWe thank the reviewer for the helpful comment. We agree that Fig. S7 refers to\nbackward scattering and is more relevant to the previous sentence. Accordingly, we\nhave revised the text on page 12 to cite only Fig. S6 in the sentence beginning with “On\nthe other hand…”.\nReviewer:\nFig. 4: we advise the authors to improve the caption, especially regarding panels c\nand d.\nReply:\nWe thank the reviewer for the suggestion. We have revised the caption of Fig. 4\n(now Fig.5) to provide clearer descriptions of all panels, especially panels c and d. The\nupdated caption now explicitly distinguishes between the forward roaming insertion\npathway (panels a and b) and the retrograde pathway (panels c and d), and clarifies the\nphysical quantities shown in each panel.\nReviewer:\nPage 12-14: It is not always clear whether the authors refer to the forward\nscattering (low impact parameter) roaming pathway or to the backward scattering (high\nimpact parameter) roaming pathway. Often, the authors just use Path 2 to indicate both\nof them or maybe only one of them. It would help to clarify this.\nReply:\nWe thank the reviewer for this valuable comment. We would like to clarify that\nthroughout the discussion on pages 12–14, Path 2 consistently refers to the roaming\ninsertion pathway associated with low impact parameters and forward (more precisely,\nsideways-to-forward) scattering. The high-impact-parameter pathway that contributes\n22\nto backward scattering, which was identified in our previous work in the H+HD\nreaction, is not the focus of the present study and is therefore not discussed in detail in\nthis section.\nTo avoid any ambiguity, we have revised the relevant paragraphs to explicitly state\nthat all references to Path 2 in this part of the manuscript concern the low-impact-\nparameter, forward-scattering roaming mechanism. We believe this clarification\nresolves the concern, and the revised text has been added on page12-14.\nReviewer: Page 13: ‘Simultaneously, the competing interaction…’. Please clarify\nwhich interaction is meant here.\nReply:\nWe thank the reviewer for pointing out this ambiguity. The term “competing\ninteraction” refers to the attractive force between the incoming H atom and the second\nD atom (D ), which weakens the original D –D bond. This interaction stretches the\nb a b\nD–D bond and lowers the barrier for the H atom to insert. To improve clarity, we have\nrevised the sentence on page 18 as follows:\n“Simultaneously, the competing interaction also extends the D-D bond, reducing\nthe steric hindrance to H atom attack, and creating a new reaction site. The H atom\nthen inserts between the D -D bond and approaches this site, ultimately forming the\na b\nreaction products. Notably, the roaming insertion pathway is more efficient and feasible\nthan the direct insertion pathway. As a result, most path 2 reactions process via the\nroaming pathway regardless of impact parameters, leading to the unconventional\nscattering direction of path 2 products. ”\nReviewer:\nPage 13: ‘Moreover, the roaming mechanism (path 2) in the H + D reaction relies\n2\non a symmetric stretch…’ we assume the authors base this on Fig. 4a and c, for times\nbetween 30 45 fs, where the H-Da and Da-Db distances are similar. If so, please clarify\nthis in the text.\nReply:\nWe sincerely thank the reviewer for this helpful comment. The interpretation that\nthe roaming mechanism involves a symmetric stretch is indeed based on Fig. 4a and 4c,\nspecifically during the interval between 30 and 45 fs, where the H–Da and Da–Db bond\ndistances become comparable. To make this clearer, we have revised the text on page\n23\n13, which now reads:\n“… This is evident in Fig. 4a and 4c, particularly during the time window between\n30 and 45 fs, where the H–Da and Da–Db bond distances become nearly equal…”\nReviewer:\nRegarding Fig. 5 and the corresponding interpretation: the authors mention that D\n2\nis excited to v=3. Does the exact vibrational level of D (so v=3) really follow from the\n2\ncalculations, is v=3 used since the mechanism was found for HD (v=3), or is this\nspeculation (maybe based on ref. 37)? We recommend the authors to clarify this in the\ntext.\nReply:\nWe sincerely thank the reviewer for this important comment and the opportunity\nto clarify this point. The assignment of D being excited to v3 is primarily based on\n2\nanalogy to the mechanism described in Ref. 38 (Greaves, S. J. et al. Nature 454, 88-91,\n(2008)), which reports that in H + D₂ collisions at similar energy regimes, symmetric\nstretching interactions lead to vibrational excitation, with v=3 being a prominently\npopulated level. While our trajectory calculations capture the vibrational stretching\nmotion of D₂ during the reaction, they do not explicitly resolve the exact vibrational\nquantum number. Nevertheless, the energetic consistency between the observed\nHD(v’=3) product state and the vibrational energy expected for D (v=3) (as discussed\n2\nin Ref. 38) supports this assignment. To avoid overstating our interpretation, we have\nrevised the text to make clear that D (v3) is inferred from the analogous mechanism\n2\nreported in Ref. 37 and is consistent with the energetic matching to the HD(v’=3)\nproduct. We hope this clarification addresses the reviewer’s concern.\nReviewer:\nMethods: ‘The experimental setup has been described in detail in a previous\narticle…’ A reference could be added here.\nReply:\nWe thank the reviewer for this helpful suggestion. In the revised manuscript, we\nhave added the appropriate reference to the sentence in the Methods section that\ndescribes the experimental setup.\nReviewer:\nWe advise the authors to here use j’=2-8, since j’=1 and 0 seem to overlap.\nReply:\nWe thank the reviewer for this helpful comment. We have revised the sentence in\nSupplementary Note 1 accordingly.\nReviewer:\nFig. S3, in the caption: ‘bule’ should be ‘blue’\nReply:\nWe thank the reviewer for carefully noticing this typographical error. The word\n“bule” in the caption of Fig. S3 has been corrected to “blue” in the revised manuscript.\nReviewer:\nSupplementary Note 4: It would be useful to provide some more detail on how the\nQCT trajectory calculations were performed (or to add references if the methods are\nsimilar to what has been used for H + HD for instance). A reference to the BKMP2 PES\nshould be included as well. Moreover, the abbreviation PES is not introduced.\nReply:\nWe thank the reviewer for this valuable comment. In response, we have added\nadditional details regarding the theoretical calculations in the Methods section.\nSpecifically, we have expanded the description of the QCT calculations and included\nthe relevant references for the PES employed in this work.\nReviewer:\nIn Supplementary Note 4, the authors use as notation H’ and D. We advise the\nauthors to here use similar notation as in the main text (H, Da and Db).\nReply:\nWe thank the reviewer for this helpful comment. We agree that consistent notation\nbetween the main text and the Supplementary Materials is important for clarity.\nAccordingly, in Supplementary Note 4 we have revised the notation, replacing H′\nand D with H, Da, and Db to match the usage in the main text.\nReviewer:\nFig. S6 and S7: it is not described what the red arrows in the left panels are\n25\n(probably the D atom movement). Moreover, labels a-e are missing in the figure itself\n(the caption refers to these labels). From the caption, it seems like the conical\nintersection (CI) is indicated by crosses in all panels. However, this is only for panel e.\nReply:\nWe sincerely thank the reviewer for the careful examination of Fig. S7 and for the\nhelpful comments. We agree with the reviewer and have addressed the issues raised.\nIn the revised version, we have clarified that the red arrows indicate the direction\nof motion of the D atom during the roaming process. The missing panel labels (a–e)\nhave now been added in the revised figure to match the references in the caption.\nWe have now corrected the caption to specify that the CI is marked only in panel e.\nReviewer:\nFig. S7, in the caption: ‘The potential energy surface contours… for a-e.’ Here, a-\ne should be a-d.\nReply:\nWe sincerely thank the reviewer for the careful reading and for pointing out this\ninconsistency. We apologize for the oversight in the caption of Fig. S7. The potential\nenergy surface contours are indeed shown for panels a–d, not a–e as originally stated,\nand we have corrected this in the revised manuscript.\nReviewer:\nIn some sentences, the English (grammar) can be improved.\nReply:\nWe thank the reviewer for this comment. Although no specific sentences were\nindicated, we have carefully re-examined the manuscript and made revisions to improve\nthe grammar, language flow, and overall clarity.\nWe would like to express our sincere gratitude to all the reviewers for their\ninsightful and thoughtful comments, which are very important for further improving\nthe quality and clarity of the manuscript. In the following, we provide detailed, point-\nby-point responses to each comment.\nWe are grateful to the reviewer for the thoughtful and valuable comments, which have\nbeen very helpful in improving the manuscript.\nReviewer:\nTitle: ”Sideways Scattering Oscillations reveal Geometric Phase Effect in the H+D →\nAurhors: Shihao Li et al.\nThe manuscript has been extensively rewritten, albeit without altering its essential\ncontent. In my opinion, the authors have satisfactorily addressed most of the comments\nfrom the four reviewers of the original manuscript. As a result, the article has improved\ngreatly and is almost ready for publication. However, I have some\ncomments/corrections that the authors should consider before the manuscript is\ndefinitively accepted.\n(All my comments and page numbering refer to the red-corrected manuscript.)\nReply:\nWe thank the reviewer very much for the positive evaluation of the revised\nmanuscript and your recognition of our efforts in addressing the reviewers' comments.\nWe greatly appreciate the detailed and constructive feedback, which is crucial for\nfurther improving the quality of our work.\nReviewer:\nOn pg.11, the sentence “The GDF, which is additive over J and includes\ncoherences among different J values, ...’ is not entirely correct as it does not explain the\ncontent of Fig.3. It should read as follows: ‘The quantum GDF represents the quantum\ncorrelation between the total angular momentum, J, and the scattering angle, and can\nbe considered as a joint quasi-probability distribution function of J and  that includes\nall coherences between different partial waves. The summation of this function over all\npartial waves recovers the exact differential cross section(see ref.33 for more\ndetails).’Incidentally, I suggest suppressing’…was first introduced by Prof. Jambrina\nand Prof. Aoiz.’ Referring to reference 33 is sufficient.\nReply:\nWe thank the reviewer for pointing out this inaccuracy and for suggesting a clearer\n2\nand more precise description. We have revised the sentence on page 11 accordingly,\nreplacing it with the reviewer’s recommended wording:\n“The quantum GDF represents the quantum correlation between the total angular\nmomentum, J, and the scattering angle, and can be considered as a joint quasi-\nprobability distribution function of J and θ that includes all coherences between\ndifferent partial waves. The summation of this function over all partial waves recovers\nthe exact differential cross section (see Ref. 33 for more details).”\nWe also agree with the reviewer’s suggestion to suppress the phrase “…was first\nintroduced by Prof. Jambrina and Prof. Aoiz,” and instead refer to Ref. 33 directly.\nReviewer:\nFig. 3 contains some inaccuracies that should be corrected:\n The caption for Fig.3 starts with ‘Topological methods were used…,’ as do the\ncaptions for Fig.S7, S8 and S9. This is very vague for what is shown in the\nfigure. As more specific alternatives, I suggest ‘Quantum correlation function\nbetween total angular momentum and scattering angle’ or ‘Joint quansi-\nprobability distribution of total angular momentum and scattering angle\n(quantum deflection function)’.\n It does not indicate which state the shown GDF belongs to.\n If it refers to a combination of states, important information is lost in QM GDF,\nsince the incoherent sum over a combination of states blurs the coherence\nbetween the different J states contributing to a given scattering angle. Note that\nthere is not interference between the partial waves from the different final states.\nOf course, there is not such interference in the QCT DF and carrying out the\nsummation is correct, although information is also lost by mixing the classical\n(J,) correlations of different states.\nI would suggest showing the QM GDF for given state (e.g. v’=3, j’=12), and\nleaving the respective plots of other states for the SM, as is currently done. The\nsame could be done with the classical GDF. Alternatively, if a global view of\nthe overall classical behavior is intended, this should be shown in a different\nfigure.\n The scale of impact parameters is incorrect. J=40 corresponds to b=1.33 Å. The\nmaximum impact parameter used in the calculations, 1.59Å, corresponds to J\n≈48.\n3\n The sentence ‘ The angular range in Fig.4 begins at 30° due to the limited\naccuracy of the QCT method in the extreme forward direction (0–30°), where\nquantum interference effects are significant’ is misleading. If the authors mean\nwhat they say in the final part of the sentence (‘where quantum interference\neffects are significant’), then something important is missing. I think the\nauthors actually meant the opposite.\nAs for the assertion that the QCT method in the extreme forward region(0-30°)\nis of limited accuracy is incorrect. For Path﹟1 there is no problem as I reckon\nthere should be ≈8106 reactive trajectories (for N =20106), and even when\ntot\nresolved into final states, there will be enough trajectories to obtain accurate\nQCT results for extreme forward and backward scattering. This is not the case\nfor path﹟2 where the total number of trajectories is less than 4,000(0.44% of\nthe reactive trajectories), distributed among all final states and scattering angles.\nTherefore the issue is not the accuracy of the QCT method, but the scarcity of\nreactive trajectories following path﹟2. This should be clarified.\nReply:\nWe sincerely thank the reviewer for the careful reading of our manuscript and for\nthe detailed and constructive comments regarding Fig. 3. We have addressed all of them\nas follows:\n1. Caption wording:\nWe agree that the previous caption starting with “Topological methods were\nused …” was too vague. By following the reviewer’s suggestion, we have added\na more specific description as follows at the beginning of the caption:\n“Quantum correlation function between total angular momentum and\nscattering angle for quantum state (v’=3, j’=12).”\n2. Indication of the state:\nWe appreciate the reviewer’s remark that the figure did not specify which state\nthe GDF corresponds to. We have clarified in the caption that the state is (v’=3,\nj’=12).\n3. Use of state-specific GDF:\nWe fully agree with the reviewer that showing the QM GDF for a specific state\nprovides clearer physical insights and avoids the blurring of coherence effects.\nIn the revised manuscript, we have explicitly indicated that the QM GDF shown\ncorresponds to (v’=3, j’=12). We also agree that there is no interference in the\nj’=12) dominates the mixed-state distribution, and the additional states are not\nessential for the GDF analysis.\n4. Impact parameter scale:\nWe thank the reviewer for identifying this mistake. To ensure the relation\nbetween J and b is properly indicated, we have corrected the unit of the impact\nparameter to “bohr”, which was mistakenly given as “Å”.\n5. Clarification of QCT accuracy in the forward region:\nWe appreciate the reviewer’s insightful clarification. The previous sentence in\nthe manuscript was indeed misleading and has been revised. It now reads:\n“The angular range begins at 30° due to the scarcity of reactive trajectories\nfollowing Path 2. The most pronounced interference structures are observed in\nthe 30°–60° range.”\nWe are grateful for these precise comments, which help us to improve both the\naccuracy and the clarity of our presentation.\nReviewer:\nIn the ‘Method’ section, the authors should explain the practical conditions that\nthey used to determine whether a trajectory was assigned to path﹟2.\nReply:\nWe thank the reviewer for pointing this out. We have added the following sentence\nto clarify the practical conditions used to determine whether a trajectory was assigned\nto Path 2. It now reads:\n“As illustrated in Figs. S10 and S11, the position of the transition state can be identified\nin hyperspherical coordinates, and the number of trajectories passing through the\ntransition state allows the assignment of Path 1 and Path 2.”\nReviewer:\nOn Pg.8, instead of ‘the partial rotational-state resolved image’ change to ‘the\nimage showing rotational states partially resolved’.\nReply:\nWe thank the reviewer for pointing this out. We have made revisions on page 8.\nReviewer:\n(sic) state’ is correct. It should be either ‘the combination of states’ or the ‘sum of the\ncontributions from the aforementioned states.\nReply:\nWe thank the reviewer for the suggestion. The text “the HD combinational state”\nin the caption of Fig. 2 has been revised to be “the combination of states”.\nReviewer:\nc). On Pg.18 referring to reference 39, the text reads ‘…by Zare et al.’; it should sat\nGreaves et al.\nReply:\nWe thank the reviewer for the suggestion. The text has been revised accordingly.\nWe sincerely thank the reviewer for the thoughtful and helpful comments, which have\nbeen very important in improving the manuscript.\nReviewer #4 (Remarks to the Author):\nWe would like to thank the authors for their detailed replies, the additional calculations\nand analysis, and the many revisions made to the manuscript.\nWe are very pleased to see that the oscillatory structures shown in panel 2b can indeed\nbe attributed to the same roaming insertion mechanism.\nWe also appreciate the comparison between the H + D and H + HD systems. The barrier\n2\nfor path 2 is higher for H + D than for H + HD, which explains why this path\n2\ncontributes less for H + D , and why the GP effect is harder to measure for this system.\nmechanism is present in H + D that is absent in H + HD. We find that the differences\n2\nbetween the systems are due to very subtle isotope effects that arise from differences in\nvibrational timescales and energy barriers between the two systems, due to their\ndifferent masses. In addition, the manuscript remains quite inconclusive about the\npresence of oscillatory structures in sideways scattering in the H + HD system, that one\ncould use to compare to the sideways oscillations observed for H + D .\nobserved here for H + D is something specific for H + D (and justifies the narrative\n2 2\nthat a new mechanism is present), but we feel that to substantiate such claim, a clearer\ndistinction must be made between an “isotope effect of the same underlying mechanism”\nand “a new mechanism”.\nIn our opinion, it is extraordinary that the authors are now able to resolve such\nsubtle isotope effects experimentally, and that they can disentangle them in great detail\nthrough rigorous theory and analysis. We feel that this is the true accomplishment of\nthis work, and we therefore encourage the authors to adjust the title, manuscript\nnarrative and conclusions to instead emphasize the subtle isotope effects that they have\nso impressively observed and explained.\nReply:\nWe sincerely appreciate the reviewers’ insightful comment. We agree with the\nreviewers that isotope effects play a crucial role in relative contributions of different\n7\nreaction pathways in the two systems, and these effects make the GP effect more\nchallenging to observe in the H+D reaction. Accordingly, following the reviewers’\n2\nsuggestion, we have adjusted the title and conclusions of the manuscript to highlight\nthe subtle isotope effects.\nThe influence of the geometric phase (GP) on chemical reactions has emerged as\na pivotal topic in chemical physics over the past decades. High-resolution scattering\nmeasurements previously revealed GP signatures in the H+HD system, manifested as\npronounced forward angular oscillations in the H product distribution. While, the GP\n2\neffect exerts barely changes in the sideways structures in the H+HD system. In contrast,\nfor the isotopic variant H+D system, the observed significant sideways angular\n2\noscillations in the HD product distribution, combined with accurate quantum\nmechanical calculations, provide unambiguous evidence of GP-induced quantum\ninterference. Thus, the sideways oscillations observed in H+D are distinctive when\n2\ncompared with the isotopic H+HD system.\nFurther analysis of the time-dependent bond length changes in the H+D reaction\n2\nrevealed that the interaction between the incoming H atom and the D molecule\n2\nelongates the D-D bond, promoting product formation via Path 2. The analysis also\ndemonstrated the role of the potential barrier height in modulating the GP effect.\nIn contrast to the reaction with HD, it reflects the isotopic effect play a key role in relative\ncontributions of Path 2 in the two systems.\nIn summary, this study provides conclusive evidence for GP effect in the H+D\n2\nreaction through the observation of sideways oscillations, thereby resolving the long-\nstanding challenge of the directly identifying GP signatures in this system. Moreover,\nthe specific influence of the synchronous elongation and potential barrier along Path 2\nare clearly identified, indicating that the geometric phase is modulated by the isotopic\neffect, which advanced our understanding of GP effects in reaction dynamics.\nThe title now is “Sideways Scattering Oscillations reveal Geometric Phase Effect\nand Isotope Effect in the H + D → HD + D Reaction”\nhighlights a pronounced isotopic effect while reflecting the unique dynamic competition\nwithin the roaming insertion pathway. Collectively, these findings not only underscore\nthe critical role of pathway interference but also provide novel mechanistic insights into\nhow the isotopic effect modulates the geometric phase in an elementary chemical\nreaction.”\n8"}]

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