The F + HD (v = 0, 1; j = 1) reaction: angular momentum correlations in the low (<1 meV) collision energy regime

Abstract

A detailed analysis of the low collision energy (0.03–10 meV) integral reaction cross-section has been carried out for the F + HD (v = 0, 1; j = 1)→ HF(DF) + D(H) reaction using accurate, fully converged time-independent hyperspherical quantum dynamics. Particular attention has been paid to the shape (orbiting) resonances and their assignment to the orbital (L) and total (J) angular momenta as well as to the product's state resolved cross-sections at the energies of the resonances. As in previous works, it has been found that the energy position of the resonances depends on the initial state, but is essentially the same for the two exit channels and the product's rovibrational states. The analysis in terms of the orbital and total angular momenta showed that each resonance is characterised by a given value of L but is contributed by several J. The main resonances are due to L = 3 and L = 5 for both F + HD (v = 0, j = 1) and F + HD (v = 1, j = 1) reactions, although they appear at different collision energies. The product's vibrationally resolved excitation functions are found to follow the same pattern as the integral cross-section summed over all final states. A more detailed analysis has been made for the rotationally resolved integral cross-sections associated with L = 3, which gives rise to the main resonance for the two reactions and both product channels, for different final j′ states, showing similar behaviour for all j′ states except for j′ = 0 due to parity conservation. The joint analysis of the final rotational and orbital angular momenta shows that L′ and j′ tend to have an antiparallel orientation.

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I. Introduction

The F + H₂ reaction and its isotopic variants have long served as a prototype in chemical reaction dynamics (see for instance ref. 1–4 and references therein). Insights into many aspects of reactivity have been gained from the detailed experimental and theoretical investigations carried out for this system over the last few decades. It has been a major benchmark for the study of resonances in reactive collisions.⁵ The first unequivocal identification of a resonance was reported by Skodje et al.^6,7 in the form of a bump in the experimental energy dependence of the integral reaction cross-section, σ_R(E_coll), at a collision energy, E_coll, of 22 meV in the HF + D exit channel of the F + HD reaction. The bump, which corresponds to an energy below the classical reaction threshold, could be well reproduced with accurate quantum mechanical (QM) calculations. Further resonances, mostly of the Feshbach type, were discovered in the following years for the F + H₂ and F + HD isotopologues and are described in a series of articles by D. H. Zhang, X. Yang and co-workers.^5,8–13 These works combined measurements based mostly on the H-atom Rydberg tagging time of flight¹⁴ technique with results from accurate QM calculations. Additional evidence for resonances was provided by joint experimental and theoretical studies on the photoelectron spectroscopy of FH₂⁻ and FD₂⁻ anions.^15–19

A major challenge since early work on the F + H₂ system was the construction of an ab initio potential energy surface (PES) of sufficient accuracy to account for the wealth of high-resolution experimental findings. Gradual refinements of the PES were reported by various groups.^20–31 A critical review of the evolution of progress in the PESs for F + H₂ till 2015 was written by Sun and Zhang.³² Although the fluorine atom has two spin orbit states, F(²P_3/2) and F(²P_1/2), in the low energy range, most experimental findings, including the resonance structures mentioned in the previous paragraph, could be well accounted for by theoretical calculations on the ground state adiabatic PES, corresponding to the reaction with F(²P_3/2), without considering the couplings between the two spin–orbit states. In a rigorous dynamical study, using diabatic PESs and including spin–orbit coupling, Alexander et al.²² concluded that, in the global F(²P_3/2) reactivity, the contribution of F(²P_1/2), which can only be due to non-adiabatic transitions, is relatively small, and this remains true for most of the available data. However, it has been found both experimentally and theoretically that, in some cases, the contribution from F(²P_1/2) could be comparable to, if not larger than that from the reaction with F(²P_3/2). In particular, at low collision energies (E_coll ⪅ 20 meV), the F(²P_1/2) + o-D₂ non-adiabatic reaction was found to dominate over its F(²P_3/2) + o-D₂ counterpart.³³ Other subtle effects caused by interference between the spin–orbit split partial waves have been seen in the rotationally state resolved differential cross-section (DCS) of the resonance mediated F + HD → HF + D reaction at 91 meV.³⁴ New diabatic PESs for the F + H₂ system including spin–orbit coupling have been recently released^35,36 and their accuracy for the calculation of cross-sections and rate coefficients has been checked.^36–38

At this point, it is worth noting the paramount role played by the F + H₂ reaction and other closely related systems in the development of both time-independent and time-dependent methods in reactive scattering theory over the last few decades. Progress in this field is discussed in ref. 36 and 39.

The F + H₂ reaction was also used by Balakrishnan and Dalgarno^40,41 in their landmark study on chemical reactivity at ultralow temperatures. These authors showed that, in the Wigner limit,⁴² when the temperature tends to zero, the rate coefficient reaches a constant value. In the calculations of Balakrishnan and Dalgarno, which were performed for the ground rovibrational state of the molecule and for a value of the total angular momentum J = 0, the Wigner limit behaviour was observed for E_coll lower than 0.02 meV. The study was extended to F + D₂ and to the j = 2 and v = 1 states of the molecule^43,44 and similar behaviour was observed in the ultracold regime. In a further theoretical study, the stereodynamics of the reaction in the cold and ultracold regimes was investigated down to 1 μeV. It was shown that the integral cross-section is not amenable to stereodynamic control in the ultracold regime.⁴⁵ A general perspective of chemistry of ultracold molecules can be found in ref. 46.

Above the ultracold limit, which is dominated by collisions with J = 0, larger values of the angular momentum become relevant and give rise to new dynamic features. Recent theoretical investigations have revealed the presence of shape resonances of the orbiting type in the cold, E_coll < 1 meV, collision energy regime.^47–50 These resonances are manifested as maxima in the integral reaction cross-section, σ_R(E_coll), that can reach tens of Ų when the reaction involves vibrationally excited molecules, and are thus good candidates for experimental investigation. Although the range of energy characteristics of orbiting resonances has not yet been experimentally reached, it could be soon accessible considering that the F + D₂ and F + HD isotopic variants of the reaction have been recently studied at collision energies of a few meV.^51–53

In classical two-particle scattering, orbiting takes place when the collision energy is equal to the height of the centrifugal barrier in the effective potential which is in turn determined by the value of the (non-quantized) orbital angular momentum, L.^54,55 In the rigorous quantum mechanical description, L is quantized and the corresponding orbital states (partial waves) determine the effective potentials and thus the observed resonances which are related with quasibound states in the van der Waals well of the reactants, behind the centrifugal barrier. L is thus the appropriate quantum number to associate with orbiting resonances. For H₂ molecules in the rotational ground state (j = 0), the total angular momentum, J, coincides with L and the resonance peaks can be directly related to specific J values. However, when H₂ molecules are rotationally excited (j > 0), the picture becomes more complex and the possible couplings between L, j and J have to be taken into account.

Rotational excitation opens the possibility of stereodynamics.⁵⁶ In a previous work, we have examined the stereodynamic properties of orbiting resonances in the F + HD (v = 0, 1; j = 1) reaction.⁵⁰ It was found that the polarization of the rotational angular momentum with respect to the relative velocity vector of the reactants had a marked influence on the reactivity and, specifically, on the resonance pattern. The preferential orientation for the reaction was found to correspond to a perpendicular alignment of the HD intermolecular axis with respect to the direction of the approach and, in fact, some of the resonance peaks were exclusively seen in collisions with this orientation.

In addition to inducing orientational (molecular alignment) preferences, the rotational angular momentum, j, has other effects on the reactivity. Here, we consider these effects for the F + HD isotopic variant of the F + H₂ reaction. The choice of this asymmetric variant is of interest because it provides two different isotopic exit channels (HF + D and DF + H), which may exhibit different dynamic properties. In fact, the well-known Feshbach resonance at E_coll = 22 meV for this reaction is only found in the HF + D channel.^6,7 Feshbach resonances are associated with quasibound levels of a specific closed channel that correlates with individual levels of the products. In contrast, orbiting resonances are largely determined by the van der Waals potential at the entrance channel, and very similar resonance patterns are expected for the two isotopic exit channels. That is indeed what was observed for F + HD (v = 0, 1; j = 0).⁵⁰ The energy location and relative intensity of the resonances were still similar for F + HD (v = 0, 1, j = 1), but some differences were found in the shapes of the resonance maxima of the HF + D and DF + H exit pathways that could be traced back to the different weights of the various J values contributing to a given resonance. Rotational excitation of the HD molecule offers thus various possibilities of angular momenta combination that lead to appreciable differences in dynamic properties (resonance peak shape) for the two isotopic exit channels, and it is conceivable that similar channel specific effects can be observed in other dynamic characteristics of the reaction.

In the present work, we further explore the connection between the various angular momenta involved in the low energy (E_coll < 1 meV) collisions of F + HD (v = 0, 1; j = 1) and the reactivity of this system in its ground state adiabatic PES. We investigate the dependence of the orbiting resonances on the orbital angular momentum quantum number L of the collision, which, as indicated above, is the “natural” quantum number for these resonances, and on the possible J values associated with each L. We also analyze and discuss the relationship between the angular momenta at the entrance and the rovibrational energy distributions and angular momenta of the products (j′ and L′) for the two isotopic exit channels.

II. Theoretical methods

Time-independent QM calculations of integral reaction cross-sections for the F + HD reaction (v = 0, 1; j = 0, 1) as a function of collision energy, σ_R(E_coll), were obtained using the recent LWA-5 PES (scaling factor parameter s = 1.05) of Li et al.²⁷

An extensive description of the methods used in these calculations is presented in ref. 57 and 58, so only a summary of the most relevant details for the present article will be presented here. All calculations were carried out using the coupled-channel hyperspherical coordinate method implemented in the ABC code of Skouteris et al.⁵⁷

The scattering matrix in the helicity representation is denoted by where v, j and v′, j′ stand for rovibrational quantum numbers of the reagents and products, respectively, and Ω, (Ω′) represents the initial (final) helicity quantum number. J is the total angular momentum quantum number. Scattering matrices, were obtained for several total energy intervals. A grid of 30 total energies ranging from 0.24581 eV to 0.25181 eV and from 0.69724 eV to 0.70324 eV were used for the calculations of the HD molecule in the vibrational states of v = 0, and v = 1, respectively.

A finer grid of 40 points with 0.02 meV spacing was also calculated starting from the same collision energies as the coarse grid. This grid covers a much narrower interval in order to highlight the very sharp resonances shown in the excitation function at very low collision energy for both v = 0 and v = 1 vibrational levels of the HD molecule.

The rest of the input parameters were identical for all the calculations. A basis set including all diatomic levels up to a cut-off energy of 3.0 eV and comprising helicity quantum numbers until a maximum value of 25 was used. The propagation of the integration was extended in 150 sectors up to 18a₀. To check the convergence of the calculations, the cumulative reaction probabilities for J = 0 calculated with the just mentioned parameters were compared with those calculated for 800 sectors with a maximum hyper-radius of 50a₀ and were found to differ by less than 0.07% for the lowest energy considered (E_coll = 0.03 meV). A different number of partial waves were employed for each energy and the vibrational state of the HD molecule. All partial waves till J_max = 16 (coarse grid) and J_max = 8 (finer grid) were considered for the v = 1 level, whereas partial waves till J_max = 9 (coarse and fine grids) were considered for the v = 0 level of the HD molecule.

Correlations involving the orbital angular momentum of the reactant or products require the scattering matrix, to be defined in the orbital angular momentum (OAM) or space-fixed representation, where L (L′) are the quantum number of the initial (final) orbital angular momenta, and represents the parity of the triatomic system. The matrix in the orbital angular momentum representation can be obtained from the scattering matrix in the helicity representation by the unitary transformation as in ref. 57:

In terms of the scattering matrix, the excitation function (i.e., collision energy dependent integral cross-section) summed over all product's states, σ_R(E_coll), for a given v, j initial rovibrational state can be written as in ref. 49:where k = (2μE_coll)^1/2/ħ denotes the initial relative wave number.

If the sum over L is omitted, the L-resolved cross-section can be written as:

eqn (2) can also be written as:

The L, J double partial cross-section can be expressed as:

Summing over L, the J partial cross-section is:

The v′-resolved excitation function can be obtained from eqn (2) if the sum over v′ is obviated:

Similarly, the j′-resolved excitation function is as follows:

The L′-resolved excitation function is given by:The total reaction cross-section can be obtained as follows:These expressions have been applied to investigate both channels of the F + HD (v = 0, 1; j = 1) reaction, as shown in the next section.

III. Results and discussion

Fig. 1 shows the total cross-section as a function of the collision energy for the F + HD (v = 0, 1; j = 1) → HF + D (left panels) and F + HD (v = 0, 1; j = 1) → DF + H (right panels) reactions. The respective excitation functions are resolved in the contributions from the orbital angular momentum, L. As expected, the cross-sections are much larger for the reactions with vibrationally excited HD (lower panels). For clarity, we have only represented σ^L_R(E_coll) for L values that give rise to maxima and contribute to resonances. Specifically, contributions of L = 0, L = 1 and L = 2, which present a monotonic decay with growing energy, are not included in Fig. 1–3. It should be noted that, in Fig. 1, each L only contributes to one resonance feature in σ_R(E_coll). For the F + HD (v = 0, j = 1) → HF + D reaction, L = 3 contributes to a maximum and a shoulder that will be discussed below. In some cases, two different L values can contribute to the same maximum in the total cross-section; this is the case for L = 4 and L = 5 in the two exit channels of F + HD (v = 1, j = 1), but a specific L value does not contribute to two separated resonance maxima, corroborating that orbiting is indeed at the root of the observed resonances. In the cases where there are contributions from two L values to a given peak in σ_R(E_coll), they exhibit different widths suggesting different lifetimes for each L component of the peak. Although the shape of the individual σ^L_R(E_coll) is similar for the two isotopic exit channels, their relative weights are different and this leads to some differences in the shape of the resonance features of the total cross-section, which are especially appreciable in the F + HD (v = 0, j = 1) reaction.

Fig. 1 Top panels: L-partial reactive cross-sections resolved in the orbital angular momentum, σ^L_R(E_coll), as a function of collision energy for the F + HD (v = 0, j = 1) reaction for the HF + D (left) and DF + H (right) product channels. Bottom panels: Same as top panels for the F + HD (v = 1, j = 1). For clarity of display, only the L contributions giving rise to maxima are represented. Note the different ordinate scale of each panel. In each panel, the total cross-section is also shown as a black line.

Fig. 2 Reaction cross-section resolved in J and L, σ^L,J_R(E_coll), for the F + HD (v = 0, j = 1) reaction. Upper panel: HF + D. Lower panel: DF + H.

Fig. 3 Same as Fig. 2 for the F + HD (v = 1, j = 1) → HF + D reaction. For clarity of display, the upper panel shows the contributions of L = 3, and the lower panel shows those of L = 4 and 5.

The contributions of the individual J values to each of the L shape resonances of Fig. 1 are shown in Fig. 2. This figure displays the double cross-sections, σ^L,J_R(E_coll), for the two exit channels of the F + HD (v = 0, j = 1) reaction. For each L value, J can take the values J = L − 1, L, L + 1, with j = 1. The σ^L,J_R(E_coll) decomposition is represented for L = 3 and L = 5. Inspection of the upper panel, corresponding to the HF + D exit channel, shows that the maximum in the L = 3 resonance is almost exclusively due to J = 3 (i.e., J = L). The shoulder, located at slightly higher energy, is mostly due to J = 4, and, to a lesser extent, to J = 2. The difference in the location of the maxima of J = L and J = L ± 1 is worth noting. In contrast to this behaviour, the resonance for L = 5 shows just a sharp maximum that is exclusively associated with J = 6 (i.e. J = L+ 1), whereas the cross-sections for J = 4 and J = 5 are much smaller and rise monotonically with collision energy. A similar behaviour is observed for the DF + H channel (lower panel) but, in this case, the contribution of J = 3 is dominant with smaller contributions from J = 2 and 4 than for the HF + D channel. The dominant L values and their relative contribution to the various maxima in σ_R(E_coll) are listed in Table 1 for the F + HD (v = 0, j = 1) reaction. It also shows the dominant J values and their relative contribution for the corresponding σ^L_R(E_coll) maxima.

Table 1 Contribution of the dominant L to the total reaction cross-section of F + HD (v = 0, j = 1) at different resonances with an indication of the fraction of the dominant J within each L

E tot (eV)	E coll (meV)	HF + D				DF + H
		L	(%)	L, J	(%)	L	(%)	L, J	(%)
0.24593	0.13	3	56	3, 3	82	3	78	3, 3	98
0.24599	0.19	3	55	3, 4	56	3	53	3, 4	44
0.24633	0.53	5	28	5, 6	96	5	48	5, 6	98

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Fig. 3 shows the J decomposition of the L = 3 (upper panel) and L = 4 and 5 (bottom panel) resonances for the F + HD (v = 1, j = 1) → HF + D reaction. For L = 3, the three J that contribute to this orbital angular momentum behave differently. The partial wave J = 4 (i.e., J = L + 1), with a prominent maximum, carries most of the L = 3 resonance peak. J = 2, with a much smaller partial cross-section, still shows an appreciable maximum at the location of the L = 3 resonance. In contrast, the cross-section for J = 3 rises steeply at the start and then stabilizes with a very slow decline towards high energies. The lower panel of this figure displays the J dependent cross-sections for L = 4 and L = 5. The resonance maxima for L = 4 and L = 5 are close in energy and both contribute appreciably to the same maximum in the total cross-section, but their shapes are very different which confirms that they are different resonances. The L = 4 curve is comparatively broad with a slow fall to higher energies and its maximum corresponds almost exclusively to J = 4 (J = L). The L = 5 partial cross-section is bimodal, with a sharp and narrow peak followed by a valley and a further rise. The peak corresponds entirely to J = 6 (J = L + 1) and the secondary rise, beyond the valley, is mostly due to J = 5 and, to a lesser extent, J = 4, but none of these contributions can be assigned as resonances. It should be noted that J = 4 participates in the cross-sections for the three L values analyzed in this figure, and carries a major contribution to the L = 3 resonance at 0.12 meV and to the maximum in the L = 4 partial cross-section. Its contribution to L = 5 is probably due to the opening of a channel. The double cross-sections, σ^L,J_R(E_coll), for the F + HD (v = 1, j = 1) → DF + H reaction show similar behaviour and are not shown for brevity. The peculiarities of σ^L,J_R(E_coll) just commented on are suggestive of subtle dynamic effects, but they are beyond the scope of this work.

Table 2 lists the predominant L values and their relative contribution to the maxima in σ_R(E_coll) for the F + HD (v = 1, j = 1) reaction. Note again that the peaks of L = 4, at E_coll = 0.29 meV, and L = 5, at E_coll = 0.31 meV, contribute to a single unresolved maximum in the total cross-section. The table also shows the dominant J values and their relative contributions for the corresponding σ^L_R(E_coll) maxima. In all cases, it is found that a single J value accounts for more than 80% of the various L maxima.

Table 2 Contribution of the dominant L to the total reaction cross-section of F + HD (v = 1, j = 1) at various resonance peaks with indication of the proportion of the major J within each L

E tot (eV)	E coll (meV)	HF + D				DF + H
		L	(%)	L, J	(%)	L	(%)	L, J	(%)
0.69734	0.11	3	79	3, 4	93	3	66	4	85
0.69752	0.29	4	40	4, 4	93	4	47	4	96
0.69754	0.31	5	43	5, 6	99	5	24	6	100

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One simple way to isolate and assign resonances is to plot the normalized L-resolved partial cross-sections, defined as follows:

for given values of L. Fig. 4 shows the results for L = 3 and L = 5. The plots of the two exit channels of F + HD (v = 0, j = 1) in the top panels of Fig. 4 show the two maxima assigned to the L = 3 and L = 5 resonances, which appear at the same collision energies for both exit pathways. The double peak in the HF + D exit channel and the peak and shoulder for the DF + H channel correspond to the contributions from J = 3, J = 4 and, to a lesser extent, from J = 2 to the L = 3 resonance. As shown in Fig. 2, the first peak is due to J = 3, whilst the second peak (or the shoulder for the DF + H channel) is due to J = 4. The sharp resonance due to L = 5 and J = 6 is clearly visible at E_coll = 0.53 meV. Since the denominator of eqn (11) increases abruptly at the L = 5 resonance, the normalized σ^L=3_norm exhibits a minimum. Similar results can be found for the two exit channels of the F + HD (v = 1, j = 1) reaction (lower panels). The first maximum in both cases corresponds to a resonance due to L = 3 and J = 4 (the contribution from J = 2 is an order of magnitude smaller), and that from L = 5 and J = 6 appears at E_coll = 0.31 meV, along with a minimum of σ^L=3_norm. Entirely analogous results can be found by plotting for a given v′ product's state, reinforcing the attribution of the resonances to shape or orbiting resonances in the entrance channel.

Fig. 4 Normalized reaction cross-sections σ^L_norm = σ^L_R(E_coll)/σ_R(E_coll) summed over all v′, j′ final states. Upper left panel: F + HD (v = 0, j = 1) → HF + D. Upper right panel: F + HD (v = 0, j = 1) → DF + H. Lower left panel F + HD (v = 1, j = 1) → HF + D. Lower right panel: F + HD (v = 1, j = 1) → DF + H.

We will now analyze the possible influence of the L, J, and j angular momenta on the internal energy states of the HF and DF exit channels. Given the large exoergicity of the reaction, many products' states are energetically accessible. We will first consider the vibrational states. Fig. 5 shows the cross-sections for the F + HD (v = 0, j = 1) and F + HD (v = 1, j = 1) reactions resolved into the vibrational states, v′, of the HF and DF product molecules. For simplicity, only the most populated vibrational states of the products have been represented. The figure reflects the different v′ state distributions obtained for the four possible output channels. In the reaction of F with HD (v = 0, j = 1), the HF(v′) distribution is dominated, by far, by HF(v′ = 2). The maximum in the DF(v′) distribution corresponds to v′ = 3, but appreciable contributions of v′ = 2 and v′ = 4 are also found. In the reaction with vibrationally excited molecules, F + HD (v = 1, j = 1), the maximum of the HF(v′) distribution corresponds to v′ = 3, with a significant contribution of v′ = 2 and a small one of v′ = 1. In the DF(v′) distribution, the v′ = 3, 4, and 5 states are predominant with, interestingly, almost the same cross-sections; there is also an appreciable contribution of v′ = 2. For all four reactions, the resonance structure is the same irrespective of the v′ state of the HF and DF molecules. The L, J analysis described in the previous paragraphs for the total cross-sections applies thus to the cross-sections of the individual v′ product states. This is not surprising since the partition into vibrational states of the products is mostly an energetic effect unrelated, in principle, to the possible coupling of angular momenta.

Fig. 5 Vibrational state distributions of the products of the F + HD (v = 0,1;j = 1) reaction. Upper left panel: F + HD (v = 0, j = 1) → HF (v′) + D. Upper right panel: F + HD (v = 0, j = 1) → DF(v′) + H. Lower left panel F + HD (v = 1, j = 1) → HF(v′) + D. Lower right panel: F + HD (v = 1, j = 1) → DF(v′) + H.

The next step of our analysis is to consider the rotational states, j′, of the products. We will focus our attention on the reactive cross-sections summed over the vibrational states. The choice of the L = 3 orbital angular momentum is motivated by its contribution to L resonances, which, as discussed above, consist of a maximum and a slightly higher energy shoulder corresponding to different J values, and increases the possibility of observing effects of specific initial angular momenta in the rotational states of the products. The L = 3 reaction cross-sections for the two isotopic exit channels of the F + HD (v = 0, j = 1) and F + HD (v = 1, j = 1) reactions are shown in Fig. 6. They have been normalized as:

where contains all the reactive flux into a given j′, to remove the influence of the relative weight of a specific j′ in the rotational state distributions. For simplicity, the analysis is restricted to j′ = 0–4, although similar results are obtained for other j′ states.

Fig. 6 Normalized reaction cross-sections for the F + HD (v = 0, 1, j = 1) reaction. The cross-section values have been selected for L = 3 and are resolved into the rotational states, j′, of the products. Upper left panel: F + HD (v = 0, j = 1) → HF + D. Upper right panel: F + HD (v = 0, j = 1) → DF + H. Lower left panel F + HD (v = 1, j = 1) → HF + D. Lower right panel: F + HD (v = 1, j = 1) → DF + H.

The two upper panels show the results for the two exit channels of F + HD (v = 0, j = 1). In both cases, the normalized reaction cross-sections show, for j′ ≠ 0, a first peak or shoulder followed by a broad maximum at higher energy. The cross-sections for j′ = 0 show only the broad maximum and thus do not contribute to the first maximum in σ^L=3_norm(E_coll). The fact that j′ = 0 does not contribute to this peak is a consequence of parity conservation, which can be expressed as (−1)^L+j = (−1)^L′+j′. In this case, with L = 3 and j = 1, parity is positive. For j′ = 0, parity is given by (−1)^L′ which is equivalent to (−1)^J due to angular momentum conservation. Consequently, for parity to be positive, J must be even. As discussed above, the first peak of σ^L=3_R(E_coll) corresponds to J = 3 and therefore has no contribution from j′ = 0, in contrast with the broad maximum which is due to J = 2 and 4. At E_coll = 0.53 meV, the normalized cross-sections for F + HD (v = 0, j = 1) have a sharp dip coincident with the location of the L = 5 resonance peak. As shown in Fig. 4, when the L = 5 resonance shows up, the denominator of eqn (12) increases giving rise to a minimum in . The corresponding results for the F + HD (v = 1, j = 1) reaction are displayed in the two lower panels of Fig. 6. Here, there is just one maximum, at E_coll = 0.11–0.13 meV, for the different j′. Parity conservation has no effect on the j′ = 0 normalized cross-section because the maximum for L = 3 is mostly caused by J = 4 which is parity allowed for j′ = 0 (see Fig. 3). The contribution of J = 3 is very small and nearly independent of energy in the range of the L = 3 cross-section maximum. At E_coll = 0.30 meV, there is a dip in the normalized cross-sections of the two isotopic exit channels that correspond to the energy of appearance of the L = 5 resonance, which is shifted to lower energies as compared with the equivalent resonance for the F + HD (v = 0, j = 1) reaction.

We will inspect now in detail the products' distributions of rotational states, j′, and orbital angular momenta, L′, for the F + HD (v = 0, 1; j = 1) reaction at the collision energies of the L = 3 resonance maxima. Fig. 7 shows the distributions of rotational states resolved into individual vibrational states for the four cases under consideration. In the upper left panel, depicting the HF + D exit channel of F + HD (v = 0, j = 1), virtually all the reactivity is concentrated on v′ = 2. The rotational distribution is asymmetric with a maximum at j′ = 9 and a shoulder at j′ = 2–3. In the DF exit channel of the same reaction (upper right panel), three vibrational states (v′ = 2, 3 and 4) contribute appreciably to the total cross-section. Due to energetic restrictions, the maxima of the j′ distributions shift toward lower values with growing v′. This trend is particularly notable for v′ = 4. The lower left panel displays the v′, j′ distributions for the HF channel of the F + HD (v = 1, j = 1) reaction. In this case, the j′ distributions of the v′ = 1, v′ = 2 and v′ = 3 states are neatly separated due to the large spacing between the individual v′, j′ levels in HF. Conservation of energy leads to colder rotational distributions for the higher v′ states. The j′ distributions for the DF exit channel of F + HD (v = 1, j = 1) are represented in the lower right panel. The total distribution is broad, with a maximum for j′ = 5, 6 and a shoulder at j′ = 10. This shape is a reflection of the rotational distributions obtained for the individual v′ levels. The cold rotational distribution for v′ = 5, which is the highest vibrational level accessible, provides the dominant contribution to the j′ = 3–4 maximum. Also, the long tail, extending beyond j′ = 20, for v′ = 3, is noted.

Fig. 7 Reaction cross-sections for F + HD (v = 0, 1, j = 1) resolved into the rovibrational states of the products summed over all L contributions. The cross-sections correspond to the collision energy of the L = 3 resonance maximum, i.e. E_coll = 0.13 meV for v = 0 and E_coll = 0.11 meV for v = 1.

Fig. 8 shows the distributions of the product's orbital angular momenta for F + HD (v, j = 1) also at the collision energies of the respective L = 3 maxima for v = 0 and v = 1. For the reaction with HD in the ground vibrational level (upper panels), the L′ distributions are similar to the corresponding j′ distributions. The HF channel is largely dominated by the contribution from v′ = 2, which has an asymmetrical L′ distribution, and the DF channel has significant contributions from three vibrational levels, whose L′ distributions become colder with growing v′. The lower left panel shows the L′ distributions for the HF + D channel of F + HD (v = 1, j = 1). The total L′ distribution shows some structures but, in contrast with the j′ distribution shown in Fig. 7, are not so neatly resolved into the individual v′ levels. For this product channel at the collision energy dominated by the L = 3 resonance, the L′ distributions for individual v′ states are bimodal with either two maxima or a maximum and a shoulder. These broad, bimodal distributions are found just for a comparatively small range of collision energies around those of the resonance maxima considered in the figure. This is in contrast with the corresponding j′ distributions shown in Fig. 7, which are not restricted to the vicinity of the resonance. For the DF + H channel of the reaction with HD (v = 1, j = 1), the global L′ distribution (the lower right panel) is broad, with a maximum at L′ = 6 and two shoulders to the sides. The L′ distributions for the individual v′ levels are very similar to the corresponding j′ distributions shown in Fig. 7. The relative L′ populations for v′ = 3 and 4 are comparable and that for v′ = 5 is notably colder.

Fig. 8 Distributions of orbital angular momenta, L′, for the F + HD (v = 0, 1, j = 1) reaction at the different vibrational, v′, states of the products. The partial cross-sections correspond to the collision energy of the L = 3 resonance maximum, i.e., E_coll = 0.13 meV for v = 0 and E_coll = 0.11 meV for v = 1.

Fig. 9 shows the build-up of the rotational distributions associated with increasing batches of L′, thus showing the correlation between the orbital and rotational angular momenta of the products of F + HD (v = 0, 1, j = 1) at the same collision energies as shown in Fig. 7 and 8. For clarity of display, batches of L′ values are represented versus individual j′ levels. For the reaction with HD (v = 0, j = 1) leading to HF + D (upper left panel), only one vibrational state of the products (v′ = 2) is significantly populated and a direct correlation between L′ and j′ is clearly seen. An increase in L′ corresponds to an increase in j′. The same trend is found for the DF + H channel (upper right panel): growing L′ is related to growing j′, although in this case the tendency is less clear-cut due to the appreciable contribution of various v′ levels with different j′ distributions. For the HF + D exit channel of the F + HD (v = 1, j = 1) reaction, the direct positive correlation between L′ and j′ is maintained across the individual vibrational levels which, as indicated above, are well separated in the j′ resolved global cross-section (lower left panel). Finally, for the DF + H channel (lower right panel), the correlation exists, but is more blurred due to the smaller energy spacing of the DF vibrational levels.

Fig. 9 Reaction cross-sections for F + HD (v = 0, 1, j = 1) resolved into rotational states, j′, of the products associated with batches of the product's orbital angular momentum, L′. The cross-sections correspond to the collision energy of the L = 3 resonance maximum, i.e., E_coll = 0.13 meV for v = 0 and E_coll = 0.11 meV for v = 1.

Fig. 7–9 show that both L′ and j′ can reach values beyond 10 for the reaction with HD (v = 0, j = 1), and around 20 for the reaction with HD (v = 1, j = 1). These results correspond to an orbital angular momentum L = 3 in the entrance channel and thus to a comparatively small maximum value of the total angular momentum J_max = . To obtain a small total angular momentum (J ≤ 4) from a sum of a large L′ and a large j′, conservation of the angular momentum dictates that the orbital and rotational angular momentum vectors of the products should have a predominantly antiparallel orientation.

The study presented in this work is restricted to the lowest adiabatic PES of the reactive system, which corresponds to the F(²P_3/2) spin–orbit state of the fluorine atom. However, as mentioned in the Introduction section, the contribution of F(²P_1/2) could be significant and even dominant at low energies.³³ To the best of our knowledge, no studies have been carried out that include the role of F(²P_1/2) in the collision energy range of orbiting resonances. It is possible that the overall resonance structure in the integral cross-section would not change much.

Regarding the angular momentum correlations, the atomic angular momentum would have to be included and the possible coupling between the orbital and the atomic angular momenta would also have to be considered (see, for example, ref. 34). The influence of the open-shell character of the fluorine atom in the low collision energy regime is an interesting research subject, but it is beyond the scope of this work.

IV. Summary and conclusions

Angular momentum correlations at the entrance and output channels were investigated for the two isotopic exit arrangements of the F + HD (v = 0, 1; j = 1) reaction in the cold regime (E_coll < 1 meV). It was found that L (orbital), rather than J (total), is the natural angular momentum quantum number characterizing the resonance maxima found in the total reaction cross-section, thus corroborating that the observed structures correspond to shape resonances. Although shape resonances are mostly determined by the properties of the entrance channel, slight differences are seen in the peak shapes of the HF + D and DF + H exit channels due to the different relative weights of the various J values within each L.

The effects of the L, J and j entrance angular momenta on the final product states have been explored. The resonance structure of the total reaction cross-section is replicated for the individual vibrational states of the products, HF(v′) and DF(v′), which is not surprising since the partition into vibrational states is mostly an energy effect, unrelated in principle with a possible coupling of angular momenta.

In contrast, distinct effects of the entrance angular momenta are observed in the distributions of product's rotational (j′) and orbital (L′) states. For simplicity, this part of the study is centered on the L = 3 resonance, which, for the F + HD (v = 0, j = 1) reaction, presents a maximum followed by a higher energy shoulder. The maximum corresponds to J = 3 and parity conservation dictates that HF(j′ = 0) or DF(j′ = 0) product molecules cannot contribute to this maximum, as demonstrated by the calculations.

Given the large exoergicity of the F + HD system, many rotational states are energetically accessible to the HF and HD product molecules and, in fact, broad rotational distributions extending to high j′ values are observed. The j′ distributions are colder for the highest vibrational states allowed due to energy conservation. The product orbital angular momenta exhibit also broad distributions reaching high L′ values, and there is a positive correlation between j′ and L′, i.e., large j′ values correspond to large L′ values. In the low energy region characteristic of orbiting resonances, only comparatively low values of the total angular momentum are possible at the entrance and, consequently, the large j′ and L′ values found for the products indicate that the rotational and orbital angular momentum vectors of the products have a preferential antiparallel orientation.

Data availability

All data supporting the results of this study are included in the article.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

This work has been funded by the MINECO, the Ministry of Science and Innovation of Spain under grants PID2020-113084GB-I00, PID2023-146415NB-I00 and PGC2018-096444-B-I00 and the Community of Madrid under the framework of the multiyear agreement with the University of Alcalá (Stimulus to Excellence for Permanent University Professors, EPU-INV/2020/008). This research was conducted within the Unidad Asociada between the Department of Physical Chemistry of the UCM and the CSIC of Spain.

References

1. D. M. Neumark, A. M. Wodtke, G. N. Robinson, C. C. Hayden and Y. T. Lee, J. Chem. Phys., 1985, 82, 3045.
2. D. M. Neumark, A. M. Wodtke, G. N. Robinson, C. C. Hayden, K. Shobatake, R. K. Sparks, T. P. Shafer and Y. T. Lee, J. Chem. Phys., 1985, 82, 3067.
3. F. J. Aoiz, L. Banares, V. J. Herrero, V. Saez-Rabanos, K. Stark and H.-J. Werner, Chem. Phys. Lett., 1994, 223, 215.
4. D. Manolopoulos, J. Chem. Soc., Faraday Trans., 1997, 93, 673.
5. T. Wang, T. Yang, C. Xiao, Z. Sun, D. H. Zhang, X. Yang, M. L. Weichman and D. M. Neumark, Chem. Soc. Rev., 2018, 47, 6744.
6. R. T. Skodje, D. Skouteris, D. E. Manolopoulos, S.-H. Lee, F. Dong and K. Liu, J. Chem. Phys., 2000, 112, 4536.
7. R. T. Skodje, D. Skouteris, D. E. Manolopoulos, S.-H. Lee, F. Dong and K. Liu, Phys. Rev. Lett., 2000, 85, 1206.
8. M. Qiu, Z. Ren, L. Che, D. Dai, S. A. Harich, X. Wang, C. Xu, D. Dai, M. Gustafsson, R. T. Skodje, Z. Sun and D. H. Zhang, Science, 2006, 311, 1440.
9. X. Wang, W. Dong, M. Qiu, Z. Ren, L. Che, D. Dai, X. Wang, X. Yang, Z. Sun, B. Fu, S.-Y. Lee, X. Xu and D. H. Zhang, Proc. Natl. Acad. Sci. U. S. A., 2008, 105, 6227.
10. Z. Ren, L. Che, M. Qiu, X. Wang, W. Dong, D. Dai, X. Wang, X. Yang, Z. Sun, B. Fu, S.-Y. Lee, X. Xu and D. H. Zhang, Proc. Natl. Acad. Sci. U. S. A., 2008, 105, 12662.
11. X. Yang and D. Zhang, Acc. Chem. Res., 2008, 41, 981.
12. T. Wang, J. Chen, T. Yang, C. Xiao, Z. Sun, L. Huang, D. Dai, X. Yang and D. H. Zhang, Science, 2013, 342, 1499.
13. T. Wang, T. Yang, C. Xiao, Z. Sun, L. Huang, D. Dai, X. Yang and D. H. Zhang, J. Phys. Chem. Lett., 2014, 5, 3049.
14. L. Schnieder, K. Seekamp-Rahn, E. Wrede and K. H. Welge, J. Chem. Phys., 1997, 107, 6175.
15. D. E. Manolopoulos, K. Stark, H. J. Werner, D. W. Arnold, S. E. Bradforth and D. M. Neumark, Science, 1993, 262, 1852.
16. C. Hock, J. B. Kim, M. L. Weichman, T. I. Yacovitch and D. M. Neumark, J. Chem. Phys., 2012, 137, 244201.
17. T. I. Yakovitch, E. Garand, J. B. Kim, C. Hock, T. Theis and D. M. Neumark, Faraday Discuss., 2012, 157, 399.
18. J. B. Kim, M. L. Weichman, T. F. Sjolander, D. M. Neumark, J. Klos, M. H. Alexander and D. E. Manolopoulos, Science, 2015, 349, 510.
19. D. Yu, J. Chen, S. Cong and D. Zhang, J. Phys. Chem. A, 2015, 119, 12193.
20. K. Stark and H.-J. Werner, J. Chem. Phys., 1996, 104, 6515.
21. B. Hartke and H. J. Werner, Chem. Phys. Lett., 1997, 280, 430.
22. M. H. Alexander, D. E. Manolopoulos and H. J. Werner, J. Chem. Phys., 2000, 113, 11084.
23. V. Aquilanti, S. Cavalli, A. Volpi and D. Cappelletti, J. Phys. Chem. A, 2001, 105, 2401.
24. V. Aquilanti, S. Cavalli, D. de Fazio, A. Volpi, A. Aguilar and M. J. Lucas, Chem. Phys., 2005, 308, 237.
25. M. Hayes, M. Gustafsson, A. M. Mebel and R. T. Skodje, Chem. Phys., 2005, 308, 259.
26. C. X. Xu, D. Q. Xie and D. H. Zhang, Chin. J. Chem. Phys., 2006, 19, 96.
27. G. Li, H.-J. Werner, F. Lique and M. H. Alexander, J. Chem. Phys., 2007, 127, 174302.
28. B. Fu, X. Xu and D. H. Zhang, J. Chem. Phys., 2008, 129, 011103.
29. G. Lique, G. Li, H.-J. Werner and M. H. Alexander, J. Chem. Phys., 2011, 134, 231101.
30. Y.-Q. Li, Y.-Z. Song and J. Varandas, Eur. Phys. J. D, 2015, 69, 22.
31. J. Chen, Z. G. Sun and D. H. Zhang, J. Chem. Phys., 2015, 142, 024303.
32. Z. Sun and D. H. Zhang, Int. J. Quantum Chem., 2015, 115, 689.
33. L. Che, Z. Ren, X. Wang, W. Dong, D. Dai, X. Wang, D. H. Zhang, X. Yang, L. Sheng, G. Li, H.-J. Werner, F. Lique and M. H. Alexander, Science, 2007, 317, 1061.
34. W. Chen, R. Wang, D. Yuan, H. Zhao, C. Luo, Y. Tan, S. Li, D. H. Zhang, X. Wang, Z. Sun and X. Yang, Science, 2021, 371, 936.
35. B. Mukherjee, K. Naskar, S. Ravi, K. R. Shamasundar, D. Mukhopadhyay and S. Adhikari, J. Chem. Phys., 2020, 153, 174301.
36. R. Wang, Z. Sun and M. H. Alexander, J. Chem. Theory Comput., 2024, 20, 3449.
37. K. Naskar, S. Ghosh and S. Adhikari, J. Phys. Chem. A, 2022, 126, 3311.
38. K. Naskar, S. Mukherjee, S. Ghosh and S. Adhikari, J. Phys. Chem. A, 2024, 128, 1438.
39. H. Zhao and Z. Sun, Phys. Chem. Chem. Phys., 2024, 26, 22790.
40. N. Balakrishnan and A. Dalgarno, Chem. Phys. Lett., 2001, 341, 652.
41. N. Balakrishnan and A. Dalgarno, J. Phys. Chem. A, 2003, 107, 7101.
42. E. P. Wigner, Phys. Rev., 1948, 73, 1002.
43. E. Bodo, F. A. Gianturco and A. Dalgarno, J. Chem. Phys., 2002, 116, 9222.
44. E. Bodo and F. A. Gianturco, Eur. Phys. J. D, 2004, 31, 423.
45. J. Aldegunde, J. M. Alvariño, M. P. de Miranda, V. Saez-Rabanos and F. J. Aoiz, J. Chem. Phys., 2006, 125, 133104.
46. N. Balakrishnan, J. Chem. Phys., 2016, 145, 150901.
47. D. De Fazio, V. Aquilanti and S. Cavalli, Front. Chem., 2019, 7, 328.
48. D. De Fazio, V. Aquilanti and S. Cavalli, J. Phys. Chem. A, 2020, 124, 12.
49. V. Sáez-Rábanos, J. E. Verdasco and V. J. Herrero, Phys. Chem. Chem. Phys., 2019, 21, 15177.
50. V. Sáez-Rábanos, J. E. Verdasco, F. J. Aoiz and V. J. Herrero, Phys. Chem. Chem. Phys., 2021, 25, 8002.
51. H. Wang, Y. Li, Z. Jiao, H. Zang, C. Xiao and X. Yang, Chin. J. Chem. Phys., 2021, 34, 925.
52. Z. Jiao, H. Wang, Y. Li, H. Zang, C. Xiau and X. Yang, Chin. J. Chem. Phys., 2022, 35, 263.
53. T. Wang, T. Yang, C. Xiao and X. Yang, J. Phys. Chem. A, 2024, 128, 3180.
54. C. F. C. J. O. Hirschfelder and R. B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, 1954.
55. J. P. Toennies, W. Weltz and G. Wolff, J. Chem. Phys., 1979, 71, 614.
56. J. Aldegunde, P. G. Jambrina, M. P. de Miranda, V. Saez-Rabanos and F. J. Aoiz, Phys. Chem. Chem. Phys., 2011, 13, 8345.
57. D. Skouteris, J. F. Castillo and D. E. Manolopoulos, Comput. Phys. Commun., 2000, 133, 128.
58. F. J. Aoiz, M. Brouard, C. J. Eyles, J. F. Castillo and V. Saez-Rabanos, J. Chem. Phys., 2006, 125, 2353837.

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