HF(v′ = 3) forward scattering in the F + H₂ reaction: Shape resonance and slow-down mechanism

Abstract

Crossed molecular beam experiments and accurate quantum dynamics calculations have been carried out to address the long standing and intriguing issue of the forward scattering observed in the F + H₂ → HF(v′ = 3) + H reaction. Our study reveals that forward scattering in the reaction channel is not caused by Feshbach or dynamical resonances as in the F + H₂ → HF(v′ = 2) + H reaction. It is caused predominantly by the slow-down mechanism over the centrifugal barrier in the exit channel, with some small contribution from the shape resonance mechanism in a very small collision energy regime slightly above the HF(v′ = 3) threshold. Our analysis also shows that forward scattering caused by dynamical resonances can very likely be accompanied by forward scattering in a different product vibrational state caused by a slow-down mechanism.

Chemical reactions occur when one reactant collides with another and some rearrangements among reactants take place along a path connecting reactants to products. The path is called the reaction coordinate for a chemical reaction, along which the reactants will go through an intimate region to reach the product side. In a typical chemical reaction with an energetic barrier, no discrete quantum structure could exist along the reaction coordinate. However, quantized states do exist along coordinates perpendicular to the reaction coordinate. For each quantized state, there is an effective, vibrationally adiabatic potential. In certain cases, transiently trapped quantum states could exist on these vibrational adiabatic potentials along the reaction coordinate. Such quasi-bound quantized states along the reaction coordinate in the intimate region of a chemical reaction are normally called dynamical resonances, or reaction resonances. Because reaction resonances are very sensitive to the potential energy surface governing a chemical reaction, they provide possibilities for probing the critical region of the potential energy surface more directly. As a result, reaction dynamics has been a central topic in the study of chemical reaction dynamics in the last few decades (1–4).

Probing of dynamical resonances experimentally is essential to the study of the resonances in chemical reactions. A key signature of reaction resonance is the product forward scattering caused by the time delay of the reaction system trapped in quasi-bound resonance states. However, forward scattering in a scattering experiment does not necessarily come from reaction resonances. Recently, Zare and coworkers (5) have attributed the forward scattering in the H + D₂ reaction to a time delay mechanism. In the study of the H + HD system by Harich et al., the forward scattering was attributed to the time delay when the reaction system passes over a specific reaction barrier with little translational speed (6, 7). Therefore, distinguishing which mechanism is causing the time delay and the forward scattering product in a specific reaction has become a key issue in assigning reaction resonances.

A textbook example for reaction resonance is the F + H₂ → HF + H reaction (8, 9). The existence of reaction resonances in the F + H₂ reaction was first theoretically predicted in the 1970s (10–12). In the middle of the 1980s, Lee and coworkers (13, 14) performed a milestone crossed-beams experiment on the F + H₂ → HF + H reaction with HF product vibrational state resolved. In this work, forward scattering of the HF(v′ = 3) product was unambiguously observed. This forward scattering was tentatively attributed to a dynamical resonance effect. But subsequent theoretical studies (15, 16) based on the Stark–Werner PES (SW-PES) (17) have suggested that the HF(v′ = 3) forward peak is likely not related to the effect of a dynamical resonance. However, due to the defect of the SW-PES on the threshold energy for HF(v′ = 3), the theoretical calculations carried out on the PES were not able to provide conclusive insights to the mechanism of the HF(v′ = 3) forward scattering at threshold collision energies. The FH₂⁻ photoelectron spectrum study provided a useful meaning to directly probe the potential energy surface near the transition state region, which might shed light on the reaction resonance in the system. Indeed, the experiment not only observed signal due to the resonance in the reaction but also discovered direct evidence that the F + H₂ reaction has a bend transition state (18). However, because of limited experimental resolution, the experiment could not resolve the signal associated with the reaction resonance and hence did not supply any new clue to the dynamical origin of the HF(v′ = 3) forward scattering product in the F + H₂ → HF + H reaction. On the other hard, joint experimental and theoretical studies on the F + HD → HF + D reaction have detected and characterized a Feshbach resonance in the reaction that has a dramatic effect on the behavior of the HF(v′ = 2) product (19–21). Recently, we have observed forward scattering HF(v′ = 2) product from the F + H₂ reaction at the collision energy of 0.52 kcal/mol. Theoretical analysis based on a newly constructed PES attributed this pronounced HF(v′ = 2) forward scattering peak to the interference of two Feshbach resonance states that are quasi-trapped in the HF(v′ = 3)–H vibrational adiabatic potential (VAP) well (22). However, after all of the endeavors, an intriguing question still remains unanswered for this important system: Is the forward scattering HF(v′ = 3) product originally observed by Lee and coworkers (13) related to Feshbach resonances in this reaction?

We have re-investigated carefully the F(²P_3/2) + H₂ (j = 0) → HF(v′ = 3) + H reaction channel both experimentally and theoretically, in an effort to clarify this issue. A full quantum state resolved crossed-beams scattering study on the F + H₂ reaction has been carried out by using the H-atom Rydberg tagging time-of-flight (TOF) method (22–26). The new experimental apparatus used for this experiment is described in ref. 27. A unique feature of the experiment is the use of a double stage pulsed discharge beam source for F atom (28), in an effort to increase the F atom beam intensity. The F atom generated from the F₂/He mixture is found to be predominantly in the ground state (²P_3/2). The H₂ beam is obtained by expanding a neat H₂ sample from the liquid N₂-cooled pulsed nozzle. Collision energy can be varied in the experiment by changing the crossing angle between the F and H₂ beams.

TOF signals of the H atom product were measured at a series of laboratory angles for the F + H₂ reaction in the range of collision energy between 0.40 and 1.2 kcal/mol. To determine the relative integral cross-section (ICS) as a function of collision energy, relative TOF signals at specific lab angles at different collision energies were measured carefully back and forth many times to reduce the measurement error. Fig. 1 shows the measured HF(v′ = 3) signal as a function of collision energy from 0.4 to 1.2 kcal/mol. From the experimental data, it is very clear that the HF(v′ = 3) appears right at the threshold of this reaction channel around 0.52 kcal/mol, suggesting that the HF(v′ = 3) channel does not have any detectable intrinsic exit barrier. The ICS for the HF(v′ = 3) channel also appears to increase as the collision energy almost monotonically after the initial step in the excitation function at the threshold of this channel.

Fig. 1.

Total excitation function for the HF(v′ = 3) channel from the F(²P_3/2) + H₂ (j = 0) reaction. The squares are the experimental data, and the error bars are the measurement errors. The solid line is the theoretical excitation function using the full quantum dynamics calculations based on the new global potential energy surface.

Differential cross-sections (DCS) in the center-of-mass (CM) frame have also been determined in the above collision energy range by converting the TOF spectra in the laboratory frame, using a standard Jacobian transformation. Fig. 2A shows the full rotational-state resolved DCS at the collision energy of 0.94 kcal/mol. Only three rotational states of HF(v′ = 3) are energetically accessible at this collision energy, and all of these rotational states, in particular the HF(v′ = 3, j′ = 0) product, exhibit pronounced forward scattering features. Furthermore, notable forward scattering of the HF(v′ = 3) product was also observed at other collision energies above the threshold energy of the HF(v′ = 3) channel, and it persists at all of the collision energies studied in this work. This observation is consistent with the previous experimental observation by Lee and coworkers (14), in which forward scattering HF(v′ = 3) was observed at higher collision energies. The intriguing question is whether the mechanism of the HF(v′ = 3) forward scattering from low to high collision energy is related to the dynamical resonances in this reaction.

Fig. 2.

Experimental (A) and theoretical (B) three-dimensional DCS contour plots for the F(²P_3/2) + H₂(j = 0) → HF(v′ = 3, j′) + H reaction at the collision energy 0.94 kcal/mol.

To understand the mechanism for the forward scattering, we have constructed a new potential energy surface based on the spin unrestricted, coupled cluster method, including single and double excitations with perturbative accounts of triple excitations, using an augmented, correlation consistent, polarized valence quintuple zeta quality basis set [UCCSD(T)/aug-cc-pV5Z]. The correlation energies for all of the ab initio data points were scaled by a factor of 1.01 to obtain the exact exothermicity (32.00 kcal/mol) in this reaction with the spin–orbit interaction energy included. A new global potential energy surface was then constructed by using the three-dimensional cubic-spline method (B.F., X.X., and D.H.Z., unpublished work)—the potential energy surface is available upon request. Finally, the spin–orbit interaction energy function used in XXZ PES (29) was added to the new PES to include the spin–orbit interaction.

Full quantum scattering calculations also have been carried out on the new PES by using the ABC program (30) to determine fully converged, product quantum state resolved ICS and DCS for the F + H₂ reaction. In comparison with our previously reported DCS for the F + H₂ reaction at 0.52 kcal, this new PES seems to be very accurate in describing the Feshbach resonances in this system. Fig. 1 shows the calculated ICS as a function of collision energy for the HF(v′ = 3) channel in the F + H₂ reaction, in comparison with the experimental results. The agreement between experiment and theory is very good. Detailed analysis of the new potential energy surface confirms that there is no exit barrier for the HF(v′ = 3) reaction channel. The calculated DCS for the HF(v′ = 3) channel at the collision energy 0.94 kcal/mol is shown in Fig. 2B. The theoretical forward scattering peaks are very similar to the experimental result. Theoretical results also show forward scattering in this reaction channel at other collision energies above the reaction threshold (0.52 kcal/mol) in agreement with the experimental observation. Furthermore, the theoretical DCS shows an intriguing oscillation in the angular distributions of the HF(v′ = 3, j′ = 0,1) products, which is believed to arise from the interference between near-side and far-side scattering amplitudes (31). The experimental result in this work does not show such fine oscillatory structures in the DCS, mainly because of the angular resolution in this experiment. To see this oscillatory structure, one will need at least 1° to 2° resolution in the CM frame in the forward scattering direction, where the oscillatory structure is most obvious. In the present experiment, we have taken experimental data every 5° in the LAB frame because the multichannel detector has an angular resolution of 3°. This is not sufficient to resolve these fine angular structures. Considering this factor, the overall agreement between theory and experiment for the DCS at this collision energy is quite good. The agreement between experiment and theory indicates that our theoretical model or the new theoretical potential energy surface should be quite accurate for this channel. This provides us a solid foundation to interpret the intriguing dynamics of the forward scattering peak for the HF(v′ = 3) product.

The ICS shown in Fig. 1 and the DCS shown in Fig. 2B comprise contributions from all relevant partial waves with different total angular momentum J. For ICS, the contributions can be calculated independently for each partial wave as (2J + 1)P(J) where P(J) is the total reaction probability for that partial wave. Whereas for DCS, we calculate the contribution for the J partial wave as the difference between DCS with 0 to J partial waves included and that with 0 to (J − 1) partial waves included; i.e., DCS(J) − DCS(J − 1). Fig. 3A shows the partial wave contributions to ICS at the collision energy of 0.94 kcal/mol. For this collision energy, partial waves of J = 4 and 5 have the biggest contributions to ICS. The contributions falls down quickly as J increases further, and becomes negligible when J = 11. In strong contrast, the partial wave contributions to DCS in the forward scattering direction are dominated by J = 9 and 10 components, in particular the J = 10 component, as shown in Fig. 3B. Obviously, the centrifugal potential creates a barrier on the reaction path that prevents the J = 11 partial wave from reacting but also causes a time delay for the J = 10 partial wave. To search the location of the barrier, we plot in Fig. 4 the one-dimensional vibrational adiabatic potentials of HF(v′ = 3)–H along the reaction coordinate for J = 0–11 partial waves, obtained from the new potential energy surface. The reaction coordinate is defined as the distance between the center of mass of HF molecule and the departing H atom. One can see from the figure that there is no exit barrier for the HF(v′ = 3) channel for J = 0. But as the total angular momentum J increases, a centrifugal barrier is gradually formed due to the effect of the centrifugal potential term. At the collision energy of 0.94 kcal/mol, the centrifugal barrier is too high for the J = 11 partial wave to tunnel through, resulting in the vanishing of the total reaction probability for J = 11 as shown in Fig. 3A. The scattering wave function can barely go through the top of the J = 9 centrifugal barrier but has to tunnel through the J = 10 barrier, which causes some time delay for the partial waves. The desired time delay is found to be 80 fs and 165 fs for the J = 9 and 10 partial waves, respectively, by use of the expression given by Smith (32). It turns out that the 165-fs delay time for the J = 10 partial wave is very close to the time needed for the reaction complex to rotate by half a circle, which is found to be ≈200 fs. Consequently, the J = 10 partial wave produces the observed forward scattering peak for the HF(v′ = 3) product shown in Fig. 3B. We carried out similar analyses for the other collision energies and obtained essentially the same conclusion as discussed above.

Fig. 3.

Partial wave contributions to integral cross-section (A) and differential cross-section (B) at the forward scattering direction, at the collision energy 0.94 kcal/mol.

Fig. 4.

One-dimensional vibrational adiabatic potentials of HF(v′ = 3)–H along the reaction coordinate for J = 0–11 partial waves.

Hence, the dynamical mechanisms for the observed forward scattering for the HF(v′ = 2) and HF(v′ = 3) products in the F + H₂ reaction are completely unfolded. Fig. 5 shows the one-dimensional vibrational adiabatic potentials of both HF(v′ = 2)–H and HF(v′ = 3)–H along the reaction coordinate at different J. One can see clearly that the resonance states that are much related to the dynamics of the F + H₂ reaction are trapped quasi-bound states on the vibration-adiabatic potential well of HF(v′ = 3)–H. For the total angular momentum of J = 0, there are two resonance states, the ground and the first excited states as shown in Fig. 5A. As J increases, the HF(v′ = 3)–H potential well could not support the excited state resonance state anymore at some point (Fig. 5 B and C), and eventually the ground resonance state also disappears above J = 9 (Fig. 5D). This is simply due to the effect of the centrifugal potential term that makes the effective potential well increasingly shallower as the total angular momentum J increases. In this vibrational adiabatic picture, the reaction via these resonance states to produce the HF(v′ = 2) channel has to occur via the nonadiabatic coupling between the two vibration-adiabatic potentials. This is clearly a typical case of Feshbach resonances in chemical reaction (33), similar to the case of molecular predissociation. The forward scattering in the HF(v′ = 2) product occurs obviously via this Feshbach resonance mechanism.

Fig. 5.

Slow-down reaction mechanisms of the HF(v′ = 3) channel for four typical cases. The four panels are the corresponding appearance vibration-adiabatic potentials for J = 0 (A), 5 (B), 9 (C), and 14 (D), including the centrifugal term.

For the HF(v′ = 3) forward scattering, the dynamical pictures involving these resonance states are remarkably different. First, it is obvious that the HF(v′ = 3) reaction dynamics occurs only on the HF(v′ = 3)–H vibration-adiabatic potential surface. The four panels in Fig. 5 show typical mechanisms that can cause time delays during reaction that could produce forward scattering HF(v′ = 3) product. In the case of J equal to or near zero (Fig. 5A), the two major resonance states (the ground and first excited resonance states) are below the HF(v′ = 3) reaction threshold and therefore could not be responsible for the HF(v′ = 3) reaction or forward scattering. There do exist a few more resonance states related to the bending excitations of the system, with their respective energies higher than the v′ = 3 threshold. These resonance states deplete the reaction in contrast to the first two resonance states, which enhance the reaction. More importantly, they change rapidly with the total angular momentum, making their effects on either integral or differential cross-sections negligible after the partial wave summation. Hence, forward scattering at the energies right above the HF(v′ = 3) reaction threshold can only be produced by delays in the reaction with very small kinetic energy in the exit channel as shown in Fig. 5A. For the case of J = 5, there is only one main resonance state left, and the resonance state is still below the threshold of the HF(v′ = 3) channel (Fig. 5B). Thus, the two main resonance states could not play a significant role in the formation of HF(v′ = 3) in the J = 0–5 partial waves. The HF(v′ = 3) product channel must proceed above the energy of the resonance states.

For some higher J partial waves, the ground resonance state trapped in the HF(v′ = 3)–H potential well could form the HF(v′ = 3) product via tunneling through the centrifugal barrier in a small energy region. Fig. 5C shows this typical case of a shape resonance in chemical reaction (33). Manolopoulos and coworkers (16) previously discussed this possible mechanism in the production of HF(v′ = 3). However, they assigned the “A” peak (the ground resonance) to a direct scattering peak; the role of the shape resonance mechanism in this channel was thus eluded. Clearly, time delay can be produced in shape resonance trapping and causes forward scattering for the HF(v′ = 3) product. More detailed theoretical analysis shows that the shape resonance mechanism can only play a significant role in the formation of HF(v′ = 3) in a very narrow collision energy regime approximately between 0.60 and 0.85 kcal/mol via the ground resonance state.

For even high J partial waves, reactions occurring on the HF(v′ = 3)–H potential energy surface to form the HF(v′ = 3) channel cannot be related to the ground reaction resonance state. Even though this shape resonance mechanism cannot occur anymore at higher collision energy >0.85 kcal/mol, time delay in the formation of HF(v′ = 3) can still occur because of a slow-down mechanism when the reaction intermediate passed over the exit centrifugal barrier (Fig. 5D) in near resonance, similar to the time-delay mechanism in the H + HD reaction in which the collision energy is in near resonance with the barrier of a specific quantized bottleneck state (6). This type of time-delay mechanism in the reaction intermediate could also cause forward scattering for the HF(v′ = 3) channel as for the shape resonance mechanism discussed above. Based on the above analysis, it is clear that the forward scattering of HF(v′ = 3) observed at higher collision energy by Lee and coworkers (13) is mainly caused by this slow-down mechanism over the centrifugal barrier in the exit channel and not by the quasi-trapped reaction resonance states in this reaction. This conclusion is in line with previous theoretical studies (15, 16) based on the SW-PES. This slow-down mechanism is also quite different from that in the H + D₂ and H + HD reactions in which the reactive thresholds are located on the reaction barrier (5–7) instead of in the exit channel in this reaction.

Through the above analysis, we have discussed all possible time-delay mechanisms that could be responsible for the forward scattering of the HF(v′ = 3) product channel at the threshold and higher collision energy. There are three types of time-delay mechanisms that are contributing in the title reaction: (i) time delay caused by near-zero kinetic energy reaction on a nearly flat surface near the threshold energy; (ii) time delay caused by a shape resonance mechanism; and (iii) time delay caused by a slow-down over the centrifugal barrier in the exit channel. We believe that the clarification of these time-delay mechanisms is crucial in understanding the rich dynamics exhibited in this benchmark reaction and provides a reasonable explanation of the long-standing issue of the intriguing HF(v′ = 3) forward scattering in the F + H₂ reaction, which is dynamically distinctive from the HF(v′ = 2) forward scattering caused by the Feshbach resonances.

However, it is important to note that the HF(v′ = 2) forward scattering caused by dynamics resonance and HF(v′ = 3) forward scattering caused by exit-channel slow-down are closely related, because they both originate on the HF(v′ = 3)-H adiabatic potential. Resonance states are the discrete quasi-bound states on the adiabatic potential for a particular channel, while the continuous states above the threshold that show slow-down effects will always exist when the channel becomes energetically open. Hence, one may anticipate that dynamics resonances are very likely accompanied by “slow” states; i.e., forward scattering caused by dynamics resonance may be accompanied by forward scattering caused by slow-down although the vibrational state for the forward scattering product is different. But the reverse may not be true, as found in the H + HD and H + D₂ reactions.