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P. G.
Jambrina
*^{a},
M.
Menéndez
^{b} and
F. J.
Aoiz
*^{b}
^{a}Departamento de Química Física Aplicada, Universidad Autonoma de Madrid, 28049, Madrid, Spain. E-mail: pablo.gjambrina@uam.es
^{b}Departamento de Química Física I, Facultad de Ciencias Químicas, Universidad Complutense de Madrid, 28040 Madrid, Spain. E-mail: aoiz@quim.ucm.es

Received
28th December 2017
, Accepted 25th April 2018

First published on 26th April 2018

A natural generalization of the classical deflection function, the functional dependence of the deflection angle on the angular momentum (or the impact parameter), is the joint probability density function of these two quantities, revealing the correlation between them. It provides, at a glance, detailed information about the reaction mechanisms and how changes in the impact parameter affect the product angular distribution. It is also useful to predict the presence of quantum phenomena such as interference. However, the classical angular momentum–scattering angle correlation function has a limited use whenever quantum effects become important. Rigorously speaking, there is not a quantum equivalent of the classical joint distribution, as the differential cross section depends on the coherences between the different values of J caused by the cross terms in the expansion of partial waves. In this article, we present a simple method to calculate a quantum analog of this correlation, a generalized deflection function that can shed light onto the reaction mechanism using just quantum mechanical results. Our results show that there is a very good agreement between the quantum and classical correlation functions as long as quantum effects are not all relevant. When this is not the case, it will also be shown that the quantum correlation function is most useful to observe the extent of quantum effects such as interference among different reaction mechanisms.

The impact parameter b (or the orbital angular momentum ) and scattering angle θ are the two main variables that are studied to discern reaction mechanisms. The former is related to the asymptote of reactants and is one of the key factors in determining the outcome of a collision,^{1,2} as it determines the regions of the potential energy surface (PES) that will be explored during the collision (head-on vs. glancing collisions). The scattering angle, in turn, is defined at the product asymptote and provides information about nuclei scrambling during the collision; besides, it is amenable to experimental measurement using cross molecular beams with mass spectrometric universal detection or, more recently, velocity-mapped ion imaging^{3–6} or single beam coexpansion such as photoloc^{7} among other techniques. Moreover, from the theoretical point of view, it is relatively straightforward to extract the reaction probability as a function of J (opacity function or P_{r}(J)), and the differential cross section (DCS or σ_{r}(θ)) as a function of the scattering angle θ. Hence, it is not surprising that P_{r}(J) and DCS are two of the most important quantities used to determine the collision mechanism. However, the knowledge of these two distributions may not be sufficient to characterize the mechanisms. To this purpose, it would be necessary to relate how the initial and final conditions are correlated; specifically, which impact parameters give rise to scattering at certain angles.

To relate the angular momentum and scattering angle, the deflection function (DF), that is, the functional of the deflection angle (Θ whose absolute value is θ) in terms of the angular momentum, in its classical, semiclassical and quantal versions, has been widely used to explain elastic and inelastic scattering, in particular to understand those features related to glory and rainbow scattering.^{1,8,9} More recently, Connor and coworkers have devised a quantum deflection function (QDF) in the context of the glory analysis of forward scattering that can also be applied to reactive scattering.^{10–12} The QDF, defined as the derivative of the argument of the scattering matrix element with respect to J, has proved to be a valuable tool to predict the presence of rainbows. Besides, it could be used to predict interference between nearside and farside scattering. However, the QDF does not consider that a single J can correlate with different θ which limits its use to predict the presence of different mechanisms.

Within a classical mechanics framework, there is no limitation in the amount of information that may relate the initial and final conditions in a collision. Thus, it is perfectly feasible to go beyond the DF and to determine the joint dependence of the reaction probability as a function of the scattering angle and the impact parameter, the J–θ correlation function. This sort of generalization of the DF not only contains all the information provided by the P_{r}(J) and the DCS but also, primarily, provides how J and θ correlate throughout the collision. For reactive scattering, a strong correlation between J and θ is expected for reactions following a direct mechanism, whereas no or very weak correlation between these variables can be anticipated if the reaction takes place through a long-lived collision complex. Furthermore, discontinuities and different trends in the J–θ correlation can be associated with different reaction mechanisms, and permit its characterization even for apparently simple reactions.^{13,14}

The classical joint probability distribution has also been used to predict interference causing oscillations in the DCS.^{15,16} Given the wave nature of quantum mechanics (QM), it is expected that when one particle may follow two different pathways with the same outcome, they will interfere. In Young's double-slit experiment,^{17} interference arises when electrons going through two different slits hit the detector. In reaction dynamics we do not need slits and the system itself acts as an interferometer whenever two different J could scatter at the same angles and final quantum state.^{15,16,18} This analogy also explains why a correlation function between θ and J cannot be calculated using pure quantum mechanical grounds, as it is done in classical calculations. In QM, the angular distribution depends on the coherences between different partial waves, and therefore something apparently as simple as obtaining a rigorous joint probability distribution as a function of J and θ cannot be computed. This would be similar to disentangling which part of the signal comes from electrons going through one or the other slit in Young's double-slit experiment.

Throughout this article, we will try to circumvent this limitation and propose a new quantum analog to the classical correlation function, Q_{r}(θ,J), that will appear as a generalized deflection function (GDF) for the interpretation of quantum scattering results. This new function is a joint quasi-probability distribution of J and θ that includes all coherences between different partial waves, and whose summation over all partial waves recovers the exact differential cross section. As will be shown Q_{r}(θ,J) emerges as a valuable tool to assist in the elucidation of reaction mechanisms, especially when quantum phenomena are important, or when several reaction mechanisms coexist.

The article is organized as follows: in Section 2 we will review the classical GDF as the joint distribution of θ and J, followed by the definition of an intuitively simple QM quasi-probability joint distribution or QM GDF, Q_{r}(θ,J), starting from the definition of the scattering amplitude. In Section 3 we will demonstrate the usefulness of the proposed Q_{r}(θ,J) function to disentangle reaction mechanisms and to unveil quantum effects such as interference for three different systems and situations: inelastic collisions of Cl + H_{2}; reactive D^{+} + H_{2} collisions; and H + D_{2} reactive scattering at a collision energy where quantum interference governs the shape of the DCSs for certain combinations of final and initial states. For all these systems, QM calculations have been carried out using the close-coupling hyperspherical method of Skouteris et al.,^{19} while quasiclassical trajectory (QCT) calculations have been performed using the procedure described in ref. 20. The reader not interested in the theoretical details can skip Section 2 and move directly to Section 3 where the potential of the GDFs is exemplified and discussed.

(1) |

Eqn (1) is valid if the sampling in J is performed discretely and uniformly and, similarly, for the orbital angular momentum in the |J − j| ≤ ≤ J + j interval (for details see ref. 21). In addition, not all reactive trajectories need to have the same weight. Sometimes it is necessary to attribute different weights to each trajectory as is done in the Gaussian binning procedure^{22–24} to make the assignment of the final rovibrational states ‘more quantal’. In those cases, N_{r}(J) in eqn (1) is replaced by S_{w}, the sum of the weights of reactive (or inelastic) trajectories into a given final manifold of states. If one wishes to calculate a property that depends on more than one variable, for example J and , the scheme is the same except that now a joint probability has to be considered (say, the number of reactive trajectories with values of J and , N_{r}(J,)).^{21} The aforementioned procedure is suitable for discrete variables, while for continuous variables it is a common practice to use histograms or, more elegantly, to fit the distributions to a series of orthogonal polynomials.^{20,21,25} Obviously, integration (or summation) over one of the variables of a given joint probability distribution leads to the probability distribution of the other variable. Moreover, if we split the original ensemble of trajectories into a series of sub-ensembles and calculate the respective joint probability distribution, it turns out that the global probability distribution can be easily recovered from the joint probability distributions for all the sub-ensembles; that is to say, the probability distributions are always additive. As we will see, this is not the case in QM scattering due to the coherences.

To illustrate the calculation of the classical correlation function, let us assume that the orbital angular momentum is sampled continuously in ∈ [0,_{max}] with a weight of 2 + 1, that is, the orbital angular momentum for the i-th trajectory is sampled as _{i}(_{i} + 1) = ξ_{i}[_{max}(_{max} + 1)], where ξ is a random number in [0,1] (this is the same as sampling the impact parameter as b = ξ^{1/2}b_{max}).

We can conveniently define a J-partial cross section, σ_{r}(J):

(2) |

The Monte Carlo normalized probability density function can be written as

(3) |

(4) |

If the sampling in J (and in ) is made continuous, the J-partial cross section can be expressed as an expansion in Legendre polynomials, P_{n}(x):

(5) |

(6) |

(7) |

Similarly, the DCS can be expressed as an expansion in Legendre polynomials:

(8) |

(9) |

The joint probability distribution of J and θ, already used in ref. 15 and 26, normalized to the integral cross section, can now be expressed as a double expansion in Legendre polynomials

(10) |

(11) |

The Monte Carlo expression of the deflection function can be expressed as a sum of Gaussian functions given by

(12) |

Integration of eqn (10) or (12) over θ and the azimuthal angle provides the J-partial cross section of eqn (5) and (3). Alternatively, integration over J in these equations gives σ_{r}(θ)sinθ. Hereinafter we will indifferently denote σ_{r}(θ,J) as the QCT generalized deflection function (QCT GDF) or the QCT J–θ correlation function.

Some observables can be readily extracted from the S-matrix. This is the case of P_{r}(J) that, for a given initial state and total energy, can be calculated as follows:

(13) |

(14) |

The derivation of vector properties such as the DCS from the S-matrix is not so straightforward, firstly, because we need to include the angular dependence, and secondly, because they involve coherences between different elements of the S-matrix. It is convenient to express the DCS in terms of the scattering amplitudes, which are defined as:

(15) |

(16) |

From eqn (15) and (16) it is clear that the DCSs for state-to-state processes are additive, even when they are resolved in Ω′ and Ω. However, the squaring of the sum over J in eqn (15) makes the DCS no longer additive in J, i.e., there are coherences (cross terms) between different J partial waves. This property is a reflection of the wave nature of quantum mechanics, so that two or more “paths” (impact parameters or J) leading to scattering at the same angle will interfere. Hence, in principle, it is not possible to separate the contribution of two partial waves to the converged DCS. It is worth noticing that usually coherences are only important between nearby values of J^{27} so, for certain cases, it is possible to extract the contributions from one or many mechanisms from the DCS.

To calculate a QM J–θ correlation function we would need to extract the contribution of each J to the total DCS. Furthermore, for a QM correlation function to be reliable it should be additive so that the sum over J leads to the converged (summed over all J-partial waves) DCS. In principle, one could compute it by neglecting all coherences between different Js. This would be equivalent to using the random phase approximation that lies in the core of the statistical model,^{28,29} giving rise to forward–backward symmetric DCSs. For non-statistical (direct) reactions, a symmetric DCS is in clear disagreement with the experimental and QM results, and hence neglecting coherences can be considered as a very inappropriate approximation to obtain a QM correlation function. Instead, to devise a QM GDF, we will start by defining a J-partial dependent scattering amplitude as:

(17) |

(18) |

The DCS can be expressed as a function of the J-partial scattering amplitudes:

(19) |

(20) |

Eqn (19) and (20) only differ in the presence of an additional sum over J in eqn (20) that is compensated with the term (δ_{J1,J} + δ_{J2,J})/2, which guarantees that both equations include the same number of cross products and hence that they are equivalent. The advantage of eqn (20) is the presence of a separate summation over J that allows us to define a function that depends on a single J and θ; that is, a quantum analog to the classical joint probability distribution, and we will denote that as Q_{r}(θ,J),

(21) |

(22) |

In the absence of coherences, that is, in the random phase approximation limit, the only surviving term would be that depending on J only. The remaining terms account for the possible interference that most of the time can be expected to be only important between partial waves in a restricted range of J in [J − ΔJ, J + ΔJ].^{15,16} However, as will be shown below, interference can also take place between partial waves that cover the full range of angular momentum leading to scattering.

The QM J–θ correlation function (or QM GDF) shares some important properties in common with its classical counterpart, σ_{r}(θ,J). As in the classical case, summing eqn (21) over J leads to the DCS given by eqn (20) multiplied by sinθ, σ_{r}(θ)sinθ. Similarly, integration over the scattering angle and the azimuthal angle

(23) |

In spite of the similarities between the classical σ_{r}(θ,J) (eqn (10) or (12)) and the quantum Q_{r}(θ,J) (eqn (21)), there are important differences between them. The latter is not a true joint probability distribution (and, hence, a GDF in the classical sense) since it includes coherences between different values of J. Moreover, it can take negative values whenever there is destructive interference between pairs of J values, although when summed over J up to J_{max}, the GDF is always positive. Notwithstanding the differences, as will be shown in Section 3, when the interference is not significant, classical and quantum correlation functions bear a close resemblance.

It is sometimes useful to calculate the angular distributions for a subset of partial waves. These angular distributions, labeled as DCS([J_{i},J_{k}]), can be calculated by restricting the sum over J in eqn (20) to a given range of J, J ∈ [J_{i},J_{k}],

(24) |

The partially summed QM DCS, DCS([J_{i},J_{k}]), includes all coherences between partial waves within the [J_{i},J_{k}] range but none outside this range. In addition, like the DCS itself, DCSs([J_{i},J_{k}]) are not additive, especially if there is interference between different groups of Js. This is again in contrast to the corresponding classical partial cross section summed over the [J_{i},J_{k}] interval.

It is also possible to define a deflection function by restricting the sum over a given [J_{i},J_{k}] range of J, Q_{r}(θ;[J_{i},J_{k}]), as

(25) |

In spite of the similarities between the partial DCS([J_{i},J_{k}]) and Q_{r}(θ;[J_{i},J_{k}]) (and the fact that in the limit of the complete convergence interval, J_{i} = 0 and J_{k} = J_{max}, both functions are identical), there are two main differences between them: (i) the latter also includes coherences between partial waves outside the [J_{i},J_{k}] range so it may take negative values (if destructive interference prevails for some scattering angles); (ii) the Q_{r}(θ;[J_{i},J_{k}]) values for different intervals defined in eqn (25) are additive as in the classical case. Hence, from the comparison between the partially summed DCSs and partial QM correlation function, it is straightforward to ascertain if interference phenomena arise and the partial waves that contribute to them. It must be noted that in the classical case both functions are the same.

The semiclassical deflection function, Θ(_{θ}), is related to the phase shift, η_{}, by

(26) |

S_{} = e^{2iη} | (27) |

Hence,

(28) |

Eqn (26) also holds if the exact-QM phase shifts are defined, leading to a quantum mechanical DF that can be applied for elastic scattering processes even for soft edge potentials.^{35}

In a series of articles, Connor and co-workers extended the semiclassical treatment and developed a quantal version of the deflection function applicable to the most general case of inelastic and reactive scattering.^{10–12} It is thus pertinent to compare our proposed GDF with the QDF devised by Connor and coworkers. In what follows, we will briefly summarize the main equations of that method for our present purposes.^{10}

For given initial and final rovibrational states the QDF, denoted as Θ˜_{Ω′Ω}, is defined as

(29) |

_{Ω′Ω}(J) = exp(iπJ)S^{J}_{Ω′Ω} | (30) |

It should be highlighted that arg_{Ω′Ω}(J) does not denote the principal value, but it is defined as a continuous function as follows:

(31) |

The differences between the QDF and Q_{r} need to be emphasised. Whilst the latter is a sort of probability density function in terms of both θ and J, which appears as a 3D plot of the scattering intensity as a function of both J and θ containing information about the scattering intensity and the presence of constructive or destructive interference, the QDF represents a one-dimensional relationship between the deflection angle (or the scattering angle) and the angular momentum J. Moreover, as shown in the previous subsection, if Q_{r}(θ,J) is summed over J, one gets the DCS, in contrast to the QDF. Another difference is that whilst the QDF is defined for each pair of Ω and Ω′ values, the GDF defined in this work can include the average over the reactant's and the summation over product's helicities as shown in eqn (21), although it can also be calculated for specific values of Ω and Ω′, as will be shown below. Apart from these differences, one would expect a confluence with regard to the relationship between the scattering angle and angular momentum, at least if each partial wave can be mostly associated with one range of scattering angles.

Regarding inelastic collisions, some interesting features emerged from previous studies.^{41,42} QM and QCT calculations using the BW2 PES^{43} showed that at relatively high collision energies (E_{coll} > 0.6 eV) and for small Δj values (Δj = j′ − j), the inelastic probabilities, P_{r}(J), exhibit two maxima separated by a minimum in the QCT and QM results. This minimum was identified as that corresponding to the glory impact parameter. The analysis of the results showed that there are two mechanisms responsible for the inelastic scattering, possibly associated with different regions of the PES and resulting in very different stereodynamical behaviours.^{41,42} Both dynamical regimes depend primarily on the value of the total (here also orbital) angular momentum: (i) for Js below the glory impact parameter, collisions seem to take place following a sort of “tug-of-war” mechanism,^{44} which indicates the stretching of the H–H bond;^{42} and (ii) for J ≳ 40 collisions can be assigned to rainbow scattering in which the attractive part of the PES is sampled.^{41} For transitions involving higher Δj, which require more head-on collisions, the contribution of high impact parameters wanes rapidly, and the second maximum in the P_{r}(J) leading to small scattering angles disappears. The semi-quantitative agreement between the classical and quantum P_{r}(J) and DCSs seems to indicate that quantum effects associated with interference between the two groups of partial waves are not expected to be important.^{41} Therefore, the Cl + H_{2}(v = 0, j = 0) inelastic scattering seems to be a good example of a collision system in which the QCT and QCT GDFs would be similar.

Fig. 1 displays the QCT and the QM deflection functions for the j = 0 → j′ = 2 and j = 0 → j = 4 transitions (top and bottom panels, respectively) at E_{coll} = 0.73 eV. The left panels show the QCT σ(θ,J). The two different dynamical regimes can be easily distinguished. For Δj = 2, the high-J mechanism is preeminent and gives rise to scattering into θ < 50°. The low-J mechanism appears in the deflection function as a narrow band that extends from θ = 40° to θ = 180° and comprises J values from 0 to 40. The negative slope, common to both regimes (although with different values), is characteristic of direct collisions, and follows the simple correlation of low (high) impact parameters leading to high (small) scattering angles. For Δj = 4, the prevailing mechanism is that corresponding to J ≤ 40 values, and the high-J mechanism appears as a small island in the J–θ map, centered at J = 50 and θ = 30°.

The equivalent QM Q_{r}(θ,J)'s, shown in the right panels of Fig. 1, bear close similarities to their classical counterparts, although with some noticeable differences. For Δj = 2, the high-J mechanism, responsible for most of the scattering, extends to larger values of J, it is broader, and it is flanked by a series of stripes, some of negative value (green colour) associated with destructive interference. The negative slope of the low-J mechanism is also observed, although in this case both mechanisms merge at J ∼ 45. There are also a series of negative stripes parallel to the main band which cause a small decrease of the DCS. It should be noticed that, for the sake of clarity in the figure, the QM GDF has been smoothed given the discrete character of J. The same procedure will be followed for all remaining 3D plots of this article. For Δj = 4, the QM-DF also extends to larger J values and the high-J mechanism covers a broader J–θ region than in the QCT case. As in the classical case, for this transition, the low-J mechanism bears away most of the scattering.

The results of the QDF for Δj = 2 are also shown as a dotted lines along with the present Q_{r}(θ,J). The points corresponding to Ω′ = 0, 1 and 2 are all included. As can be seen, the QDF follows almost exactly the middle line (reproducing the two different slopes) of the present QM GDF and is also in good agreement with the corresponding QCT function. More detailed information is shown in Fig. 2, where the Q_{r}(θ,J,Ω′) is plotted separately for each of the three possible Ω′ values along with the corresponding QDF. As can be seen, the agreement is excellent and the QDF matches almost exactly the most probable dependence of θ with J found with the present Q_{r}(θ,J). It should be pointed out, however, that the latter also carries information on the intensity of scattering for each J–θ region, and about the presence of constructive and destructive interference. Indeed, the information conveyed by the present Q_{r}(θ,J,Ω′) goes well beyond that obtained by the QDF. As can be seen, most of the intensity of the high-J mechanism corresponds to Ω′ = 2, indicating that the product's j′ rotational angular momentum lies preferentially along the recoil velocity, whilst that corresponding to low-J is more isotropic with some preference for Ω′ = 1.^{42}

The partial DCS, eqn (24), and the QM Q_{r}(θ,ΔJ) summed over the indicated range of J, eqn (25), are shown in the left and right panels of Fig. 3, respectively, for Δj = 2 and 4. The two J intervals have been chosen to comprise partial waves corresponding to low-J (J ≤ 41 for Δj = 2 and J ≤ 45 for Δj = 4) and high-J (J > 41 for Δj = 2 and J > 45 for Δj = 4). Therefore, the two magnitudes are broken down in their contributions from the two intervals for comparison purposes. It should be recalled that if the whole range of J is included, both magnitudes become identical, corresponding to the converged (including all partial waves) DCS. However, whilst the partial DCS only encompasses those coherences within the chosen interval, the partially summed QM GDF comprises all possible coherences internal and external to that interval.

Fig. 3 Comparison of the DCS partially summed over the indicated J interval, DCS([J_{i},J_{k}]) (defined in eqn (24)) (left panels) and the QM deflection functions summed over the same J interval Q_{r}(θ;[J_{i},J_{k}]) (defined in eqn (25)) (right panels) for the inelastic collisions between Cl and H_{2}(v = 0, j = 0) at E_{col} = 0.73 eV and Δj = 2 (top panels) and Δj = 4 (bottom panels). |

The first consideration to be held is the similarity of the respective decompositions of the partial DCSs and the summed QM GDFs Q_{r}(θ;ΔJ), of the left and right panels. As a second consideration, for Δj = 2, the incoherent sum of σ(θ;J ≤ 42) and σ(θ;J > 42) reproduces fairly well the converged DCS (recall that the partial DCSs are not additive), evincing that interference between the two mechanisms is practically negligible. A similar analysis was performed in ref. 42 leading to the same conclusion. This is further confirmed by inspection of Q_{r}(θ;ΔJ), shown in the right-top panel, which is almost identical to the partial DCSs, except for a few differences in the forward region. For the case of Δj = 4 the situation is much the same as that for Δj = 2. The only main difference between partial DCSs and Q_{r}(θ;ΔJ) can be observed at forward scattering angles θ = 10–30°. As can be seen, there is a peak centred at θ = 12° in the Q_{r}(θ;J < 46) which is absent in the respective σ_{r}(θ,J < 46). This implies that there are some, relatively unimportant, interference between the two groups of partial waves. Returning to Fig. 1, it is possible to associate this effect with the feature that appears with a ‘hook’ at the top corner of the right-bottom panel of that figure.

It must be pointed out that the above discussion does not imply that for Δj = 2 there is no interference within one of those groups of partial waves. By the inspection of the right-bottom panel of Fig. 1, it is obvious that in the forward region and at high J > 40 there is high constructive and destructive interference that is the origin of the oscillations observed at θ < 30° in Fig. 3.

We will focus on the results at a sufficiently low energy, E_{coll} = 150 meV and HD(v′ = 0, j′ = 1) formation, where the statistical (ergodic) assumption seems to hold.^{52–55} Indeed, at this energy, the D^{+} + H_{2} reaction proceeds through the formation of a long-lived complex, and the shape of P_{r}(J) and the product state distributions follow the trend predicted by statistical methods.^{55} Hence, this seems to be a good example to test the reliability of the Q_{r}(θ,J) in statistical reactions. In Fig. 4 three GDFs are shown: the classical function, the QM Q_{r}(θ,J) and the quantum one under the assumption of the random phase approximation. The latter implies that there are no correlations between different Js, so that it only includes the |f^{J}_{Ω′Ω}(θ)|^{2} terms in eqn (22). In all three cases, as expected for a statistical reaction, there is no clear correlation between J and θ: all Js seem to contribute to every scattering angle. The only remarkable feature in the classical J–θ correlation function is the largest probabilities found at high J, due to the fact that the P_{r}(J) is flat until it decreases abruptly when reaching J_{max}. The Q_{r}(θ,J), shown in the middle panel of Fig. 4, indicates the presence of high destructive (green) and constructive (red/yellow) interference that will give rise to multiple oscillations in the DCS over the whole range of scattering angles. However, coherences even if they occurred between partial waves with separated J values are so numerous that their effect is smoothed out to some extent. This is the basic assumption in the random phase approximation,^{29,58} which allows one to calculate coarse-grained product's state distribution DCSs and other vector correlations^{59} by neglecting the coherences between different total angular momenta. The right panel of Fig. 4 shows the random phase approximated DF, where all the coherences have been neglected by only keeping the diagonal terms. Apart from the discrete character of J, the similarity with the QCT GDF is remarkable. For this reaction, the QDF results in a highly oscillating function due to the superposition of nearside and farside scattering.^{60}

The partial and total DCSs, as well as the Q_{r}(θ,[J_{i},J_{k}]) summed over limited ranges of J, are shown in the top and middle panels, respectively, of Fig. 5. The J dividing value between low-J and high-J values has been chosen somewhat arbitrarily as J_{max}/2, since no hint of change of mechanism seems to be appreciable in either the QCT or the QM GDF. As expected from the QM GDF, the DCSs with the full QM calculation exhibit many oscillations in the whole range of scattering angles, reflecting the high interference that is apparent in Fig. 4. The partial DCSs and their respective Q_{r}(θ,ΔJ) summed in [0,18] and [19,35] are fairly similar with some interesting differences. Specifically, the inspection of the partially summed Q_{r}(θ,ΔJ) makes it possible to identify the dip in the converged DCS at 100–110° as a result of destructive interference between the partial waves with ΔJ ≤ 18 (negative value) and ΔJ > 18, information that cannot be extracted from the partially summed DCS, that only includes interference within each interval. This destructive interference can be also seen in the QM GDF shown in the middle panel of Fig. 4 as the green stripes (negative value) at those angles.

Fig. 5 Comparison of the Q_{r}(J,θ) and the DCS for the D^{+} + H_{2} → HD(v′ = 0, j′ = 1) + H^{+} reaction at E_{col} = 0.15 eV. |

The partial DCSs, which under the random phase approximation coincide with the Q_{r}(θ,ΔJ), are shown in the bottom panel of Fig. 5. There are still some oscillations that are basically the result of the summation of reduced rotation matrix terms, [d^{J}_{ΩΩ′}(θ)]^{2} (eqn (17)). The resulting random phase DCSs are strictly symmetric, peaking at forward and backward angles (recall that the represented DCSs have been multiplied by sinθ). Although at first glance there seems to be a poor approximation to the actual DCSs, it must be borne in mind that the observed oscillations change rapidly with the collision energy and initial states, and hence they would be barely discernible under experimental conditions.

Let us first turn our attention to those collisions leading to HD(v′ = 3, j′ = 0) whose QCT and QM GDFs are depicted in Fig. 6. The QCT σ_{r}(θ,J) shows the typical profile of a direct reaction mechanism, similar to that observed for the inelastic collisions between Cl and H_{2}, that is, a band running diagonally across the θ–J map with low J giving rise to backward scattering and high J correlating with forward scattering. In this case, the mechanism covers the whole range of scattering angles with one maximum in the forward and another in the backward region. Moreover, there seems to be no other mechanism to compete with it. Not surprisingly, QCT and QM GDFs are very similar, showing the same structure moving from backwards to forwards. However, although the QM results were somewhat smoothed out for the sake of clarity, we can still observe a series of constructive and destructive interference manifested as stripes, especially in the forward scattering region. In addition, the main band is flanked by two small green stripes (destructive interference) that will give rise to oscillations in the DCS.

Fig. 7 depicts the partial DCS and the Q_{r}(θ,ΔJ) for three subsets of partial waves that, according to the J–θ correlation of Fig. 6, can be associated with backward (J ∈ [0,10]), sideways (J ∈ [11,21]) and forward (J > 22) scattering. There is a remarkable similarity between the partial DCSs and the corresponding Q(θ,[J_{1},J_{2}]) for each of the three intervals, implying that there is essentially no interference between the partial waves belonging to the different subsets. Only at forward scattering angles there is some appreciable interference between partial waves associated with J values pertaining to the [11,21] and J > 22 intervals. There is one more aspect that deserves a comment. The maxima and minima that can be observed in the DCS can be easily inferred from the positive and negative values of the QM GDF. In particular, the minima at 70°, 115° and 150° correspond to the negative (green colour) contributions in the QM GDF. These minima (and the precedent or subsequent maxima) cannot be deduced from the classical GDF.

Fig. 7 Comparison of the partial DCS (upper panel) and the QM Q(θ,ΔJ) summed over the same J contributions (bottom panel) for the H + D_{2} → HD(v′ = 3, j′ = 0) + D reaction at E_{col} = 1.97 eV. |

Let us now move to the collisions leading to HD(v′ = 1, j′ = 0). The QCT and QM GDFs are shown in the top panels of Fig. 8. As discussed in previous work,^{15} and can be seen by the inspection of the QCT GDF, there are two main, distinct mechanisms that are likely to interact with each other giving rise to the interference pattern observed experimentally. One of them corresponds to the main band with a negative slope, similar to that we have found for v′ = 3; the other mechanism, confined in a small region of the J–θ map, between 110 and 160° and low J values, accounts for most of the reactivity. Between them, as a sort of bridge, there is still a third mechanism with a positive slope that comprises low values of J and θ > 160°. Using the QCT GDF it could be predicted that interference will take place,^{15} since different paths with different Js are leading to the same scattering angles. However, the QCT GDF cannot resolve the interference pattern: the number of oscillations and what would be their positions. In previous examples, we have shown that the QM GDFs were akin to their QCT counterparts. In this example, however, we will see that the quantum Q_{r}(θ,J) provides additional and most valuable information.

The first observation is that the QM GDF shown in the top-right panel of Fig. 8 is rather different to its classical counterpart. Only with the help of the superimposed contour of the classical σ_{r}(θ,J) and leaving aside the destructive coherences, we could see that they share the main gross features. Even then, the QM GDF is broader, and the region corresponding to the diagonal band almost merges with the mechanism confined between 110 and 160° and J < 10. But the main source of discrepancy lies in the presence of negative, destructive (green colour) and positive, constructive (red colour) interference that does not flank the main band – as in the case of HD(v′ = 3, j′ = 0) scattering – but it is transversal to it, cutting the diagonal band into several slices. Since Q_{r}(θ,J) is additive, it is easy to realize that each of the slices corresponds to the various peaks in the DCS, whilst the vertical green stripes correspond to minima in the DCS. Therefore, just by looking at the QM GDF we could discern (i) that there will be three peaks in the backward hemisphere, (ii) which will be their positions, as well as those of the respective minima, and (iii) the partial waves that contribute to each of the peaks.

Not surprisingly, the partial DCS and the QM GDFs summed over a range of J values, Q_{r}(θ,ΔJ), calculated for subsets of partial waves and shown in Fig. 9 do not look alike. The DCS(J ≤ 8) can be associated with the confined mechanism and, although it carries most of the reactivity, it shows a broad, blunt shape with no hint of the three finger-like peaks present in the total DCS in the 100–180° range. In stark contrast, the Q_{r}(θ,0 ≤ J ≤ 8), that accounts for all the coherences for which the J ∈ [0,8] range participates, looks similar to the overall DCS. The partial DCSs calculated for J > 8 (J ∈ [9,14] and J ∈ [15,21]) are very small throughout the whole range of scattering angles, whereas their respective Q_{r}(θ,ΔJ) is not that small. On top of that, at some angles they are negative, a consequence of the negative contours shown in Fig. 8.

The third scenario corresponds to collisions leading to HD(v′ = 1, j′ = 5) whose QCT and QM GDFs are portrayed in the bottom panels of Fig. 8. As can be seen, the structure that appeared at low Js for HD(v′ = 1, j′ = 0) has almost merged into the diagonal band and is considerably less confined. In addition, QCT and QM GDFs look now more alike. Yet, the main band is cut by the signature of destructive interference (the green slice at θ ∼ 115°) that can be expected to give rise to a minimum in the backward DCS.

The comparison of the partial DCS and the Q_{r}(θ,ΔJ) confirms these findings and clarifies the role of interference. The choice of J = 16 for the decomposition seems to be a sensible choice in light of the deflection functions shown in Fig. 8. In contrast to the results for HD(v′ = 1, j′ = 0), the DCS(J ≤ 16) is similar to Q_{r}(θ,J ≤ 16), although the latter is somewhat more structured. However, the Q_{r}(θ,J ≥ 17) displays some oscillations and a negative contribution at θ ≈ 115° (as expected from the green slice commented on above) which reveals coherences with the low subset of partial waves. The effect of these partial waves is to sharpen the shape of the DCS, defining more clearly the two maxima and the intermediate minimum.

Finally, it is worthwhile to compare the results obtained using the formalism devised in this work with the QDF. In Fig. 6 and 8, superimposed to the Q_{r}(θ,J), the respective QDFs for v′ = 3, j′ = 0 and v′ = 1, j′ = 0 are represented as open squares. For v′ = 3 the agreement is fairly good, covering the regions occupied by the present QM GDF. In particular, the oscillations observed in extreme forward, which could be predicted by the Q_{r}(θ,J), can be also foreseen using the QDF (different Js leading to the same θ). In fact, using the QDF it can be concluded that they are caused by interference between nearside and farside reactive flux.^{65} For the v′ = 1 case, however, the sole analysis of the QDF barely accounts for the confined, predominant mechanism. It must be pointed out that even if we could observe the various mechanisms in the QDF, it would not have been possible to predict either the number of peaks and dips or their position since, because of its construction, it only provides one single value of the deflection angle per partial wave.

Throughout this article, we propose a conceptually simple quantum GDF, analogous to the classical J–θ correlation function, which does account for the coherences between J partial waves and whose interpretation is rather intuitive. Moreover, the QM GDF presented here not only relates scattering angles to angular momenta but also accounts for the scattering intensity. As such, summing over the whole set of angular momenta for convergence yields the DCS, and integrating over scattering gives the reactive (or inelastic) partial cross section. The calculation of the QM GDF does not require any additional computational effort for the calculation of the DCS or J reaction probability.

In this article we have exemplified the proposed QM GDF with several case studies comprising inelastic collisions of Cl + H_{2}, the barrierless (and presumably statistical) D^{+} + H_{2} reaction, and the direct H + D_{2} reaction for different final states. Our results show that classical and QM GDFs are essentially coincident whenever quantum interference is not preeminent, although the latter is capable of adding valuable details. When quantum phenomena are present, the QM GDF arises as a powerful tool and makes it possible to observe the interference pattern at first sight and to disentangle the J partial waves that contribute to constructive and destructive interference. It also provides information on the number and position of the peaks in the DCS, which something that cannot be extracted from the classical GDF.

The methodology devised here is completely general and can be used to obtain GDF functions for any reaction including those involving more than three atoms. Moreover, due to its quantum mechanical nature, it can be used to analyse reaction mechanisms that do not have a classical analog or under conditions where the classical deflection cannot be calculated, such as at energies below the barrier or whenever either resonances or diffraction phenomena are observed. In summary, we deem the QM GDF as a valuable tool for the analysis of the results obtained using quantum dynamics that can be easily implemented by any researcher in reaction dynamics.

- R. D. Levine, Molecular Reaction Dynamics, Cambridge University Press, Cambridge, 2005 Search PubMed.
- Tutorials in Molecular Reaction Dynamics, ed. M. Brouard and C. Vallance, RSC Publishing, 2010 Search PubMed.
- A. T. J. B. Eppink and D. H. Parker, Rev. Sci. Instrum., 1997, 68, 3477–3484 CrossRef CAS.
- A. Gijsbertsen, H. Linnartz, G. Rus, A. E. Wiskerke and S. Stolte, J. Chem. Phys., 2005, 123, 224305 CrossRef CAS PubMed.
- J. J. Lin, J. Zhou, W. Shiu and K. Liu, Rev. Sci. Instrum., 2003, 74, 2495–2500 CrossRef CAS.
- M. N. R. Ashfold, N. H. Nahler, A. J. Orr-Ewing, O. P. J. Vieuxmaire, R. L. Toomes, T. N. Kitsopoulos, I. A. Garcia, D. A. Chestakov, S.-M. Wu and D. H. Parker, Phys. Chem. Chem. Phys., 2006, 8, 26–53 RSC.
- N. E. Schafer, A. J. Orr-Ewing, W. R. Simpson, H. Xu and R. N. Zare, Chem. Phys. Lett., 1993, 212, 155 CrossRef.
- K. W. Ford and J. A. Wheeler, Ann. Phys., 1959, 7, 259–286 Search PubMed.
- R. B. Bernstein, Adv. Chem. Phys., 1966, 10, 75–134 Search PubMed.
- J. N. L. Connor, Phys. Chem. Chem. Phys., 2004, 6, 377–390 RSC.
- X. Shan and J. N. L. Connor, Phys. Chem. Chem. Phys., 2011, 13, 8392–8406 RSC.
- C. Xiahou and J. N. L. Connor, J. Phys. Chem. A, 2009, 113, 15298–15306 CrossRef CAS PubMed.
- S. J. Greaves, D. Murdock and E. Wrede, J. Chem. Phys., 2008, 128, 164307 CrossRef PubMed.
- S. J. Greaves, D. Murdock, E. Wrede and S. C. Althorpe, J. Chem. Phys., 2008, 128, 164306 CrossRef PubMed.
- P. G. Jambrina, D. Herraez-Aguilar, F. J. Aoiz, M. Sneha, J. Jankunas and R. N. Zare, Nat. Chem., 2015, 7, 661–667 CrossRef CAS PubMed.
- P. G. Jambrina, J. Aldegunde, F. J. Aoiz, M. Sneha and R. N. Zare, Chem. Sci., 2016, 7, 642–649 RSC.
- R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Addison Wesley Longman, 1970 Search PubMed.
- M. Sneha, H. Gao, R. N. Zare, P. G. Jambrina, M. Menendez and F. J. Aoiz, J. Chem. Phys., 2016, 145, 024308 CrossRef PubMed.
- D. Skouteris, J. F. Castillo and D. E. Manolopoulos, Comput. Phys. Commun., 2000, 133, 128–135 CrossRef CAS.
- F. J. Aoiz, L. Bañares and V. J. Herrero, J. Chem. Soc., Faraday Trans., 1998, 94, 2483–2500 RSC.
- F. J. Aoiz, V. Sáez-Rábanos, B. Martinez-Haya and T. González-Lezana, J. Chem. Phys., 2005, 123, 094101 CrossRef CAS PubMed.
- L. Bonnet and J. C. Rayez, Chem. Phys. Lett., 1997, 277, 183–190 CrossRef CAS.
- L. Bañares, F. Aoiz, P. Honvault, B. Bussery-Honvault and J.-M. Launay, J. Chem. Phys., 2003, 118, 565 CrossRef.
- L. Bonnet and J. C. Rayez, Chem. Phys. Lett., 2004, 397, 106–109 CrossRef CAS.
- F. J. Aoiz, V. J. Herrero and V. Sáez-Rábanos, J. Chem. Phys., 1992, 97, 7423 CrossRef CAS.
- P. G. Jambrina, M. Menendez and F. J. Aoiz, Phys. Chem. Chem. Phys., 2017, 19, 16433–16445 RSC.
- A. N. Panda, D. Herraez-Aguilar, P. G. Jambrina, J. Aldegunde, S. C. Althorpe and F. J. Aoiz, Phys. Chem. Chem. Phys., 2012, 14, 13067–13075 RSC.
- E. J. Rackham, T. Gonzalez-Lezana and D. E. Manolopoulos, J. Chem. Phys., 2003, 119, 12895–12907 CrossRef CAS.
- E. J. Rackham, F. Huarte-Larranaga and D. E. Manolopoulos, Chem. Phys. Lett., 2001, 343, 356–364 CrossRef CAS.
- W. H. Miller, Adv. Chem. Phys., 1974, 25, 69–177 Search PubMed.
- M. S. Child, Molecular Collision Theory, Dover publications, Mineola, N.Y. 11501, 1996 Search PubMed.
- J. N. L. Connor, Semiclassical theory of elastic scattering, in Semiclassical methods in Molecular Scattering and Spectroscopy, ed. M. S. Child, Reidel Publishing Company, Dordrecht (Holland), 1979, ch. p. 45 Search PubMed.
- H. Friedrich, Scattering Theory, Lecture Notes in Physics 872, Springer, 2013 Search PubMed.
- J. Burgdörfer, C. O. Reinhold, J. Sternberg and J. Wang, Phys. Rev. A, 1995, 60, 1248–1252 CrossRef.
- J. Herb, P. Meerwald, M. J. Moritz and H. Friedrich, Phys. Rev. A, 1999, 60, 853–860 CrossRef CAS.
- S. C. Althorpe and D. C. Clary, Annu. Rev. Phys. Chem., 2003, 54, 493–529 CrossRef CAS PubMed.
- M. Alagia, N. Balucani, L. Cartechini, P. Casavecchia, E. H. van Kleef, G. G. Volpi, F. J. Aoiz, L. Bañares, D. W. Schwenke, T. C. Allison, S. L. Mielke and D. G. Truhlar, Science, 1996, 273, 1519–1522 CAS.
- P. Casavecchia, Rep. Prog. Phys., 2000, 63, 355–414 CrossRef CAS.
- X. G. Wang, W. R. Dong, C. L. Xiao, L. Che, Z. F. Ren, D. X. Dai, X. Y. Wang, P. Casavecchia, X. M. Yang, B. Jiang, D. Q. Xie, Z. G. Sun, S. Y. Lee, D. H. Zhang, H. J. Werner and M. H. Alexander, Science, 2008, 322, 573–576 CrossRef CAS PubMed.
- N. Balucani, D. Skouteris, L. Cartechini, G. Capozza, E. Segoloni, P. Casavecchia, M. H. Alexander, G. Capecchi and H. J. Werner, Phys. Rev. Lett., 2003, 91, 013201 CrossRef CAS PubMed.
- L. Gonzalez-Sanchez, J. Aldegunde, P. G. Jambrina and F. J. Aoiz, J. Chem. Phys., 2011, 135, 064301 CrossRef CAS PubMed.
- J. Aldegunde, F. J. Aoiz, L. Gonzalez-Sanchez, P. G. Jambrina, M. P. de Miranda and V. Sáez-Rábanos, Phys. Chem. Chem. Phys., 2012, 14, 2911–2920 RSC.
- W. S. Bian and H. J. Werner, J. Chem. Phys., 2000, 112, 220–229 CrossRef CAS.
- S. J. Greaves, E. Wrede, N. T. Goldberg, J. Zhang, D. J. Miller and R. N. Zare, Nature, 2008, 454, 88 CrossRef CAS PubMed.
- A. Aguado, O. Roncero, C. Tablero, C. Sanz and M. Paniagua, J. Chem. Phys., 2000, 112, 1240–1254 CrossRef CAS.
- L. Velilla, B. Lepetit, A. Aguado, J. A. Beswick and M. Paniagua, J. Chem. Phys., 2008, 129, 084307 CrossRef PubMed.
- D. Gerlich, J. Chem. Soc., Faraday Trans., 1993, 89, 2199–2208 RSC.
- D. Gerlich, Phys. Scr., 1995, T59, 256–263 CrossRef.
- D. Gerlich and S. Horning, Chem. Rev., 1992, 92, 1509–1539 CrossRef CAS.
- D. Gerlich and S. Schlemmer, Planet. Space Sci., 2002, 50, 1287–1297 CrossRef CAS.
- T. Gonzalez-Lezana and P. Honvault, Mon. Not. R. Astron. Soc., 2017, 467, 1294–1299 Search PubMed.
- P. G. Jambrina, F. J. Aoiz, C. J. Eyles, V. J. Herrero and V. Sáez-Rábanos, J. Chem. Phys., 2009, 130, 184303 CrossRef CAS PubMed.
- P. G. Jambrina, J. M. Alvariño, D. Gerlich, M. Hankel, V. J. Herrero, V. Sáez-Rábanos and F. J. Aoiz, Phys. Chem. Chem. Phys., 2012, 14, 3346–3359 RSC.
- P. G. Jambrina, F. J. Aoiz, N. Bulut, S. C. Smith, G. G. Balint-Kurti and M. Hankel, Phys. Chem. Chem. Phys., 2010, 12, 1102–1115 RSC.
- P. G. Jambrina, J. M. Alvariño, F. J. Aoiz, V. J. Herrero and V. Sáez-Rábanos, Phys. Chem. Chem. Phys., 2010, 12, 12591–12603 RSC.
- Z. Zanchet, O. Roncero, T. Gonzalez-Lezana, A. Rodriguez-Lopez, A. Aguado, C. Sanz-Sanz and S. Gomez-Carrasco, J. Phys. Chem. A, 2009, 113, 14488 CrossRef PubMed.
- R. F. Lu, T. S. Chu and K. L. Han, J. Phys. Chem. A, 2005, 109, 6683 CrossRef CAS PubMed.
- R. A. White and J. C. Light, J. Chem. Phys., 1971, 55, 379 CrossRef CAS.
- P. G. Jambrina, J. Aldegunde, M. P. de Miranda, V. Sáez-Rábanos and F. J. Aoiz, Phys. Chem. Chem. Phys., 2012, 14, 9977–9987 RSC.
- M. Hankel and J. N. L. Connor, AIP Adv., 2015, 5, 077160 CrossRef.
- F. J. Aoiz, L. Bañares and V. J. Herrero, Int. Rev. Phys. Chem., 2005, 24, 119–190 CrossRef CAS.
- D. X. Dai, C. C. Wang, S. A. Harich, X. Y. Wang, X. M. Yang, S. D. Chao and R. T. Skodje, Science, 2003, 300, 1730–1734 CrossRef CAS PubMed.
- J. Jankunas, M. Sneha, R. N. Zare, F. Bouakline, S. C. Althorpe, D. Herraez-Aguilar and F. J. Aoiz, Proc. Natl. Acad. Sci. U. S. A., 2014, 111, 15–20 CrossRef CAS PubMed.
- A. I. Boothroyd, W. J. Keogh, P. G. Martin and M. R. Peterson, J. Chem. Phys., 1996, 104, 7139–7152 CrossRef CAS.
- X. Shan and J. N. L. Connor, J. Chem. Phys., 2012, 136, 044315 CrossRef PubMed.

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