Reaction Dynamics and Mechanism of the Cl + HD(v = 1) Reaction: A

Article pubs.acs.org/JPCA

Reaction Dynamics and Mechanism of the Cl + HD(v = 1) Reaction: A Quantum Mechanical Study L. González-Sánchez,† J. Aldegunde,*,† P. G. Jambrina,‡ and F. J. Aoiz‡ †

Departamento de Química Física, Facultad de Química, Universidad de Salamanca, 37008 Salamanca, Spain Departamento de Química Física I, Facultad de Química, Universidad Complutense de Madrid, 28040 Madrid, Spain

‡

ABSTRACT: Time-independent quantum mechanical calculations have been performed in order to characterize the dynamics and stereodynamics of Cl + HD reactive collisions. Calculations have been carried out at two different total energy values and for various initial states using the adiabatic potential energy surface by Bian and Werner [J. Chem. Phys. 2000, 112, 220]. Special attention has been paid to the reaction with HD(v = 1) for which integral and differential cross-sections have been calculated and the effect of vibrational vs translational energy on the reactivity has been examined. In addition, the reactant polarization parameters and polarization-dependent differential cross-sections have been determined. From these results, the spatial preferences of the reaction and the extent of the control of the cross sections achievable through a suitable preparation of the reactants have been also studied. The directional requirements are tighter for the HCl channel than for the DCl one. Formation of the products takes place preferentially when the rotational angular momentum of the HD molecule is perpendicular to the reactants approach direction. Cross-sections and polarization moments computed from the scattering calculations have been compared with experimental results by Kandel et al. [J. Chem. Phys. 2000, 112, 670] for the reaction with HD(v = 1) produced by stimulated Raman pumping. The agreement so obtained is good, and it improves the accordance found in previous calculations with other methodologies and potential energy surfaces.

1. INTRODUCTION Few reactions have received the attention of the scientific community for so many years and have been as thoroughly studied as that of Cl + H2 and its isotopical variants (see ref 1 and references therein). An updated summary of the main advances on the understanding of the dynamics of this system can be found in refs.2,3 Although one could consider that our knowledge of these collisions is almost complete, it is worth mentioning that most of the theoretical and experimental efforts devoted to its study concentrated on the reactive channels involving reactants in their ground vibrational state,4−8 while the reactive processes where the reagents are vibrationally excited9,10 have drawn much less attention. In particular, Kandel et al.10 studied the reaction of Cl with HD in v = 1 and j = 1, 2 by means of excitation with stimulated Raman pumping and resonance−enhanced multiphoton ionization (REMPI) detection of the H and D fragments resulting from the DCl and HCl formation, respectively, and HCl(v′ = 1) molecules using REMPI via the E ← X transition. Through the analysis of the ion time-of-flight profiles, they were able to determine the rotational distribution of the HCl product, an experimental estimate of the product branching ratio, as well as the differential cross-sections (DCSs) for the formation of HCl(v′ = 0) and HCl(v′ = 1) from the Cl + HD(v = 1, j = 1) reaction. Moreover, using the S(0) transition they were able to produce HD(v = 1, j = 2, m = 0) aligned in the laboratory frame. From the analysis of the ion time-of-flight profiles under © 2013 American Chemical Society

different geometries, the authors were able to extract the intrinsic polarization moments for the HCl(v′ = 1) formation.10 The experimental results were compared with quantum mechanical (QM) and quasiclassical (QCT) calculations on one of the potential energy surfaces (PESs) available at that time, but the agreement was rather poor. In this work, we aim to obtain a more detailed and rigorous knowledge of this reaction through a purely QM analysis of the dynamics and mechanism of the Cl + HD reaction using the adiabatic BW2 potential energy surface,11 with special attention to those processes where the reactants are vibrationaly excited. Two values of the total energy, 0.76 and 1.00 eV, will be considered. This is, to the best of our knowledge, the first time that this PES is used to perform exact QM calculations for Cl + HD(v≠0) collisions. Previous calculations10 were carried out on the G3 PES,12 a PES that is not as accurate as the BW2 one and that could not account for the experimental results with the different molecular hydrogen isotopologues in their ground vibrational states.4,7 In this work, we will compare the outcome of our QM calculations with the results of the mentioned experiment of Kandel et al. for the Cl + HD(v = 1, j = 1−2) → HCl(v′ = 0, 1) + D reaction.10 Special Issue: Joel M. Bowman Festschrift Received: December 26, 2012 Revised: March 10, 2013 Published: March 11, 2013 7030

dx.doi.org/10.1021/jp312758r | J. Phys. Chem. A 2013, 117, 7030−7041

The Journal of Physical Chemistry A

Article

Our methodology13,14 to unravel the mechanism, briefly summarized below, is conceptually simple: we use the scattering matrices obtained from the dynamical calculations to determine the preferred polarization adopted by the reagents when the products are formed. This offers not only a clear description of the directional preferences of the reaction but also makes possible to appraise the extent to which the reaction observables can be controlled through a suitable preparation of the reactants. The article is organized as follows. Section 2 presents a concise summary of the theoretical concepts and ideas used throughout the manuscript. Section 3, in turn, describes briefly the time-independent QM calculations that serve as a starting point for the computation of the polarization moments and reaction observables in which our analysis of the reaction is based. Such analysis, together with the discussion of the results and a comparison with previous experimental data, is unfolded in sections 4 and 5. Finally, section 6 summarizes the main conclusions of the work.

The integral and differential cross-section (ICS and DCS, respectively) for a collision A + BC(v,j) → products can be written in a general case as14 2j

σ = σiso ∑ (2k + 1)s0(k)a0(k)

(1)

k=0

and σ dσ = iso dω 2π

2j

k

∑∑

(2k + 1)[Sq(k)(θ )]aq(k) (2)

k = 0 q =−k

where σiso is the integral cross-section found when the reactants (k) are unpolarized, θ is the scattering angle, s(k) 0 and Sq (θ) are the intrinsic polarization moments that quantify the directional preferences determined only by the collision dynamics, a(k) 0 and a(k) are the extrinsic polarization moments that quantify the q experimental (dynamically independent) polarization of the HD molecule before the collisions take place; that is, the directional preparation of the molecule. (k) The two sets, s(k) q and Sq (θ), of intrinsic moments are called polarization parameters (PPs) and polarization dependent differential cross-sections (PDDCSs), respectively. They are related through the expression14

2. THEORETICAL BACKGROUND When two molecules collide, their relative orientation can completely change the outcome of the process. Therefore, the analysis of the dependence of this outcome on the relative orientation of the collision partners can lead to a large amount of useful and distinctive information about the collision. This section aims to address the issue of how the directional effects in chemical reactions can be quantified, which requires to discuss the mathematical dependence of the reaction observables on the relative arrangement adopted by the reactants. Our presentation will be particularized to the case under study; i.e., to collisions between an atom and a diatomic molecule where both partners are described as having closed electronic shells. An angular momentum is said to be polarized when, instead of pointing along any direction in the space with equal probability, it displays a preference for being directed along certain directions. This concept is crucial because the analysis of the polarization of the angular momenta that characterize the atoms and molecules involved in a collision makes possible to determine the directional preferences of the process. The extent of such polarizations and of the corresponding directional effects can be quantified through the so-called polarization moments,15−17 which are spherical tensors characterized by a rank and a component and whose values depend on (i) the extent and sign of the type of polarization they measure and (ii) on the frame of reference that has been chosen to perform the analysis. The mechanism and stereodynamics of a chemical reaction can be studied either through the analysis of the moments that measure the polarization of the reactants and/or products or, equivalently, through the consideration of the changes in the values of the differential and integral cross-sections caused by a certain preparation of the reactants. A detailed explanation of how both methodologies can be applied to a A + BC(v,j) → products collision has been described in detail elsewhere.13,14,18,19 Our presentation will be limited to a brief summary of the method, with special attention to those concepts and ideas that will be used in the next section to characterize the dynamics and mechanism of the Cl + HD reactive collisions.

sq(k) =

1

∫−1 Sq(k)(θ)d(cos θ)

(3)

and account for the HCl polarization that is most likely to lead to product formation at a well-defined value of the scattering angle (PDDCSs) or averaged over the same angle (PPs). The PDDCSs and PPs are in general imaginary quantities that transform under rotation as spherical tensors. However, the interpretation of the directional information they contain is simpler if the moments are transformed into real form13,14 in exactly the same way as the complex spherical harmonics are transformed into their real counterparts: Sq{+k} =

1 [(− 1)q Sq(k) + S−(kq)] 2

1≤q≤k

Sq{−k} =

1 [(− 1)q Sq(k) − S−(kq)] i 2

1≤q≤k

S0{k} = S0(k)

(4)

s(k) q .

and similar expressions for the PPs, While the extrinsic moments can in principle be selected at will, the intrinsic ones cannot be arbitrarily chosen. From the theoretical point of view, they can be directly calculated from the scattering matrix14 as they reflect the directional preferences of the reaction. Please notice that the integration over all the spatial directions inherent to the measurement or calculation of the ICS makes this observable to depend exclusively on the interplay between the q = 0 polarization parameters and extrinsic moments (see eq 1). The control of the cross-sections through the experimental preparation of the reactants (eqs 1 and 2) reflects the idea that, in order to maximize the probability of product formation, the extrinsic polarization of the reactants should reproduce as closely as possible the features of the intrinsic polarization. In other words, as the reactant molecule approaches its collision partner, the molecular polarization should be as similar as possible to the one that is preferred by the collision dynamics. The contrary applies if one wishes to hinder the reaction in 7031



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The values of the angles β and α that specify the direction of the electric field (or the polarization vector of the Raman pumping laser), and therefore the position of the Z axis, on the scattering frame can be externally controlled in experiments that involve angle-resolved product detection.14 As the Z direction specifies, within quantum restrictions, the position of the j and r vectors makes it possible to control the polarization of j and r with respect to the scattering frame. When the experiment does not resolve scattering angles, the axis preparations depend exclusively on β because the position of the scattering plane and, in consequence, the value of the angle α become undefined. Figure 2 exemplifies the kind of extrinsic alignment of the HD internuclear axis that can be achieved through the

which case the reactants should be polarized as differently as possible from the prescription given by the intrinsic moments. In those cases in which the reactants are unpolarized prior to the collision, the rotational angular momentum and therefore the internuclear axis of the HD molecule would be isotropically distributed. Such extrinsic polarization can be represented through the a(k) q = δk0δq0 moments, in such a way that the DCS (0) would be given by (σiso/2π)S(0) 0 (θ). The S0 (θ) polarizationdependent differential cross-section is therefore the normalized DCS, and the corresponding polarization parameter s(0) 0 takes the value 1. The values and the directional meaning of the other polarization moments depend on the reference frame chosen to perform the dynamical and mechanistic analysis of the collisions. Unless otherwise stated, the polarization moments employed in this work will be referred to the scattering frame of Figure 1. In this frame of reference, the origin coincides with

Figure 1. The xyz scattering frame, defined by the reactants approach and the products recoil directions (k and k′, respectively). The origin is at the center of mass of the collision system, and the z axis is parallel to k. The direction of the Z axis of the laboratory frame is highlighted in blue, together with the two angles (β and α) that describe its position.

the center of mass of the Cl + HD system, the z axis is parallel to the reactants relative velocity (k), and the y axis is parallel to the vectorial product k × k′, where k′ is the products recoil direction. The xz plane is thus the scattering plane containing k and k′. Once the intrinsic moments are known, they can be used to unravel the reaction mechanism in two complementary ways. In the intrinsic viewpoint, the directional information contained in the PDDCSs and PPs is directly analyzed in order to determine how the reaction takes place. Alternatively, one could consider the effect of certain extrinsic preparations on the reaction observables (eqs 1 and 2) as a tool to study the mechanism. This approach, called extrinsic viewpoint, turns out to be simpler and more meaningful if the extrinsic reactant polarizations are easy to visualize and experimentally achievable. A way14 to satisfy both requirements would be to use optical Raman pumping to prepare the reactant molecule in the |j = 2,m = 0⟩ state, where the projection of the rotational angular momentum is referred to a laboratory axis Z (see Figure 1) whose position in the scattering frame is determined by the polar and azimuthal angles β and α, respectively. As m = 0, the HD rotational angular momentum (j) is preferentially perpendicular to the Z axis, and the HD internuclear axis r is preferentially parallel or antiparallel to the same axis. In other words, j has a negative alignment with respect to Z, while r displays a positive alignment.

Figure 2. Extrinsic polarizations (external preparations) of the internuclear axis of the HD(v = 0, j = 2) molecule referred to the xyz scattering frame of Figure 1. It is assumed that the HD molecule is prepared in a |j = 2,m = 0⟩ state with respect to the XYZ laboratory frame, whose Z axis is defined by α and β in the scattering frame. The distribution on the top-left panel (ISO) occurs when the molecule is unpolarized. The other distributions occur when the molecule is polarized as described in section 2, and the α and β values are as indicated on the figure.

experimental setup described in the former paragraph. Several examples, corresponding to well-defined values of β and α, are presented. In the β = 0 case, the HD internuclear axis is aligned parallel to the reactant-approach direction k. As β increases to β = 54.74° (magic angle) and then to β = 90°, the internuclear axis is tilted until it becomes perpendicular to the reactantapproach direction. This tilting may occur with the alignment direction in the scattering plane (α = 0° or α = 180°) or outside the scattering plane (other α values). Finally, two examples of the kind of axis preparation that characterizes those experi7032



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maximum value of the total angular momentum J was equal to 70 for the highest Etot. Please notice that, although for the largest J values considered in the calculations, the number of possible |Ω|s rises well above 6, convergence is already achieved with a value |Ωmax| = 6.

ments where the scattering angle is not resolved are given in the bottom line of Figure 2; the corresponding r distributions display axial symmetry with respect to the k direction due to the integration over the azimuthal α angle, as it was mentioned above. When the HD reactant is extrinsically polarized according to this experimental scheme, the values of the ICS and DCS for a Cl + HD(v,j = 2) → products reaction are given by14

4. COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS Reference 10 where Kandel et al. studied the Cl + HD(v = 1, j = 1,2) reactions, is the only work devoted to the Cl + HD system that combines theoretical and experimental results while, simultaneously, analyzing the stereodynamics of the process. Before describing the mechanism of these collisions in next section, we compare the outcome of our QM calculations with the experimental results presented in ref 10 for the ICS and DCS. As already mentioned, the main difference between the theoretical simulations of the present work and that of ref 10 lies in the PES used in the calculations. The G3 PES12 was used in ref 10, while the BW2 PES11 has been used in the present calculations. The differences between the two PESs and their dynamical effects have been previously discussed in detail,5−7 and it has been shown that the performance offered by the BW2 PES when it comes to reproducing the experimental features of the Cl + H2 reaction and their isotopical variant is much better than that obtained by the G3 PES.4−7,22−24 Kandel et al.10 measured total and vibrationally resolved cross-sections, both integral and differential, as well as rotational distributions for the Cl + HD(v = 1, j = 1) reaction at a mean total energy of 0.76 eV (Ecol = 0.065 eV). Most of the experimental results pertained to the HCl channel, as the detection of the H atoms was problematic under the experimental conditions. The effect of the rotational alignment on the Cl + HD(v = 1, j = 2) reactive collisions was also investigated, following an experimental scheme similar to that presented in section 2. As a summary of our theoretical calculations, Table 1 shows the integral cross-sections, total and vibrationally resolved, for the energies and initial states that will be considered in this

2j = 4

σ β = σiso

∑ (2k + 1)s0(k)Ck0(β , 0)A 0(k)

(5)

k=0

and σ dσαβ = iso dω 2π

2j = 4

k

∑ ∑

(2k + 1)[Sq(k)(θ )]Ckq(β , α)A 0(k)

k = 0 q =−k

(6) 16

A(k) 0

where Ckq(β,α) is a modified spherical harmonic and is the extrinsic j polarization moment with respect to the XYZ laboratory frame. The latter vanish when k is odd and their values for a |j = 2,m = 0⟩ state are A 0(0) = 1,

A 0(2) = −

2 , 7

A 0(4) =

2 7

(7)

Such preparation can be achieved by stimulated Raman pumping to prepare HD(v = 0, j = 2, m = 0) molecules via the S(0) transition with parallel laser polarization.10 This corresponds to negative alignment of the rotational angular momentum vector (j preferentially perpendicular to E) and to positive alignment of the HD internuclear axis (r preferentially parallel/antiparallel to E). As commented on above, the use of different experimental geometries leads to internuclear axis distributions as those portrayed in Figure 2 and hence enabling, at least in principle, the determination of experimental values for the intrinsic HD alignment moments. A similar experimental approach but using IR pumping have been recently realized for the Cl + CHD3(ν1 = 1) reaction20 for which all the second rank alignment PDDCS were experimentally extracted.

3. TIME-INDEPENDENT QUANTUM MECHANICAL CALCULATIONS The values of the cross-sections and intrinsic moments on which we have based our analysis of the Cl + HD reactive collisions were worked out14 from the scattering matrices in the helicity representation rendered by time-independent QM calculations. Such calculations were performed using the BW2 potential energy surface (PES)11 and the hyperspherical method of Skouteris et al.21 Two total energies, 0.76 and 1.00 eV, were chosen for the study. While the first one roughly coincides with the value of the energy in the experimental study of Kandel et al.,10 for HD(v = 1, j = 2), Etot = 1.00 eV was selected in order to extend the analysis of the reaction to larger energies. The convergence for reactive scattering, less demanding than for inelastic collisions, was assured by a suitable selection of parameters: the cutoff energy for truncation of the basis was 2.5 eV at Etot = 0.76 and 3.0 eV at Etot = 1.00 eV, the rovibrational basis set included all the states below that energy and characterized by an absolute value of the helicity quantum number (|Ω|) up to 6, the maximum hyperradius and steps of propagation were ρmax = 25a0 and 625 steps, respectively. The

Table 1. Quantum Mechanical Integral Cross-Sections (in Å2) for the Cl + HD(v = 0, j = 2) and Cl + HD(v = 1, j = 1− 2) Reactive Collisions at Etot = 0.76 and 1.0 eV and Indicated Collision Energies, Ecoll; Energies in eV Etot = 0.76 eV channel

Ecol

H2 initial state

*HCl

0.49 0.06 0.04 0.49 0.06 0.04

(v (v (v (v (v (v

channel

Ecol

H2 initial state

*HCl

0.73 0.30 0.28 0.73 0.30 0.28

(v (v (v (v (v (v

*DCl

*DCl

7033

= = = = = =

= = = = = =

0, 1, 1, 0, 1, 1,

0, 1, 1, 0, 1, 1,

j j j j j j

j j j j j j

= 2) = 1) = 2) = 2) = 1) = 2) Etot = = = = = = =

2) 1) 2) 2) 1) 2)

σtot

σv′=0

σv′=1

σv′=2

0.511 0.521 0.687 1.908 0.125 0.039 1.00 eV

0.499 0.144 0.171 1.778 0.053 0.017

0.012 0.377 0.516 0.128 0.064 0.020

0.002 0.008 0.002

σtot

σv′=0

σv′=1

σv′=2

0.771 1.942 2.856 2.741 3.774 3.850

0.732 0.915 1.211 2.331 1.058 1.084

0.038 1.026 1.642 0.373 1.519 1.459

0.001 0.001 0.003 0.037 1.197 1.307



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4 presents the theoretical and experimental DCSs for the Cl + HD(v = 1, j = 1) → HCl(v′ = 0, 1) + D reactions. The agreement between both sets of results is good, almost quantitative, and improves with respect to that obtained with the G3 PES.10 Broadly speaking, the discrepancies observed between both sets of results can be due to a variety of effects such as the no consideration in the theoretical calculations of the spin−orbit interaction or the possible contribution of nonadiabatic effects.8 It also has to be borne in mind the limited resolution of the measurements, which used the photoloc technique and the coexpansion of Cl2 and H2.10 The general features of the angular distributions, predominantly backward with a significant contribution from sideways scattering and a very small one from the forward directions, are well reproduced. The dip displayed by the experimental DCS in the backward region is not accounted for by the theoretical calculations. As it was explicitly acknowledged by Kandel et al.,10 the presence of this drop of reactivity in the backward region, absent also in the calculations performed on the G3 PES10 and in other experimental studies carried out at different energies for the same system,6,25 may be a consequence of the assumptions involved in the conversion of experimental product speeds into scattering angles. As it was discussed in section 2, the polarization parameters (PPs) are the intrinsic moments that summarize the directional information contained in the PDDCSs by integrating them over the scattering angle (see eq 3). Kandel et al. extracted10 the {2} {2} values of the s{2} 0 , s1+ , and s2+ PPs for the Cl + HD(v = 1, j = 2) → HCl(v′ = 1) + D reaction. The experimental values of the {2} {2} PPs,10 s(2) 0 = −0.04 ± 0.27, s1+ = 0.37 ± 0.14, and s2+ = −0.07 ± 0.15, correspond to a weak alignment of j perpendicular to the z axis of the scattering frame (s{2} 0 ≲ 0), a strong alignment > 0) instead of along the x − parallel to the x + z direction (s{2} 1+ z one and a weak alignment parallel to the y axis as opposite to the x one (s{2} 2+ ≲ 0). At Etot = 0.76 eV (Ecol = 0.04 eV), the values of the moments obtained from our QM calculations (s{2} 0 {2} = −0.20, s{2} 1+ = 0.58, and s2+ = −0.12) are within error bars and reproduce the overall magnitude of the alignment and its sign. In fact, the agreement between both sets of values represents an improvement with respect to that presented in ref 10 that were calculated using a QCT approach.

work. The experimental results for the intramolecular kinetic isotope effect, Γ(DCl/HCl), and vibrational branching ratio, Γ[HCl(v′ = 1)/HCl(v′ = 0)], corresponding to the Cl + HD(v = 1, j = 1) reaction at a mean collision energy of 0.065 eV (Etot = 0.76 eV) were 0.9 ± 0.2 and 3.5, respectively. The theoretical values obtained from the present QM calculations are Γ(DCl/ HCl) = 0.24 and Γ[HCl(v′ = 1)/HCl(v′ = 0)] = 2.62 (see Table 1). Although the agreement is slightly better than that obtained through QM calculations or QCT based on the G3 PES,10 it is still poor, in particular when it comes to the kinetic isotope effect. It should be kept in mind that the experimental results correspond to collisions that take place at a mean collision energy of 0.065 eV, while their theoretical counterparts were obtained from QM calculations performed at exactly this energy. Although averaging the theoretical results over the experimental energy distribution would possibly lead to a better accordance between the branching ratios,10 this would probably not suffice to obtain a quantitative agreement between both sets of values. The comparison between theoretical and experimental rotational distributions for the Cl + HD(v = 1, j = 1) → HCl(v′ = 1) + D collisions is shown in Figure 3. In turn, Figure

Figure 3. Experimental (black open squares) and theoretical (red filled circles) state-to-state integral cross-sections for the Cl + HD(v = 1, j = 1) → HCl(v′ = 1, j′) + D reaction. Experimental results10 were measured at 0.06 eV mean collision energy (0.76 eV mean total energy). The experimental data have been scaled to their theoretical counterparts. The quantum mechanical scattering calculations were performed at a single energy corresponding to the mean value.

Figure 4. Experimental (black circles and line) and theoretical (red lines) solid angle differential cross-sections for the Cl + HD(v = 1, j = 1) reactive collisions leading to formation of HCl(v′ = 0) (left panel) and HCl(v′ = 1) (right panel). The experimental angular distributions,10 which were measured at 0.06 eV mean collision energy (0.76 eV mean total energy), have been scaled to the theoretical DCSs ignoring the backward scattered point. The theoretical results correspond to calculations performed at a single energy. 7034



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5. DYNAMICS AND MECHANISM OF THE Cl + HD REACTION Figure 5 shows the total and vibrationally resolved reaction probability, P(J), as a function of the total angular momentum

third of the total cross-section (see also Table 1). The inspection of the reaction probabilities for those collisions where DCl is formed (right panels) leads to similar conclusions, although these processes exhibit a lower degree of vibrational adiabaticity than their HCl counterparts, especially for HD molecules in v = 1 for which the total reactivity into v′ = 0 is very close to that into v′ = 1. While the probability of forming DCl is larger than for HCl when the reactants are in the ground vibrational state, the opposite holds when the reactants get vibrationally excited. This change can be rationalized in terms of two opposite effects.10 The fact that the center of mass is closer to the deuterium atom determines that the cone of acceptance for the attack of the Cl atom on this side of the molecule is wider than for collisions where the incoming atom approaches the H side of the reactant molecule. This, in principle, favors the formation of DCl. However, for collisions at a given value of J, the centrifugal barrier will be lower when the attack occurs on the H side of the molecule, as the distances to the center of mass involved in the process are larger than for those collisions where the Cl atom approaches the deuterium atom. The formation of HCl is therefore favored by this effect. When plenty of translational energy is available, as it happens for vibrationless reactants, the width of the cone of acceptance is the main factor when it comes to determining the reactivity. The reaction probability is therefore larger for the DCl channel. However, at the same total energy, the vibrational excitation of the reactants reduces drastically the energy that can be employed to surmount the barrier, which is essentially translational in its origin, in such a way that the cones of acceptance get narrower and concentrate around those atomic arrangements closer to the transition state geometry. Under these conditions, the reactivity is conditioned by the centrifugal barrier, and that channel where its height is smaller (HCl) concentrates the largest probability values. When the total energy increases up to 1.0 eV (see Figure 6), the reaction probability displays significant changes, which go beyond the expected increase in the number of J values that contribute to the reaction. In particular, the collisions become less vibrationally adiabatic, and the probability of forming DCl is now consistently larger than for HCl. The first change manifests itself more intensely for the Cl + HD(v = 1) collisions, where the v′ = 1 products only predominate for the largest values of J. The second change can be understood in terms of the discussion unfolded in the previous paragraph if one considers that, at Etot = 1.00 eV, a sufficient amount of translational energy is available to overcome the barrier even for vibrationally excited initial states of the reactants. In consequence, the largest cone of acceptance makes the DCl channel more reactive than its HCl counterpart. It should be also remarked that, at sufficiently high Ecol, vibrational excitation at the same total energy is more effective promoting the reactivity in both channels than the translational energy, as shown in Table 1. Rotational energy plays also an important role enhancing the reactivity, most especially in the HCl product channel, as it was previously shown for reactants in v = 0.7 As can be seen in Table 1, a combination of vibrational and rotational excitation at Etot = 1.00 eV can enhance the reactivity into HCl most considerably, whereas the effect is much smaller for the DCl channel. We will focus next on the analysis of the DCS and the PDDCS(2,0) moment for the collisions under study (see Figures 7 and 8). At Etot = 0.76 eV, the angular distributions (top panels in Figure 7) for the DCl channel concentrate in the

Figure 5. Reaction probability as a function of J for the Cl + HD reaction at Etot = 0.76 eV and indicated collision energies. Total and vibrationally resolved results for the HCl (left panels) and DCl (right panels) channels and HD(v = 0, j = 2) (top panels), HD(v = 1, j = 1) (middle panels), and HD(v = 1, j = 2) (bottom panels) initial states are presented. The probability values have been multiplied by (2J + 1) such that the integral of each curve is proportional to the respective integral cross-section. Please, notice the different y axis scales of each panel.

quantum number, J, for both channels of the Cl + HD reaction at Etot = 0.76 eV. Notice that P(J) has been multiplied by 2J + 1 such that the areas under the curves are proportional to the respective reaction cross-sections. Different initial states, HD(v = 0, j = 2) and HD(v = 1, j = 1, 2), are considered, and the different vibrational contributions (broken lines) are also included. All the probability distributions are dominated by a single maximum and barely display any kind of structure. The range of J values necessary to converge the quantum mechanical calculations is larger for the HCl channel, as it corresponds to the fact that the center of mass of the reactant molecule is closer to the deuterium atom. At the same total energy, an increase in the vibrational level of the reactants, and hence a decrease Ecol, reduces the number of J values that contribute to reactivity. Collisions leading to formation of HCl and involving reactants in v = 0 (top-left panel) are almost vibrationally adiabatic. However, when the HD molecule is in the v = 1 state, middle and bottom left panels, there is a significant production of HCl(v′ = 0) products, almost one7035



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sideways regions. The relative magnitude of the DCSs reflects the values of the reaction probabilities discussed in connection with Figure 5. An increase of the total energy (Etot = 1.0 eV, top panels in Figure 8) causes significant changes in the shape of the DCSs. For the DCl channel, the shape of the angular distributions is almost independent of the initial state, and the products spread over a larger range of values of the scattering angle than for the lowest energy. The profile of these curves is characterized by a maximum located in the backward region followed by a monotonous decrease until the forward region, where a small peak becomes apparent. The DCSs for the HCl channel are nearly constant functions of the scattering angle except in the forward region, where the distributions corresponding to the Cl + HD(v = 1, j = 1, 2) collisions display a conspicuous maximum. The PDDCS(2,0) moment (bottom panels in Figures 7 and 8) tells us about the preferred alignment of the reactants angular momentum j when the products are formed. Positive/ negative values of this moment correlate with an alignment of j parallel/perpendicular to the reactants approach direction k and with a preference for side-on/head-on collisions. The moment is renormalized;14 that is, the S(2) 0 (θ) function is divided by the angular distribution S(0) 0 (θ), so as to isolate the directional information and to make its values for different scattering angles directly comparable. The y axis scale in the bottom panels of Figures 7 and 8 reflects the theoretical limits of the PDDCS(2,0) moment, in such a way that, as the value of the moment gets closer to the upper or lower limit, the polarization of the angular momentum becomes stronger. The analysis of the alignment experienced by the HD molecule during the course of the reaction leads to a picture of the mechanism that is strongly dependent on the conditions under which the process takes place and on the channel. Except for those collisions leading to formation of DCl and involving HD(v = 1) molecules (blue dashed lines in the bottom-middle and bottom-right panels of Figures 7 and 8), the profiles of the

Figure 6. Same as Figure 5 but for Etot = 1.00 eV.

backward and sideways regions and display a maxima at 180°. The DCS for collisions leading to formation of HCl show a similar profile, except in the Cl + HD(v = 0, j = 2) case, where the scattering distributes evenly over the backward and

Figure 7. Differential cross-sections (top panels) and renormalized PDDCS(2,0) moments (bottom panels) for both reactive channels of the Cl + HD collisions at Etot = 0.76 eV and indicated collision energies. Results for HD(v = 0, j = 2) (left panels), HD(v = 1, j = 1) (middle panels), and HD(v = 1, j = 2) (right panels) initial states are presented. 7036



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Figure 8. Same as Figure 7 but for Etot = 1.00 eV.

Figure 9. Differential cross-sections for the Cl + HD(v = 1, j = 2) → H + DCl (left panels) and Cl + HD(v = 1, j = 2) → D + HCl (right panels) collisions at Etot = 0.76 eV (Ecol = 0.04 eV) and 1.00 eV (Ecol = 0.28 eV) (top and bottom panels, respectively). Results corresponding to unpolarized (ISO) and various preparations of polarized reactants (see Figure 2) are depicted.

PDDCS(2,0) moments are very similar for Etot = 0.76 and 1.0 eV, which indicates that the directional preferences of the corresponding collisions barely depend on the energy. The aforementioned exceptions to this behavior can be explained through the combination of the larger centrifugal barrier that characterizes the DCl channel with the lack of translational energy available to surpass it when the reactants are vibrationally excited and Etot = 0.76 eV. These two effects impose very tight spatial requirements on the system that render the values of the PDDCS consistently close to their lower theoretical limit, which indicates a strong preference for head-on collisions; not a surprising result given the linear

geometry that characterizes the transition state for this reaction. When the energy available to surmount the barrier is large enough (collisions at Etot = 1.00 eV or involving reactants in v = 0 at any of the two total energies), the stereodynamical constraints are tighter for the HCl channel than for its DCl counterpart, in good accordance with the fact that the cone of acceptance is wider for the DCl collisions. Accordingly, the corresponding PDDCS(2,0) for the DCl channel oscillates around zero without getting close to the theoretical limits for any value of the scattering angle. On the contrary, scattering in the backward region clearly correlates with head-on collisions (negative values of the moment) for those processes leading to 7037



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Figure 10. State-to-state integral cross-sections for both channels of the Cl + HD(v = 1, j = 2) reactive collisions at Etot = 0.76 eV (Ecol = 0.04 eV). Results are presented for unpolarized (ISO) and polarized reactants (β = 0°, 90°, and 54.74°).

direction (β = 54.74° and α = 180°), the boost experienced by the backward scattering vanishes and the DCS diminishes rapidly. In particular, for the preparation implying β = 90° and α = 0° and the DCl channel, the magnitude of the DCS lies well below that corresponding to an isotropic preparation. The effect is similar in the HCl channel at Etot = 0.76 eV (top, right panel) except for a slight tendency of the DCS to shift into smaller values of θ when the preparation axis becomes perpendicular to the k direction (β = 90°, α = 0°). At higher total (collision) energy (bottom panels), the sensitivity to the preparation of the reactants of the DCS depends largely on the nature of the products: scant for the DCl channel and conspicuous changes for the HCl one. Backward scattering is also augmented with β = 0° preparation but to a much smaller extent than at the low energy. Sideways scattering is promoted by β ≠ 0 polarization. At this total energy, forward scattering is present in the DCS of both channels, although for the DCl it only gives rise to a small peak. For the HCl channel, however, forward scattering is the dominant feature, which is boosted (suppressed) by the β = 90°, α = 0° (β = 0°) polarization. For this channel, head-on collisions almost suppress the reactivity in the forward region, while their side-on counterpart causes a noticeable increase in the corresponding DCS. It is clear from the discussion unfolded in this paragraph that, except in the forward region, it is not possible to alter significantly the HCl/DCl ratio at a welldefined value of the scattering angle through the polarization of the reactants; the mechanism of both types of collisions is too similar as to make possible a separate control of the processes leading to HCl and DCl. A separate analysis of the PDDCSs and DCSs for each final vibrational level leads to results that are remarkably similar to those extracted from the analysis of the total (summed over final states) properties discussed in Figure 9 and has therefore not been included in the article. It is worth pointing out that the forward scattering found for the HCl channel at Etot = 1.00 eV is due exclusively to collisions into HCl(v′ = 1), with barely

formation of the HCl molecule. This preference evolves smoothly as the scattering angle decreases and shifts into the forward region, where the preferred collisions are side-on and the PDDCS(2,0) adopts positive values. The importance of the intrinsic moments goes beyond informing us about the directional preferences of the reaction. As discussed in section 2, they also make it possible to quantify the changes experienced by the cross-sections for a certain experimental preparation of the reactants. The rest of the article will be devoted to the analysis of this control for the Cl + HD(v = 1, j = 2) reaction. In general terms, the degree of sensitivity exhibited by the reaction to the polarization of the HD molecule can be assessed through the values of the renormalized PDDCSs; the closer are these moments to the theoretical limits (zero), the larger (smaller) will be the extent of the control achievable through a suitable preparation of the reactants. Figure 9 exemplifies this control by presenting the DCSs for both channels of the Cl + HD(v = 1, j = 2) reaction at Etot = 0.76 and 1.00 eV (Ecol = 0.04 and 0.28 eV, respectively) and for different preparations of the reactants. The initial rotational state of the reactants (j = 2) has been selected so as to follow the experimental scheme described in section 2, although almost identical results were obtained for j = 1. Except for those collisions leading to DCl formation at Etot = 1.00 eV (bottom-left panel), it is possible to change significantly the value of the differential cross-section by polarizing the reactant molecule. In the backward and sideways regions, the response of the DCS to this preparation is nearly independent of the nature of the final products and, to some extent, also of the value of the energy. Head-on collisions (β = 0°) significantly increase the reaction probability around θ = 180°, while causing a slight decrease for sideways scattering. This enhancement is particularly important at Etot = 0.76 eV (top panels), as under these circumstances the PDDCS(2,0) moment adopts very negative values (see Figure 7) in the backward region regardless of the channel, which indicates a clear preference for head-on collisions. As the preparation axis tilts with respect to the k 7038



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Figure 11. Same as Figure 10 but for Etot = 1.00 eV (Ecol = 0.28 eV).

any scattering in that region corresponding to products in v′ = 0. A similar effect was found in classical calculations on the G3 PES.9 Regarding the stereodynamics and control of the processes, the PDDCSs turn out to be almost independent of v′, and accordingly, the differential cross-section shows an identical response to the reactants preparation no matter the final vibrational state. Such response coincides with that described in Figure 9. The integral cross-section is also amenable to control through the polarization of the reactants. Such control for the state-to-state processes associated to both reactive channels of the Cl + HD(v = 1, j = 2) collisions is graphically presented in Figures 10 (Etot = 0.76, Ecol = 0.04) and 11 (Etot = 1.00 eV, Ecol = 0.28 eV). The integration over all the spatial directions that is inherent to the calculation of the integral cross-section renders the position of the scattering plane undefined, in such a way that the experimental preparations of the reactants are just characterized by the β angle (see section 2 for more details). At the lowest of the two energies considered in this work, Figure 10, the isotropic distributions (black, solid squares) display a single maximum and no further structure. Although the different preparations change noticeably the values of the cross-sections, they do so without altering significantly the shape of the distributions. As expected from the values of the PDDCS(2,0) (see Figure 7), the DCl channel (top panels) exhibits a larger sensitivity to the polarization of the reactants than its HCl counterpart. Head-on (β = 0°) collisions lead to an enormous increase of the probability of forming DCl, reaching values that are relatively close to the theoretical limit of 5σiso;26 while side-on (β = 90°) collisions barely change the isotropic distribution, and tilted (β = 54.74°) encounters give rise to a slight decrease in the cross-section values. For the HCl channel, head-on/side-on collisions cause the cross-sections to increase/decrease moderately, less than for the other reactive channel. The β = 54.74° collisions, in turn, lead to distributions almost indistinguishable from the isotropic ones. As the total

energy increases up to 1.00 eV (Figure 11), the picture changes drastically. The same polarization of the reactants has now a different effect on the rotational distributions depending on the final vibrational state. In any case, none of the axis preparations causes a change with respect to the isotropic j′ resolved crosssections comparable to those observed at the lowest energy. The only possible exception to this statement comes from the β = 0° preparation, which, although it only leads to moderate changes in the values of the cross-sections, distorts the shape of some of the HCl angular distributions with respect to the isotropic one and renders them bimodal. A certain degree of control of the DCl/HCl branching ration is possible at Etot = 0.76 eV, where the β = 54.74° preparation causes a considerable decrease of the cross-sections for the DCl channel, while it leaves almost unchanged the rotational distributions for the HCl channel. Equally, the increase caused by the β = 0° collisions is slightly larger for the DCl collisions. Please notice that this finding does not contradict the conclusions drawn from the analysis of the control of the differential cross-section, as the preparations considered in Figure 9 involved that the α angle was well-defined, which is no longer the case when the dependence of the rotational distributions with the polarization of the reactants is considered.

6. CONCLUSIONS We have performed a purely quantum mechanical study of the Cl + HD reactive collisions for different initial states, HD(v = 0, j = 2) and HD(v = 1, j = 1−2), and two values of the total energy; namely, 0.76 and 1.00 eV. Our analysis was based on scattering calculations performed on the BW211 potential energy surface; the scattering matrices so obtained were used to calculate values of the integral and differential cross-sections as well as the polarization moments that characterize the spatial arrangement adopted by the reagents when the reaction takes place. 7039



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The comparison with experimental results10 for the HCl channel at Etot = 0.76 eV lead to an agreement extremely dependent on the property considered: almost quantitative for the differential cross-sections and angular distributions and really poor for the branching ratio between the HCl and DCl channels. The agreement between experimental and present theoretical polarization parameters for the HCl(v′ = 1) formation is acceptable considering the degree of uncertainty of the measurements. Overall, the agreement was slightly better than that obtained with the G3 PES.10 Those collisions involving vibrationless reactants display a considerable degree of vibrational adiabaticity, slightly more marked for the HCl channel at the lowest total energy. This feature vanishes as the energy increases and the reactants gets vibrationally excited. The relative extent of the reactivity into the HCl and DCl channels has been rationalized in terms of the interplay between two factors: the width of the cone of acceptance and the height of the centrifugal barrier. While the first one dominates when plenty of energy is available to overcome the barrier, the second one becomes preeminent when this energy gets scarce. In general terms, the DCl channel, which possesses the widest cone of acceptance, turns out to be more reactive than the HCl one. The only exception to this behavior takes place when the collision energy gets drastically reduced, i.e., for vibrationally excited reactants at Etot = 0.76 eV. The differential cross-sections at Etot = 0.76 eV are dominated by backward and, to a lower extent, sideways scattering, with a minimum contribution from the forward region. This picture changes drastically when the total energy increases to 1.00 eV. The corresponding DCl distributions are still dominated by backward scattering, but they become wider and spread over the whole range of scattering angle values, including a small contribution in the forward region. The differential cross-section for the HCl collisions become nearly isotropic except in the forward region, where they display a conspicuous peak when the reactants are vibrationally excited. Interestingly, this peak comes almost exclusively from v′ = 1 collisions. The spatial preferences of the reaction and the extent of the control of the cross-sections achievable through a suitable preparation of the reactants were also studied. The directional requirements are tighter for the HCl channel than for the DCl one, with the exception of those collisions with low collision energy. In any case, formation of the products takes place preferentially when the rotational angular momentum of the HD molecule is perpendicular to the reactants approach direction. Although the differential cross-sections and the angular distributions display a noticeable sensitivity to the polarization of the reactants, in particular at the lowest of the two energies considered in this work, the two channels respond in a very similar way to such polarization, and no differential control is possible, except for forward scattering at Etot = 1.00 eV.

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ACKNOWLEDGMENTS We acknowledge funding by the Spanish Ministry of Science and Innovation (grants CTQ2008-02578, CTQ2012-37404, and Consolider Ingenio 2010 CSD2009-00038). L.G.S. and J.A. acknowledge funding by the Junta de Castilla y León (grant SA244B12-1). P.G.J. acknowledges the FPU fellowship AP2006-03740.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (J.A.); [email protected] (F.J.A.). Notes

The authors declare no competing financial interest. 7040



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Reaction Dynamics and Mechanism of the Cl + HD(v = 1) Reaction: A

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