Bimolecular Reactions of Vibrationally Excited Molecules. Roaming

Article pubs.acs.org/JPCA

Bimolecular Reactions of Vibrationally Excited Molecules. Roaming Atom Mechanism at Low Kinetic Energies Á kos Bencsura† and György Lendvay*,‡ †

Institute of Organic Chemistry, Research Centre for Natural Sciences, Hungarian Academy of Sciences, H-1525, Budapest, P.O. Box 17, Hungary ‡ Institute of Materials and Environmental Chemistry, Research Centre for Natural Sciences, Hungarian Academy of Sciences, H-1525, Budapest, P.O. Box 17, Hungary S Supporting Information *

ABSTRACT: Quasiclassical trajectory calculations have been performed for the H + H′X(v) → X + HH′ abstraction and H + H′X(v) → XH + H′ (X = Cl, F) exchange reactions of the vibrationally excited diatomic reactant at a wide collision energy range extending to ultracold temperatures. Vibrational excitation of the reactant increases the abstraction cross sections significantly. If the vibrational excitation is larger than the height of the potential barrier for reaction, the reactive cross sections diverge at very low collision energies, similarly to capture reactions. The divergence is quenched by rotational excitation but returns if the reactant rotates fast. The thermal rate coefficients for vibrationally excited reactants are very large, approach or exceed the gas kinetic limit because of the capture-type divergence at low collision energies. The Arrhenius activation energies assume small negative values at and below room temperature, if the vibrational quantum number is larger than 1 for HCl and larger than 3 for HF. The exchange reaction also exhibits capture-type divergence, but the rate coefficients are larger. Comparisons are presented between classical and quantum mechanical results at low collision energies. At low collision energies the importance of the exchange reaction is enhanced by a roaming atom mechanism, namely, collisions leading to H atom exchange but bypassing the exchange barrier. Such collisions probably have a large role under ultracold conditions. The calculations indicate that for roaming to occur, longrange attractive interaction and small relative kinetic energy in the chemical reaction at the first encounter are necessary, which ensures that the partners can not leave the attractive well. Large orbital angular momentum of the primary products (equivalent to large rotational excitation in a unimolecular reaction) is favorable for roaming.

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INTRODUCTION A general condition for a typical bimolecular chemical reaction to take place is that the system has sufficient energy to surpass a threshold, which is often associated with a potential energy barrier. The energy to be utilized to surmount the barrier can be provided in various degrees of freedom of the reactants: in the form of kinetic energy of relative motion or by excitation of vibrational and/or rotational degrees of freedom. Some degrees of freedom can often be more efficient in stimulating reaction than in others, and, analogously, in many reactions there is a clear preference for energy deposition in certain degrees of freedom. The origin of the role of various degrees of freedom lies in the shape of the potential energy surface (PES) governing the motion of atoms during the reaction, in particular, the location of the potential barrier. Polányi and Evans1 have laid down the foundations of the theory that explains this connection. The general principles of the connection have been worked out by Polanyi2−6 and Marcus.7 According to these rules, generally referred to as Polanyi’s rules, reactions with an early barrier tend to produce vibrationally © 2012 American Chemical Society

excited products and are more efficiently induced by large initial relative kinetic energy than by an equal amount of vibrational energy, and vice versa for reactions with a late barrier. The main features of the dynamics underlying these rules can be understood in terms of the bobsled effect. Significant interest has been paid to the role the vibrational energy plays in bimolecular abstraction reactions. Earlier, the way of energy deposition was the question of interest,8,9 more recently, the possibility of inducing site-selective chemistry by vibrationally exciting various vibrational modes of the reactant before the reaction has been the major direction of study, providing an alternative to actively controlling the system during the reaction.10−12 Crim,13−18 Zare,19−22 Honda,23 and their co-workers have shown that when selectively exciting the OD or OH stretch vibration of HOD, the molecule, when colliding with H atoms, will selectively produce either HD or Received: February 7, 2012 Revised: April 7, 2012 Published: April 9, 2012 4445

dx.doi.org/10.1021/jp301243a | J. Phys. Chem. A 2012, 116, 4445−4456

The Journal of Physical Chemistry A

Article

same attraction it does when the diatom does not change its orientation, and the attraction may not necessarily be strong enough to ensure that the diatom, as it rotates, could drag the attacker. As the experimental study of reaction dynamics with a vibrationally excited reactant is rather difficult, it would be important to see whether and how the enhancement of the reactivity due to reactant vibrational excitation is reflected in the state-to-all thermal rate coefficients. In addition, it would be informative to find out how vibrational excitation of the reactant influences a thermoneutral reaction. The work presented in this article addresses these questions. We selected the reaction of H atoms with HCl,

HH. The magnitude of the enhancement of the rate constant of the reaction has been determined by Smith and co-workers.24−26 They found that local excitation of the O−H bond in water by four vibrational quanta increases the rate coefficient of the reaction by 16 orders of magnitude as compared with the unexcited reactant. The rate coefficient at room temperature was measured to be 1.5 × 10−10 cm3 molecule−1 s−1, a remarkably large value for a reaction characterized by a potential barrier of about 21 kcal mol−1. Theoretical work, including the development of global analytical potential surfaces,27−30 quantum dynamical calculations,31 and quasiclassical trajectory (QCT) calculations32−35 supported these observations and closely reproduced not only the reactive rate coefficient but also the vibrational removal of excited water molecules in collisions with H atoms.26 We have also studied the phenomenon of extreme vibrational enhancement of the reaction rate coefficient in some other systems.36−38 In the highly endothermic reaction of H with HF, which is analogous to the H + H2O reaction, excitation of the HF vibration by 3 or 4 quanta was found to yield a reactivity enhancement similar to that observed in water. The calculations indicated that the extreme enhancement of the rate coefficient is due to a switch from activated (characterized by a nonzero threshold energy) to capture-like behavior, also observed earlier27,35 for the H + H2O reaction. More precisely, the low-collision-energy dynamics of the system are qualitatively different at low and high vibrational excitation. By setting the initial vibrational excitation to noninteger vibrational quanta, we found that the switch from activated to nonactivated behavior takes place as soon as the initial vibrational energy exceeds the height of the potential barrier to hydrogen abstraction. The capture-type dynamics arises because at large separations the potential influencing the motion of the attacking H atom is attractive if the H−F vibration is in the stretched phase. In addition to the abstraction reaction, we found that at very low collision energies the cross sections for the exchange channel also exhibits capture-type behavior. Moreover, surprisingly, there is exchange reaction even at very low vibrational excitations, so low that the total energy is lower than the exchange barrier. Detailed investigation of the dynamics revealed that the lowenergy exchange reaction takes place according to a kind of roaming mechanism involving two successive passages over the energetically available abstraction barrier, which we shall return in the discussion of the current results. The switch between activated and capture-type dynamics has also been observed at high reactant vibrational excitation for the reverse F + H2 reaction, which is a highly exothermic reaction with a low, early barrier. This is in contrast to the expectation, based on Polanyi’s rules, that vibrational excitation is not the favorable way of inducing reaction for this kind of reaction. From this observation and from the mechanism observed for the H + HF reaction we inferred that the phenomenon is beyond the range of applicability of Polanyi’s rules, basically because the extreme vibrational excitation takes the system to physical conditions not considered when the rules are derived. All of the studies mentioned above were performed for a nonrotating reactant molecule. Rotational excitation has been proposed to reduce the rate of some bimolecular reactions with vibrational ground-state reactants,39 but for many other reactions the opposite effect has been reported (see, e.g., refs 40−44). This aspect requires consideration in vibrationally highly excited molecules also: one can expect that if the diatomic reactant rotates, the attacking atom will not feel the

H + H′Cl(v) → Cl + HH′ (R1a) → H′ + HCl (R1b)

for studying the dynamics of an almost thermoneutral reaction, including vibrational and rotational excitation of the reactant diatom. Reaction R1a, H atom abstraction has a low barrier which is located close to the corner region of the potential surface. The H-atom exchange channel R1b has a higher barrier, which, due to symmetry, is located right at the point where the minimum energy path has the largest curvature. If the HCl molecule is supplied by one quantum of vibrational energy, its internal energy is higher than the potential barrier to abstraction. Two vibrational energy quanta exceed even the exchange barrier. On the potential surface of the reaction there is a shallow van der Waals potential well at relatively small reactant separation and another on the product side of the abstraction barrier, at very large separations. We found that these features are important in the reaction dynamics. An analytical potential surface developed by Bian and Werner45 is available for this reaction, which is based on a large number of almost accurate ab initio data. This surface proved to be a very good basis for reproducing the measured dynamics of the Cl + H2 reaction in exact quantum scattering calculations.46 For the H + HCl reaction with ground-state reactants, in QCT calculations Aoiz et al.47 found this PES is less adequate for the reproduction of the experimental data of Wolfrum et al. but reported a very good agreement for the exchange channel, and made the same observations for the H+DCl isotopic variant.48,49 The time-dependent quantum scattering calculations reported by Yao et al.50 indicated some disagreement with the QCT results and a discrepancy for the exchange channel. On the other hand, the time-independent calculations of Weck and Balakrishnan51 extended to ultracold translational temperatures (restricted to zero total and zero reactant angular momentum) produced very reasonable results. We report the results of quasiclassical trajectory calculations on reaction (R1) performed systematically for various vibrational and rotational states of the diatomic reactant. In addition, we extend our earlier work on the H + H′F(v) → F + HH′ (R2a) → H′ + HF (R2b)

system to investigate how rotational excitation of the H′F molecule influences this reaction. In the rest of the paper, we first summarize the computational details; then we present and discuss the results on the H + HCl reaction, followed by those on the H + HF reaction. Finally, we make some remarks concerning the features of the roaming mechanism. 4446



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METHODS

Table 2. Energies of the Vibrationally Excited States of the HCl and HF Molecule Measured from the Minimum of the Potential Well, Calculated Using the Nonrigid Rotating Morse Oscillator Model (eV)

Reactive cross sections were determined in standard quasiclassical trajectory (QCT) calculations. We used an adaptation of the 1988 version of VENUS52 that was parallelized53 and streamlined for the present purpose. In all calculations the connection between the orbital angular momentum and the initial impact parameter was considered to be purely classical. At every combination of translational energy, as well as vibrational and rotational quantum number we determined the opacity function to make sure that the proper maximum impact parameter is used in the cross section calculations. The initial center of mass separation was 10 Å except for cases when the opacity function exceeded 8 Å: then the initial separation was set 2 Å larger than the maximum impact parameter. The rotational quantum number j of HCl and HF was set to quasiclassical integer values from 0 to 10. At each combination of initial v, j and translational energy, 100 000 trajectories were run. The integration time step was 1 fs that ensured that the energy was conserved better than 0.01 eV. In test calculations with long, roaming trajectories we found that the same phase space path was traced if the time step was varied up and down by a factor of 2. In the studies of the H + HCl reaction, we used the Bian− Werner BW2 potential surface45 with numerical derivatives. For the H + HF reaction we found earlier that the Stark−Werner54 (SW) PES does not support the capture-type behavior even at large vibrational excitations due to the known, unphysical bump it has in the H + HF valley. We made some attempts to use a modified version of the SW PES developed by Skodje et al.55 and found that on this PES the unphysical barrier, although reduced, is not eliminated. Consequently, the modified version is also not appropriate for the study of capture-type behavior. As shown in our earlier work,38 the 6-SEC PES developed by Mielke et al.56 does not suffer from unphysical distortion of the long-range part of the potential surface and can be an appropriate basis for drawing conclusions on reaction dynamics in the low-collision-energy range. We used the 6-SEC PES, with numerical derivatives, in the H + HF calculations. The key parameters of the two potential surfaces are summarized in Tables 1 and 2.

H + HCl

H + HF

−0.112 0.330 0.981 1.431 180 0.776 1.480 180

1.375 1.418 0.757 1.640 104 1.619 1.161 86

HCl

HF

0.188 0.552 0.900 1.233 1.551

0.261 0.767 1.249 1.709 2.145

number.57,58 This scheme usually changes the excitation function in the threshold region, which is a key factor when rate coefficients are calculated by shifting the threshold to higher values. We note, however, that this scheme is biased as it focuses on “quantization” of only the zero-point motion, but it inherently is unable to consider quantization of higher vibrational states. In our calculations the selection of the binning schemes does not influence the conclusions, the two schemes result in very similar excitation functions, because generally the product is also vibrationally excited. The state-to-all thermal rate coefficients were obtained by numerical integration from the excitation functions according to the equation 1/2 1 ⎛ 8 ⎞ kvj(T ) = ⎜ ⎟ kBT ⎝ πμkBT ⎠

∫0

∞

σvj(Etr)e−Etr / kBT Etr dEtr (1)

where σvj(Etr) is the reactive cross section at collision energy Etr, kB is the Boltzmann constant, μ is the reduced mass of the reactant pair, and T is the temperature.

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RESULTS H + HCl Reaction. Opacity Functions. The opacity functions for the abstraction reaction R1a calculated at low, medium, and high collision energies are shown in Figure 1. The reactivity is very sensitive both to the vibrational excitation of the reactant HCl molecule and to the collision energy. If the reactant is vibrationally unexcited, the small impact parameter collisions are the most reactive, and the reactivity drops as the impact parameter b increases. The maximum impact parameter where reactive collisions are observed does not exceed 3 Å. Similar is the behavior if the HCl vibration is excited and the collision energy is large. A completely different opacity function characterizes the reaction of vibrationally excited HCl at low collision energies. The reaction probability is very large at small impact parameters and increases as b is set to larger values until a limit where it suddenly drops. The smaller the collision energy the larger the maximum impact parameter where reactive trajectories occur; the value of bmax is determined by the interplay of the effective attraction potential and the centrifugal barrier. Excitation Functions. Figure 2 shows the excitation functions (the state-to-all reactive cross sections versus the initial relative translational energy of the reactants, Etr) obtained using the simple QCT and the QCT-ZP schemes for both channels of the H + HCl(v,j=0) reaction for v = 0, 1, and 2. If the HCl molecule is unexcited, both channels show activated behavior, the threshold being 0.15 eV for abstraction and 0.75 eV for exchange. As expected, the simple QCT scheme provides larger cross sections at any initial energy for both

Table 1. Properties of Stationary Points on the BW2 and 6SEC Potential Surfaces for Reactions (R1) and (R2), Respectively ΔEreact/eV Ebarr(abstraction)/eV rabs‡(H−H)/Å rabs‡(H−F)/Å φabs‡(H−H−F)/degree Ebarr(exchange)/eV rexch‡(H−F)/Å φexch‡(H−F−H)/eV

v 0 1 2 3 4

In the calculation of cross section, we used two commonly used binning schemes. In what we shall refer to as simple QCT, all reactive trajectories are considered when the reactive cross sections or rate coefficients are evaluated; no considerations and exclusions are applied to the final actions. In the QCT-ZP scheme,57,58 we discarded trajectories that result in a product diatom with less action than what corresponds to the zeroth quantum level, irrespective of the rotational quantum 4447



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Figure 1. Opacity function for the reaction of H with HCl(v,j) at various initial relative kinetic energies at (a) v = 0, j = 0 and (b) v = 2, j = 0.

abstraction and exchange than QCT-ZP which yields threshold energies higher by more than 0.05 eV for both channels. The QCT-ZP cross sections show a smooth switch-on in contrast to those obtained with the simple QCT scheme. In this respect the QCT-ZP excitation function resembles more the usual threshold behavior of the quantum mechanical excitation functions than that calculated using the simple QCT scheme. Beyond the threshold region the QCT-ZP cross sections remain about one-half of those obtained with the simple QCT scheme. We found that in most abstraction trajectories discarded according to the QCT-ZP scheme, the energy of the H2 product is very close to the zero-point level, whereas in HCl produced in channel R1b they are dominantly well below it. In other words, in the abstraction channel vibrational adiabaticity is maintained better than in exchange. This can be connected to the properties of the PES. The abstraction barrier is shifted slightly into the reactant well (as expected for an exothermic reaction), whereas the exchange barrier is located symmetrically between the reactant and product limits. In collisions leading to abstraction, the relatively early barrier can be overcome by the small translational energy without significant assistance from vibration, and after the barrier, in the curved region of the PES, only a few trajectories lose vibrational energy to translation (most of them gain energy instead). In the exchange process the barrier is located symmetrically between the reactant and product valley, which, if compared with the abstraction barrier, is closer to the product valley. Among the H + HCl collisions going to the exchange channel, those will lead to reaction in which more vibrational energy is converted to translation and assists in overcoming the barrier (as it is now more in the curved part of the PES than for

Figure 2. Excitation functions for the R1a abstraction (top panels) and R1b exchange channel (bottom panels) of the H + H′Cl(v,j = 0) reaction calculated with the simple QCT (circles) and QCT-ZP (squares) schemes. (a) v = 0; large circles; experimental data from refs 42 and 62; (b) v = 1; (c) v = 2. The error bars of the cross section values are smaller than the symbol size.

the abstraction channel). The more significant reduction of product vibrational energy observed for exchange compared with abstraction is the consequence of this energy requirement. These facts represent an interesting manifestation of Polanyi’s 4448



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rule, which indicates that the latter appropriately interprets the dynamics at low vibrational energies. Parts a and b of Figure 2 display the excitation functions for reaction of vibrationally excited HCl. For v = 1 one can see that the simple QCT and the QCT-ZP results are much closer to each other than for the vibrational ground state of HCl. For v = 2 it is obvious that in this case the simple QCT and QCT-ZP schemes yield excitation functions that are very similar. This indicates that very few trajectories end in H2 or HCl below the zero-point level, which is not surprising as the vibrational quantum number would be reduced by 2 in the reaction. The fact that there are, although relatively few, reactive encounters that involve such a large reduction of the vibrational quantum number indicates significant vibrational nonadiabaticity, which means that vibrational energy is converted either to translation and utilized for passing the barrier or, after the “point of no return”, to translation and rotation. In the following we shall discuss the results concerning the excitation functions and cross sections that are obtained with the QCT-ZP scheme because they are more reasonable in the threshold region when the reactant is vibrationally unexcited (note again that at high vibrational excitation QCT-ZP produces essentially identical results with simple QCT). Excitation Functions for Various Quantum States of the Reactant HCl. Concerning the qualitative features of the excitation functions, panels a and b of Figure 2 indicate that the abstraction cross sections at v = 1 show a capture-type divergence at low and a rising excitation function for large translational energies. When the collision energy increases, the excitation function first drops quickly just like in reactions between two partners governed by an attractive potential. Still at relatively small Etr (0.005 eV) it passes a minimum and rises relatively fast until it reaches a plateau of about 0.6 Å2 at about 0.15 eV. At v = 2 the capture-type behavior extends to the whole collision energy range: the cross sections decrease monotonically from extremely high values observed at low collision energy (for example, 84 Å2 at around 10−7 eV) and remain as high as about 1 Å2 even at large Etr. The exchange excitation function shows a threshold energy of about 0.7 eV at v = 0. This is 0.55 eV higher than for abstraction, although the difference of the barrier heights is only 0.446 eV, indicating that the exchange channel has more severe dynamical constraints. The exchange threshold energy decreases by 0.28 eV to about 0.42 eV when the vibrational state of HCl changes from v = 0 to v = 1. The shift is smaller than the invested vibrational quantum, 0.365 eV, which indicates that the vibrational energy is only partially utilized for reaction. The threshold disappears when the reactant vibrational excitation is further increased. Moreover, the exchange channel also displays the capture-type divergence seen for abstraction already at v = 1. Comparing this observation with that seen for the H + HF and F + H2 reactions, we can conclude that sufficient vibrational excitation induces a switch from activated to capture-type behavior independently of the location of the potential barrier. The dependence of the reactivity on the rotational state of the reactant HCl molecules at v = 0, 1, and 2 is shown in Figure 3. The excitation function for abstraction from vibrationally unexcited, rotationally excited HCl to about j = 7 are essentially indistinguishable, in agreement with the observation of Aoiz et al.47 Reactant rotation starts to induce a visible change of the excitation function only at j = 10; when starting from the same threshold, the excitation function rises faster than for lower j. It

Figure 3. Excitation functions for the R1a abstraction (top panels) and R1b exchange channel (bottom panels) of the H + H′Cl(v,j) reaction calculated with the QCT-ZP scheme at j = 0 (filled squares), j = 2 (circles), j = 3 (filled triangles), j = 5 (open triangles), and j = 10 (filled diamonds) and (a) v = 0, (b) v = 1, and (c) v = 2.

is remarkable that the total internal energy of the reactant (which at j = 10 is as much as 0.332 eV, out of which 0.144 eV is rotational energy) does not reduce the threshold for abstraction. At v = 1, the capture-type divergence at low collision energies is quenched by rotational excitation of the reactant by as little as one rotational quantum. The threshold for abstraction is at about 0.005 eV at j = 1 and increases with the increase of the 4449



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rotational excitation of the reactant HCl but remains small (0.05 eV at v = 1, j = 10). If the HCl molecule is vibrationally more excited (v = 2 or 3 and j = 0), rotational excitation of HCl gradually suppresses the capture-type behavior at v = 2, but not at v = 3. At v = 2, diverging cross sections can be seen up to j = 3; above that, the reaction is activated, but the threshold for abstraction remains below 0.01 eV even at j = 10. The capturetype divergence returns at j = 20. At v = 3 the excitation function for abstraction remains capture-type as j grows at least up to 10, only the magnitude of the cross sections decreases. At low HCl vibrational quantum numbers 0 and 1, rotational excitation of HCl increases the reactivity for the exchange channel (bottom panels of Figure 3), by gradually shifting the excitation function to lower energies and keeping its shape essentially unchanged. At v = 0, the reduction of the threshold energy is as large as 0.15 eV at j = 10, which is larger than the additional energy provided for the system in the form of Erot (0.144 eV). The effect is even more expressed when v = 1: Ethr goes down to essentially zero due to the same rotational excitation, a reduction of more than 0.25 eV. This means that fast rotation of the HCl molecule makes more efficient the utilization of the vibrational energy in overcoming the exchange barrier, in sharp contrast to the abstraction channel. Increasing v to 2, at j = 0 the very large cross sections drop very quickly with increasing Etr all the way to zero at about Etr = 0.01 eV and start to slowly increase at Etr = 0.03 eV. Above 0.4 eV, the increase is linear in the entire energy range we studied (up to about 2 eV) and its slope is essentially independent of j. The low-energy divergence observed for j = 0 is quenched by the first rotational quantum and only the rising part of the excitation function remains. The threshold energy for exchange slowly increases with increasing j but remains very small, below 0.02 eV. The initial increase of the cross sections is steeper the larger j(HCl) is, but in the high-energy range the excitation functions run parallel, shifted to lower energies with the increase of j (by about 0.09 eV at j = 10). This indicates that in spite of quenching the capture-type behavior, rotational excitation is favorable for the exchange channel even when the vibrational energy is larger than the barrier height. At v = 3 the exchange excitation functions are capture-type even at large reactant rotational excitation, just like those for abstraction, but the cross sections are 3−5 times smaller than for the latter. Thermal Rate Coefficients. The Arrhenius plots corresponding to the state-to-all thermal rates calculated for the abstraction channel from the excitation functions of Figure 3 are shown in Figure 4. We plotted the entire temperature range in which our cross sections enable us to derive reliable rate coefficients. This range extends down to 100 K, which allows some conjecture about the behavior of the reaction at ultracold temperatures. In the calculations we used the QCT-ZP cross sections. The thermal rate coefficients for each initial vibrational quantum number were obtained by averaging over the thermal distribution of rotational states corresponding to the given temperature. The vibrational enhancement of the roomtemperature rotationally averaged rate coefficient is sizable: k(v) is 5.4 × 10−14, 6.7 × 10−11, and 4.9 × 10−10 cm3 molecule−1 s−1 at v = 0, 1, and 2, respectively. This is much smaller than the 16-order of magnitude increase observed for the H + H2O(v) reaction. However, whereas the latter, being highly endothermic, has a very small room-temperature reactivity, reaction R1a has a sizable rate coefficient at room temperature and the 4order magnitude enhancement already brings the rate coefficient to the gas kinetic limit. The activation energies for

Figure 4. Rate coefficients for the R1a abstraction channel of the H + H′Cl(v,j) reaction calculated form the excitation functions shown in Figure 3 at individual rotational quantum numbers j. The thick continuous lines represent the average over the rotational states at each v.

reaction R1a depend on the temperature (Table 3 and Figure 4). As one can expect from the low-energy part of the excitation Table 3. Activation Energy of the H + H′Cl(v) → Cl + HH′ Reaction Averaged over the Initial Rotational States of the HCl Reactant (eV) T/K

v=0

v=1

v=2

v=3

1000 300 100

0.250 0.177 0.145

0.090 0.031 0.010

0.033 0.002 −0.003

0.010 −0.002 −0.005

functions, at low reactant vibrational energy, v = 0 the rate coefficients drop quickly with the reduction of the temperature, and the rotational excitation has hardly any influence on the Arrhenius plots. The activation energy drops by more than 0.105 eV between 2000 and 300 K, corresponding to significant curvature in the plots. For v = 1 the rate coefficients vary in a much narrower range, and the activation energy actually decreases less (the Arrhenius plots for v = 1 are virtually more curved than for v = 0 only due to the difference in scales). At v = 1, j = 0 the cross sections diverge at low translational energy, but this region is so narrow that the activation energy is found to be positive even at temperatures as low 100 K. The disappearance of the capture-type part and the gradual increase of the threshold energy with the increase of the rotational quantum number of HCl are reflected in the reduction of the rate coefficient and the steeper slope of the Arrhenius plots. The effect of the rotational excitation is so large that it extends to the thermal rate coefficient averaged over the rotational states. The corresponding thermal Arrhenius plot is curved due 4450



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to two factors. One factor is that the ln k − 1/T lines at individual j’s are not straight themselves, the other comes from the fact that reaction of HCl with lower rotational excitation has significantly larger rate coefficients than those corresponding to large rotational quantum numbers. The equilibrium population of the low-j states increases as T is reduced, inducing a gradually smaller rate reduction as compared with those at high temperatures. At v = 2 the rate coefficient at j = 0 starts to increase at about 500 K with the reduction of the temperature; the activation energy has a small negative value (−4.6 meV). At higher j the activation energies are positive in the entire temperature range we studied, due to the diminution of the width, and the eventual disappearance of the capturetype part of the excitation functions. The thermal average rate coefficient for the v = 2 vibrational state of HCl starts to increase at 200 K with decreasing temperature. Below 200 K the large reactivity of the j = 0 state combined with the enhanced population of this state dominates the thermal average. This means that, to have a chance to experimentally observe the macroscopic consequences of the capture-type excitation function for the reaction of HCl excited by two vibrational quanta, the rate coefficient needs to be measured at temperatures at and below 200 K, which is certainly not an easy task but is not hopeless because k is extremely large. If the HCl reactant is excited more (v ≥ 3), the activation energy is negative already at 2000 K, but its magnitude is very small (around −0.1 kcal mol−1) even at 100 K. This means that the Arrhenius plots are essentially horizontal, the rate coefficients are at or above the gas kinetic limit: the room-temperature thermal average rate coefficient is as high as 10−10 cm3 molecule−1 s−1. The Arrhenius plots for the exchange channel shown in Figure 5 display much larger activation energies and much lower rate coefficients. In contrast to abstraction, the rate coefficients become larger as the rotational quantum number of HCl increases. The curves corresponding to the individual initial rotational states of HCl reflect the properties of the excitation functions. Note that at j = 10, with respect to j = 0, the Arrhenius activation energy is reduced by 0.09 eV at 2000 K and by 0.22 eV at 300 K, indicating that the shift of the threshold energy (−0.15 eV) is not mapped directly onto the activation energy. At v = 1 the dispersion of the rate coefficients and of the activation energies according to j is more visible than for v = 0. The 300 K activation energies are 0.35, 0.25, and 0.105 eV at j = 0, j = 3, and j = 10. At v = 2 the respective values are 0.057, 0.060, and 0.037 eV, reflecting that the reaction requires much smaller collisional activation. At low temperatures the j = 0 rate coefficients are larger than at j = 1 and j = 2 because the capture-type divergence of the cross sections is quenched by rotational excitation. This effect will be more expressed at ultralow temperatures (T < 1 mK). Overall, at higher initial vibrational excitation of HCl the range of the rate coefficients is reduced, between 100 and 2000 K its breadth being 36, 12, 3.5, and 1.5 orders of magnitude for v = 0, 1, 2, and 3, respectively. The activation energies are not negative even at v = 3 and low T. The numerical comparison of the absolute magnitudes of the rotationally averaged rate coefficients for abstraction and exchange is shown in Table 4. The large difference between the abstraction and exchange barriers is reflected in the large abstraction/exchange selectivity even at 2000 K, which is not annulled even if the vibrational excitation is as large as v = 3. The lower the temperature, the larger the importance of the

Figure 5. Rate coefficients for the R1b exchange channel of the H + H′Cl(v,j) reaction calculated form the excitation functions shown in Figure 3 at individual rotational quantum numbers j. The thick continuous lines represent the average over the rotational states at each v.

Table 4. Ratio of the Abstraction and Exchange Rate Coefficients kabs/kexch at Various Temperatures and Vibrational Quantum Numbers T/K

v=0

v=1

100 300 2000

2.61 × 10 9.66 × 108 18.6

29

1.21 × 10 2.14 × 105 14.8

12

v=2

v=3

9952 325 5.7

87.0 31.1 4.3

abstraction channel. Nevertheless, at low temperatures vibrational excitation causes a spectacular reduction of selectivity. H + HF Reaction. Excitation Functions for Rotationally Excited HF. The excitation functions for the abstraction and exchange channels R2a and R2b are collected in Figure 6 for various rotational excitations at HF vibrational quantum numbers v = 3 and 4, calculated according to the QCT-ZP scheme. We note that reactive collisions often yield HH′ or H′F products with only a little smaller energy than the zpe that are discarded according to this scheme. We think this is too strict a restriction and provides excitation functions at low Etr that should be considered with caution. However, the capture-type or activated nature is not changed by this effect. With respect to the relation of reactant vibrational energy to the barrier height, HF vibrational states v = 3 and v = 4 are more-or-less analogous to the v = 1 and v = 2 states of HCl. As Figure 6 shows, if the rotational quantum number of HF increases from 0, capturetype behavior is gradually switched off in this reaction at both v = 3 and v = 4, similarly to the abstraction reaction R1a at v = 1 and v = 2, but the low-kinetic-energy divergence remains observable at both v = 3 and v = 4 at j = 0, 1, and 2. At v = 3, at 4451



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the HF molecule increases, but at high j the tendency is reversed, in agreement with the changes of the excitation functions. At v = 4, below about 300 K the activation energy has a small negative value for both channels if the reactant molecule is in the rotational ground state; at v = 3 the abstraction rate constants keep decreasing with decreasing T. On the other hand, the rotationally averaged rate coefficients increase as the temperature is reduced below room temperature both for channels at v = 4 and for exchange at v = 3. The magnitude and the origin of the negative activation energy is the same as discussed for the H + HCl reaction.

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DISCUSSION The excitation functions for the two reactions follow a similar qualitative tendency. The large vibrational excitation of the reactant molecule induces a large increase of the reactivity. This is manifested in nonactivated behavior for certain, relatively high initial rovibrational states, namely, in the shift of the threshold energy to zero. In addition, in vibrationally excited states at low reactant rotational excitation the reactive cross sections diverge as the collision energy tends to zero, which is characteristic to capture reactions. As found earlier for the H + H2O abstraction reaction27,35−37 and for reactions R2a and R2b,38 the capture-type behavior is due to a long-range attraction between the partners if the diatomic reactant is stretched. Figure 8 illustrates this for the abstraction and exchange channel of reaction (R1). During the vibration of the diatomic reactant, near the outer turning point the bond is “halfway” broken, and if the attacking H atom arrives in this phase of HX vibration, the system can pass over the barrier on a downhill energy path. Classical versus Quantum Comparisons of Reactivity at Ultracold Temperatures. The capture-type divergence of the cross sections is manifested in negative activation energies, but the magnitude of EA is small, remaining below 0.05 eV. The effect could be spectacular at ultracold collisional temperatures. Unfortunately, our excitation functions do not cover a dense enough grid to get reliable rate coefficients at ultracold temperatures. We can compare, however, the low-energy excitation functions obtained with the quasiclassical trajectory method for reaction (R1) with those derived in the exact quantum scattering calculations of Weck and Balakrishnan.51 According to the quantum mechanical results the j = 0 cross sections are inversely proportional to the square root of the collision energy if the latter is below the lowest-energy resonance that is around 0.002 eV. The inverse square root energy dependence means that the reactive cross section is inversely proportional to the relative velocity of the reaction partners, a relationship that is independent of the initial vibrational state. This behavior is assigned to the system reaching the Wigner limit of near-threshold scattering, an intrinsically quantum phenomenon. Very interestingly, the limited data we have for very low collision energies (10−5 to 10−3 eV) for the v = 2, j = 0 state of HCl also represent the same kind of velocity dependence, even though in our calculations no quantum effects except the quasiclassical setting of the initial state of the reactant are taken into account; i.e., no tunneling or interference is included. The QCT-ZP abstraction cross section at v = 2, j = 0 is smaller than the quantum mechanical at very low collision energies (for example, at 10−5 eV they are 20 vs about 60 Å2). The agreement is better at v = 1 (the quantum vs quasiclassical difference is below 20%). At v = 0 the quantum mechanical calculations do detect reaction, with

Figure 6. Excitation functions for the R2a abstraction (top panels) and R2b exchange channel (bottom panels) of the H + H′F(v,j) reaction calculated with the QCT-ZP scheme at j = 0 (filled squares), j = 2 (circles), j = 4 (triangles), and j = 10 (filled diamonds) and (a) v = 3 and (b) v = 4.

higher rotational excitation no capture-type excitation function can be seen. At v = 4 and above, however, the divergence at low collision energies returns at j = 9 and its importance increases with the rotational excitation of the reactant HF molecule. This is similar to what was observed for reaction R1a at v = 2. The exchange channel R2b also exhibits capture-type behavior at low rotational excitation at both vibrational levels of HF. It is noteworthy that the capture-type divergence for reaction R2a is observable already at relatively low vibrational excitation, v = 3, where the vibrational energy exceeds the exchange barrier height by only 0.1 eV. Here we recall that low-energy divergence was observed earlier38 for noninteger vibrational quantum numbers (v = 2.6, 2.7, 2.8, or 2.9) where not only the vibrational energy but even the total energy is below the exchange barrier. This indicates that exchange can take place via bypassing the barrier: roaming. We return to this effect later. Thermal Rate Coefficients. The state-to-all abstraction and exchange thermal rate coefficients are shown in Figure 7 for HF initial states v = 3 and v = 4, 0 ≤ j ≤ 10. The rate coefficients are in a narrow range for the H + HF system at this high excitation, at room temperature the smallest and largest differing by only factor of 15. For the abstraction channel the rate coefficients first drop as the initial rotational excitation of 4452



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Figure 7. Rate coefficients for the (a) abstraction channel R2a and (b) the exchange channel R2b of the H + H′Cl(v,j) reaction calculated from the excitation functions shown in Figure 6 at individual rotational quantum numbers j. The thick continuous lines represent the average over the rotational states at each v.

cross sections around 10−6 Å2, whereas with the quasiclassical method no reactive collisions are seen (because it is energetically forbidden), indicating that in this quantum state the reaction takes place purely by tunneling. In the case of exchange cross sections, the quasiclassical vs quantum agreement is worse, in the ultralow collision energy range the former are generally 2 or more orders of magnitude smaller as compared with the quantum result. On the other hand, in the collision energy range 0.01−0.1 eV the QCT-ZP abstraction cross sections are larger than those provided by the exact quantum method. The large vibrational enhancement of the reactivity has been noted in the quantum scattering studies,51,59 but the attractive interaction has not been connected to this fact. The good qualitative agreement between the low-energy cross sections calculated quantum mechanically and quasiclassically is remarkable for the following reason. In ultracold systems the atoms move so slowly that their de Broglie wave wavelength can attain macroscopic dimensions. These conditions are ideal for quantum effects to be manifested, and many theoretical studies of chemical reactions are devoted to detect these quantum effects.60−63 The close correlation of our quasiclassical results with the quantum mechanical ones indicates that not all of the observed phenomena are of quantum mechanical origin. This suggests that more classical mechanical studies are needed to understand what the intrinsic quantum effects under these conditions are. Note that the attraction felt by the attacking atom when approaching a vibrationally excited HCl molecule and itself induces lowenergy divergence of cross sections is a factor that is not included in Wigner’s treatment of near-threshold scattering or in the J = 0 quantum scattering calculations.51 Atom−diatom attraction is essential in ultracold systems where molecules are formed by, for example, photoassociation64,65 and J ≫ 0 conditions require more attention on quantum scattering for

such systems. Such calculations are in progress in our laboratory. Roaming Mechanism at Low Collision Energies. Another factor that needs to be considered in the investigation of reactions at ultracold temperatures is related to the lowenergy behavior of the exchange reactions R1b and R2b. The exchange excitation functions also display capture-type divergence. What is even more remarkable is that the pure classical cross sections for exchange reaction are not zero even when surpassing the exchange barrier is energetically not allowedwhen the reactant is only slightly vibrationally excited and the collision energy is low. More precisely, exchange is observed when the total energy in the system is below the exchange but higher than the abstraction barrier. Under such conditions the exchange reaction takes place by bypassing the exchange barrier via a route that is energetically available. The mechanism is the following: the slowly approaching H atom abstracts the H′ of the H′X partner. Very often the relative translational energy of the products remains small so that the HH′ molecule and the halogen atom X spend a long time at small separation, but sometimes they depart to very large distances (exceptionally as large as 8 Å, illustrated in Figure 9). This time is long enough for the HH′ molecule to perform several complete rotations. If during the rotation the H atom of the newly formed HH′, which often has a large vibrational excitation, points toward the X atom when the HH′ bond is in the stretched phase of its vibration, X can abstract the H from HH′. The same mechanism allows reabstraction of the H′, the original partner of X before the reaction. This latter process manifests as a nonreactive collision, masking two actual reactive steps. In the Supporting Information we present movies depicting trajectories corresponding to exchange reaction according to this mechanism. The “double abstraction” mechanism is made possible because the long-range attractive 4453



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Figure 9. Interatomic distances as functions of the integration time during a roaming H + H′Cl trajectory started at 4.5 × 10−5 eV initial relative kinetic energy (only a part of the whole collision is shown). In the first encounter rotational energy transfer takes place, in the second abstraction of H′ by H, in the third, reabstraction of H by Cl from the primary product HH′. The reason the slowly departing particles return to each other from extreme large interparticle separation is because of the combined effect of the long-range attraction and the centrifugal barrier provided by the large orbital angular momentum.

participates in a unimolecular decomposition and is initially highly rotationally excited, large orbital momentum for the decomposing partners is ensured by angular momentum conservation. The aforementioned conditions are fulfilled in many systems, in particular, in complex-forming bimolecular reactions and in unimolecular decomposition of molecules that can decompose into several pairs of products. This means that roaming can be expected to be common in gas-phase kinetics, for which several observations have been made.72 The importance of such processes is that they ensure formation of products unexpected according to the commonly used picture of chemical reactions, which is based on the assumption that the motion of the atoms of the reacting molecules follows the minimum energy path of the potential surface. Accurate quantum mechanical treatment of roaming is a serious challenge to reaction dynamicists. Such collisions take place on a long time-scale, making time-dependent calculations less practical. In addition, the selection of coordinates that can adequately describe the chattering in such systems also requires care. As the atoms move very slowly in ultracold systems, there is a good chance for the kind of roaming observed in reactions (R1) and (R2) to occur, so that it is probably a very common contributor to reactivity under such conditions. The timeindependent quantum scattering calculations applied so far in the studies of reactions involving ultracold atoms seem not to have been able to take into account this special dynamical phenomenon, which calls for novel reaction dynamical technologies.

Figure 8. Two sections of the potential surface of the H + H′Cl → Cl + HH′ reaction. The potential energy is shown as a function of the location of the attacking H atom around the H′Cl. The origin of the coordinate system is at the H′ atom, the H′−Cl is oriented toward the negative y axis. The H′−Cl distance is (a) 1.4 Å, its value at the saddle point of the PES, and (b) 1.6 Å, corresponding to the outer turning point in the highly vibrationally excited H′Cl molecule. In the upper half of (a) the van der Waals potential well is visible. In (b) the energy scale is much wider to make sure that the deep attraction near the saddle point is visible. The energy is measured in electronvolts from the separated reactant limit with the H′Cl bond stretched to 1.4 and 1.6 Å, respectively.

forces allow the system to switch from one channel to another by bypassing the barrier to the direct transition from reactants to products. The mechanism of reactions bypassing the direct barrier is termed roaming.66 The roaming mechanism has been observed in theoretical and experimental studies of several reactions.67−71 Our results on the H + HX reactions as a case study display several features necessary for roaming: (1) a longrange attractive interaction has to act between the products formed in the first encounter; (2) the primary products formed have to depart generally slowly; (3) there has to be another possible reaction channel that is energetically available when the products of the primary encounter come together again as reactants of a secondary chemical step. Large orbital angular momentum is favorable for roaming because the centrifugal barrier helps the primary products stay together and sample the potential surface. In bimolecular reactions the orbital angular momentum can be large if the initial impact factor is large, overcompensating for the relative velocity being low (because otherwise the partners have too much energy to follow the slight attractive interaction). If a molecule (or even radical)

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CONCLUSION The dynamics of reactions of vibrationally excited diatomic molecules were studied by the quasiclassical trajectory method. The reactions include a highly endothermic and an almost thermoneutral one (H + HF as well as H + HCl). The calculations were extended not only to vibrational but also to rotational excitation of the reactant HX molecule. One can expect that if the vibrational excitation of the reactant provides enough energy for the system to surmount the potential barrier 4454



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(without negligible assistance from translational energy), the reaction becomes unactivated; in other words, the threshold energy for reaction is shifted to zero. According to the calculations, however, in addition to annulling the threshold energy, reactant vibrational excitation induces a switch of mechanism. Namely, with the reduction of the collision energy, the reactive cross sections diverge for both the abstraction and the exchange channels. This switch is induced by a specific feature the interaction between the attacking atom and the vibrating molecule as follows. If the diatomic molecule is highly stretched, the energy of the system is so large that the attacker observes an almost free H atom, which is manifested in the reduction of the potential energy as the attacker approaches closer; in other words, the partners attract each other. As the initial energy is very high, passing the potential barrier is actually a downhill motion on the PES. Just like in truly attractive systems, the slower the approach, the larger the chance that the attacker is captured and reacts. Additional rotational excitation of the reactant reduces the magnitude of cross sections in the low kinetic-energy regime and eventually quenches their divergence. The probable reason for this is that, as the molecule rotates, it turns its different sides toward the slowly approaching partner, which often results in hitting the molecule from the side. A further increase of the rotational excitation results in the return of the low-energy divergence, because, as the molecule rotates fast, the attacker now observes an averaged potential, which is attractive if the vibrational excitation of the reactant is large enough. From the capturetype excitation functions corresponding to the rotationally unexcited reactant, thermal rate coefficients characterized by a small negative activation energy at and below room temperature can be derived. Experimental detection of a negative activation energy could be a proof of the proposed capture-type behavior. However, if the thermal rate coefficients are averaged over the rotational states, the chance that negative activation energy can be discerned is generally reduced, except in some specific cases. In particular, at very large vibrational excitation, when the low rotational states are significantly more reactive than the higher states, the thermal average rate coefficient increases when the temperature decreases, faster, than that of the j = 0 state. The reason is that, because the population of the slower-reacting higher rotational states decreases with the reduction of the temperature, there will be more room for the higher reactivity of the rotationally cold states to be manifested. If the relative motion of the reaction partners is slow, a unique possibility is offered for the exchange reaction to occur: the roaming mechanism can operate. In this process the exchange barrier is bypassed at the price of surmounting the abstraction barrier twice. The attacking atom first abstracts the hydrogen from the molecule. If the relative kinetic energy of the formed products is also small, the products stall nearby for one or more rotational periods, and if the phases of motion are right, the halogen can abstract the original attacker from the temporarily formed H2 molecule. For roaming to occur, longrange attraction between the primary partners and small relative kinetic energy release in the first encounter are necessary, and large orbital angular momentum, providing a sizable centrifugal barrier is favorable. Such conditions hold in reactions of ultracold atoms with vibrationally excited diatomic molecules, where, accordingly, the consequences of roaming would probably be observable. The accurate quantum mechanical description of the roaming mechanism will be a challenge even for triatomic systems.

Article

ASSOCIATED CONTENT

S Supporting Information *

Movies of two trajectories depicting double-exchange trajectories resulting in exchange reaction via roaming at total energy lower than the exchange barrier. This material is available free of charge via the Internet at http://pubs.acs.org

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by the Hungarian National Research Fund (grant No OTKA T77938).

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REFERENCES

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Bimolecular Reactions of Vibrationally Excited Molecules. Roaming

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