A Phase Space Theory for Roaming Reactions - The Journal of

Article pubs.acs.org/JPCA

A Phase Space Theory for Roaming Reactions Duncan U. Andrews, Scott H. Kable,* and Meredith J. T. Jordan* School of Chemistry, University of Sydney, NSW 2006, Australia S Supporting Information *

ABSTRACT: We describe a new, simple theory for predicting the branching fraction of products in roaming reactions, compared to the analogous barrierless bond dissociation products. The theory uses a phase space theory (PST) formalism to divide reactive states in the bond dissociation channel into states with enough translational energy to dissociate and states that may roam. Two parameters are required, ΔEroam, the energy difference between the bond dissociation threshold and the roaming threshold, and the roaming probability, Proam, the probability that states that may roam do roam rather than recombine to form reactants. The PST-roaming theory is tested against experimental and theoretical data on the dissociation dynamics of H2CO, NO3, and CH3CHO. The theory accurately models the relative roaming to bond dissociation branching fraction over the experimental or theoretical energy range available in the literature for each species. For H2CO, fixing ΔEroam = 146 cm−1, the midpoint of the experimental bounds for the roaming threshold, we obtain Proam = 1. The best-fit value, ΔEroam = 161 cm−1, is also consistent with the experimental bounds. Using this value, the relative roaming to dissociation branching ratios are predicted to be similar in D2CO and H2CO, consistent with experimental observation. For NO3, we fix ΔEroam = 258.6 cm−1, the experimental threshold for NO + O2 production, and we model low-temperature experimental branching fractions using the experimental rotational and vibrational temperatures of Trot = 0 K and Tvib = 300 K. The best fit to the experimental data is obtained for Proam = 0.0075, with this very small Proam being consistent with the known geometric constraints to formation of NO + O2. Using Proam = 0.0075, our PST-roaming theory also accurately predicts the low-temperature NO yield spectrum and quantum yield data for roomtemperature NO3 photolysis. For CH3CHO, we fix ΔEroam = 385 cm−1, based on theoretical calculations, and obtain a best-fit value of Proam = 0.21, fitting to reduced dimensional trajectory calculations. These values of ΔEroam and Proam yield PST-roaming theory results that are also consistent with two experimental room-temperature data points. The combination of other kinetic theories and the PST-roaming theory will provide rate coefficients for roaming reactions. trajectories was shown in the QCTs to be “frustrated”; although there is sufficient energy for dissociation, some of the energy is constrained in HCO internal degrees of freedom and not available in the reaction coordinate. In this case, the H atom escapes into the van der Waals (vdW) region, where it undergoes large-amplitude “roaming” around the periphery of the HCO before abstracting the other H atom to yield molecular products. The characteristic fingerprints of the roaming atom mechanism in the QCTs were very low recoil kinetic energy, low angular momentum of the CO fragment, and extraordinarily large vibrational energy in the H2 fragment, exactly as measured in the experiment. Roaming was next inferred10 and then demonstrated11 in the photolysis of acetaldehyde, CH3CHO. In this instance, the roaming moiety was shown to be the methyl (CH3) group,11,12 and therefore, the mechanism became known more generally as just roaming. The same signatures of roaming were observed, small angular momentum and recoil velocity in the CO

1. INTRODUCTION The term “roaming atom” was first coined to describe an unusual mechanism in the photodissociation of formaldehyde, H2CO.1−4 A bimodal rotational distribution in the CO photofragment had been noticed in the early 1990s.5 The high angular momentum component was subsequently assigned to dissociation of H2CO via a high-energy, tight, three-center transition state leading to H2 + CO products.6 However, it was not until ion imaging experiments were performed that the low angular momentum component of the CO rotational distribution was correlated with a very slow recoil kinetic energy.1 These coupled measurements allowed inference of the energy deposited into the H2 cofragment, demonstrating that it was very highly vibrationally excited. Interpretation of these results was accomplished through comparison with quasiclassical trajectory (QCT) calculations performed by Bowman and co-workers on a high-quality ab initio full dimensionality potential energy surface (PES).1 The QCT results confirmed that the “conventional” route to the H2 + CO molecular products occurs over the conventional transition state (TS), which confers high angular momentum and kinetic energy to the molecular products.7−9 The second pathway was found to be associated with the simple bond fission channel that produces H + HCO. A small fraction of the bond fission © 2013 American Chemical Society

Special Issue: Joel M. Bowman Festschrift Received: June 5, 2013 Revised: June 17, 2013 Published: June 17, 2013 7631

dx.doi.org/10.1021/jp405582z | J. Phys. Chem. A 2013, 117, 7631−7642

The Journal of Physical Chemistry A

Article

fragment,10,13,14 while the CH4 fragment was measured to have enormous vibrational excitation.11 Importantly, while roaming accounted for just 10% of the CO + H2 yield in H2CO, it was the dominant mechanism for CH4 + CO production in CH3CHO, accounting for up to 85% of the product yield at 308 nm.11,13 Roaming is now a well-recognized mechanism for the unimolecular dissociation of closed-shell species.15 In addition to H2CO and CH3CHO, roaming has been measured or postulated, experimentally, in the photodissociation of acetone,16 methyl formate,17 and nitrobenzene.18 Roaming has also been observed in shock-tube experiments studying CH3CHO,19 propane,20 and dimethyl ether21 decomposition, indicating that roaming occurs in thermal as well as photochemical reactions. Theoretical calculations suggest that roaming occurs in a variety of alkanes.21 In addition to the wealth of data on the dissociation of closed-shell species, evidence is now accumulating that roaming also occurs in open-shell species, NO322,23 and the 2hydroxyethyl radical.24 Roaming has also been implicated in the isomerization of CH3NO2 to cis-CH3ONO25 and C6H5− NO2 to C6H5−ONO.18 In the case of NO3 → NO + O2, competing roaming mechanisms have been observed and calculated on both the ground and first excited state of NO3,26−28 indicating that roaming in not solely a ground-state phenomenon but may also occur in excited electronic states. Theoretically, roaming seems quite well understood. QCT calculations1,4,11,29−35 and reduced dimensionality trajectories (RDTs)36 agree that the key criterion for roaming is the presence of a barrierless bond dissociation pathway. An effectively barrierless self-reaction (an “intramolecular abstraction”) occurs between the incipient bond dissociation products late in the bond dissociation product channel. This is a common circumstance in organic photochemistry, and as a result, there is a growing consensus that roaming is nearly universal.37,38 The far-reaching outcome of the roaming mechanism is that roaming produces less reactive, more stable products at the expense of more reactive products, typically molecules at the expense of radicals, with resulting consequences in atmospheric and combustion modeling.4,11,19,39−41 Dynamical modeling, like QCT and RDT calculations, requires a significant effort to first build a full or reduced dimensionality PES and then to run a statistically significant number of trajectories. If roaming proves to be as ubiquitous as the rapidly emerging experimental and theoretical data seem to indicate, then a theoretical method is needed to predict the rate coefficient and branching for roaming, in a similar fashion to the way that transition-state theory (TST) and variational TST (VTST) are used for more conventional mechanisms.42,43 Klippenstein and co-workers have taken steps toward this goal, developing a statistical theory of roaming44 based on generalizations of the variable reaction coordinate transitionstate theory approach (VRC-TST).45−47 This theory accurately reproduces the low-energy branching ratios obtained from the trajectory simulations, using a relatively complex TS dividing surface and high-accuracy analytic potentials obtained from fits to multireference ab initio electronic structure calculations. The reliance on an accurate description of the PES, however, means that this theory is not transferable. In this paper, we present a simple, transferable theory of roaming and use it to calculate the branching ratio of roaming reactions, showing how it can be used to calculate rate

coefficients for roaming reactions. The theory is inspired from two sources, first that roaming is integrally associated with a barrierless bond dissociation pathway and that the dissociation products are well described by statistical theories.42,43 Second, on the basis of evidence from QCT calculations, when the incipient bond dissociation products orbit, there is very little energy exchange between them.1,11,12,29−37 Therefore, our theory has two central premises: (i) Roaming is a post-VTST mechanism. This means that the underlying kinetics can be calculated by a VTST calculation of the bond dissociation channel. The total VTST rate is then partitioned into a dissociation fraction and a roaming fraction. (ii) The bond dissociation and roaming fractions are determined by the relative phase space theory (PST)48−50 sum of states for each pathway.

2. PHASE SPACE THEORY (PST) OF ROAMING In the PST-roaming theory, we consider explicitly roaming to be a branching from the barrierless bond dissociation channel. The energetics of the mechanism are illustrated schematically in Figure 1, using a parent aldehyde for illustrative purposes. The barrierless dissociation, in this case to radical products, R + HCO, is indicated by the thick blue curve, with an energy threshold, Ediss, relative to the ground state of the parent. A TS pathway to molecular products is shown as a thin green line for illustrative purposes. Our theory implicitly assumes that the roaming and conventional TS pathways to molecular products are dynamically independent, and as such, the rate coefficients of each process can be calculated separately. In contrast, we assume that the barrierless dissociation and roaming pathways are dynamically linked and that the rate coefficients need to be calculated together. Shepler et al. have recently calculated modest barriers between the roaming and conventional TSs on the H2CO and CH3CHO PESs.51 For the present theory, however, we assume that these two pathways remain kinetically independent. Roaming is illustrated schematically in Figure 1 by the red dotted line, which diverges from the radical dissociation channel at long-range. The roaming threshold, Eroam, lies slightly below Ediss by an amount that we call ΔEroam. When the nascent fragments enter the roaming region, the system ceases to be ergodic in the sense that energy is not exchanged between the two fragments. This divides phase space into two regions. For a fixed total energy, Etot, there will be (i) fragments with a sufficient kinetic energy, Etrans, to escape into

Figure 1. Schematic of the photodissociation of an aldehdye, RCHO, showing the conventional TS pathway as the thin green line, the radical dissociation channel as a thick blue line, and the roaming pathway as the red dashed line. 7632



Article

simple bond dissociation products and (ii) fragments in which the internal energy reservoir is too large, that is, there is too little kinetic energy for the simple products to form. In our theory, these fragments form roaming products. We use PST48−50 to determine the number of states that result in bond dissociation or roaming products. PST is a statistical method for calculating the sum of vibrational and rotational quantum states, wtot(E), as a function of energy, E, and has been demonstrated to model the barrierless radical dissociation channel of many molecules.50−56 Our application of PST simply moves the threshold from Ediss to Eroam. However, we also introduce some assumptions for calculation convenience. Foremost is that we assume that the incipient bond dissociation fragments, in their mutual roaming region (the vdW and/or the long-range attractive part of the PES), behave essentially as separated fragments. We model them as free, rigid rotors. We have not considered centrifugal barriers at this stage, implicitly assuming that a centrifugal barrier will have the same effect on both bond dissociation and roaming states. We have also made the harmonic oscillator assumption for the fragment vibrations. A PST calculation for the roaming and dissociation channels is therefore the same, involving the same rotational constants and vibrational diss frequencies, except that the available energy is Eroam avail not Eavail (see Figure 1). This obviously makes calculation much simpler, and we note that the neglect of vibrational anharmonicity, for example, will likely have a similar affect on both the roaming and dissociative states. We revisit the remaining assumptions later in the paper. In essence, we perform a standard PST calculation at the available energy for roaming, Eroam avail (see Figure 1). All states, including degeneracy, allowed by conservation of energy and angular momentum (excluding the effect of a centrifugal barrier) are calculated and separated into “roaming” states or “dissociative” states according to the criteria

kdiss(E) = fdiss (E) × k VTST(E)

We note, however, that any VTST-type calculation will overestimate kVTST and, hence, both kroam and kdiss. The quality of any predicted rate coefficient depends first on the quality of kVTST and second on the accuracy of the PSTroaming theory. The roaming branching ratio, f roam, will also be an upper limit for three reasons: (i) the roaming rotational sum of states will almost certainly be lower than that calculated using our assumption of noninteracting rigid rotors; rotation is likely hindered rather than free. Moreover, (ii) there may be regions of the 4π steradians of orientational space that are not available for roaming due to steric effects, that is, intramolecular abstraction may require a specific orientation of the roaming fragments. Essentially, this is the same argument as that for the inclusion of an A-factor in Arrhenius kinetics. Finally, (iii) not all roaming states will necessarily lead to roaming products; some may re-form the reactant molecule. This is pertinent as we are performing the state count at finite rather than infinite separation. Indeed, the concept of “recrossing” the TS dividing surface is implicit in VTST and is why TST rate coefficients are upper bounds. We have included all three possibilities into a single, energyindependent parameter, Proam, which can be interpreted as reducing the roaming channel sum of states relative to the dissociative channel sum of states. In this way, Proam becomes somewhat analogous to the Arrhenius A-factor. Equation 3 then becomes froam (E) =

(1)

Dissociative states have: Etrans > ΔEroam

(2)

The predicted branching ratio for roaming, f roam, compared to that of dissociative products, fdiss, is simply the ratio of state counts froam (E) =

wroam(E) wroam(E) + wdiss(E)

(3)

and fdiss (E) =

wdiss(E) wroam(E) + wdiss(E)

(4)

where w is the state count for either roaming or dissociation, as indicated by the “roam” or “diss” subscript. The microcanonical rate coefficient for roaming and dissociation, kroam and kdiss, can then be calculated from the VTST rate coefficient for dissociation, kVTST, (or any other method for calculating this rate coefficient) by scaling the rate coefficient by the fractions k roam(E) = froam (E) × k VTST(E)

Proamwroam(E) Proamwroam(E) + wdiss(E)

(7)

We treat Proam as the only adjustable parameter in the PSTroaming theory and discuss its magnitude in several benchmarking examples below. The spirit of the PST-roaming theory is that it encompasses the key physics of roaming, is simple, is transferable, and does not rely on detailed knowledge of the molecular PES. The mechanism of carrying out a calculation is as follows: (i) ΔEroam is selected from experiment or theory. (ii) A PST calculation is performed at the available energy for roaming, using the internal states (rotational and vibrational) of the two bond dissocation products. (iii) Product states with sufficient translational energy to dissociate go on to dissociative products, and these states are counted. Product states with insufficient translational energy to dissociate may either roam or recombine. These states are counted, and a fraction, given by Proam, are assigned as roaming states. (iv) The roaming rate coefficient is calculated as the VTST rate coefficient multiplied by the fraction of roaming to total states. A more detailed description of the PST-roaming theory, as well as all parameters used in our calculations, can be found in the Supporting Information.

Roaming states have: Etrans ≤ ΔEroam

(6)

3. RESULTS AND DISCUSSION The quantity from the PST-roaming theory that lends itself to comparison with experiment is the branching ratio between the roaming (generally molecular) and barrierless bond dissociation (generally radical) products as a function of excitation energy and/or temperature. Unfortunately, there are few experimental measurements of this quantity. The majority of roaming

(5)

and 7633



Article

experiments have measured the branching of molecular products formed from the roaming pathway compared to that of the conventional TS pathway. Indeed, a bimodal product state distribution in these products has often been the first indication that roaming may be present. While there are many earlier experimental measurements of the branching between bond dissociation and molecular products, these did not distinguish molecular products formed from the TS pathway versus those formed from roaming. Comparison of the branching between roaming and bond dissociation channels then requires a measurement or calculation of the branching between bond dissociation and TS pathways to connect the observed roaming to TS ratio to the required roaming to bond dissociation ratio. There are few systems that satisfy these criteria; we have found three that can be used for comparison with the PST-roaming theory developed above, the photolysis of H2CO/D2CO, NO3, and CH3CHO, which are considered in this order below. A. H2CO and D2CO. The best-characterized roaming reaction is in the photodissociation of H2CO. In the nearultraviolet region of the spectrum, irradiation of H2CO leads to several chemical pathways, in increasing order of threshold energy H 2CO + hν → H 2 + CO (TS channel, 28150 cm−1, ref 57)

Table 1. H2CO Experimental Branching Fractions as a Function of Energya

(R2)

→ H + HCO (dissociation channel, 30327.6 cm−1, ref 58)

(R3)

→ H + HCO (triplet channel, 31450 cm−1, ref 55)

(R4)

→ H + H + CO (triple fragmentation, 35370 cm−1, refs 58, 59)

(R5)

F(R3)

F(R1)

F(R2)

fdiss

F(R2)

ref

30120 30241 31150 31750 32000 32150

0.00 0.00 0.54 0.72 0.75 0.75

1.00 0.82 0.31 0.18 0.15 0.15

0.00 0.18 0.10 0.10 0.10 0.10

0.00 0.783 0.878 0.882 0.882

1.00 0.217 0.122 0.118 0.118

3 3 4 4 3 4

a

Fractions: F represents overall branching fractions; f, represents relative branching fractions (see text).

froam =

F(R2) F(R2) + F(R3)

(8)

and fdiss =

F(R3) F(R2) + F(R3)

(9)

for direct comparison with the roaming and radical dissociation fractions in eqs 3 and 4. The PST-roaming theory requires the spectroscopic constants of the radical products. The vibrational and rotational constants of HCO are well-known and are given in the Supporting Information. The two roaming parameters are ΔEroam and Proam. As reported above, ΔEroam = 87−205 cm−1, and we choose the midpoint of this range, ΔEroam = 146 cm−1, and choose, initially, Proam = 1. The roaming to radical dissociation branching fractions predicted by the PST-roaming theory, with no adjustable parameters, are shown in Figure 2, in comparison with the available experimental data from Table 1, where the conventional TS channel, reaction R1, is not represented in Figure 2. The theoretical results pass comfortably through the experimental data. To test the sensitivity of the two roaming parameters, each was adjusted in turn and compared with the experimental data. The global best fit was for ΔEroam = 161 cm−1 and Proam = 0.99. Varying each in turn until the root-mean square (RMS) error was 50% worse provided a range of ΔEroam = 132−196 cm−1. Both upper and lower bounds lie within the experimental limits for ΔEroam. Using the same approach, the range of Proam was determined to be Proam = 0.75−1.0. The results for the PST-roaming theory in Figure 2 show distinct structure, on both a high- and a low-frequency scale. The broad structure arises from different vibrational states of the HCO product as they become energetically available. The roaming fraction increases as each vibrational level opens due to the increased number of rovibrational states available. The same HCO vibrational level becomes “available” to the radical dissociation sum of states somewhat higher in energy (indeed higher by ΔEroam). The fine structure in the branching fractions arises as each rotational level becomes available. The roaming, wroam, and radical dissociation, wdiss, sums of states are shown explicitly in Figure 3 as a function of the energy available to the dissociation channel. The inset to Figure 3 shows wroam and wdiss near the dissociation threshold with higher resolution. Below the dissociation threshold, only states that may roam will be enumerated; in Figure 3, for Ediss avail less than zero, wdiss is zero. In this case, all states counted have translational energy less than ΔEroam and are therefore potential roaming states. At energies above but near this threshold, the roaming channel still dominates, as shown in the inset to Figure

(R1)

→ H 2 + CO (roaming channel, 30180 cm−1, ref 3)

Etot/cm−1

Roaming, reaction R2, has been explored in H2CO from below the dissociation threshold up to the energy for triple fragmentation, reaction R5.1−4,30 Experimentally, the threshold for roaming has been established to lie between 30123 and 30241 cm−1, that is, 87 < ΔEroam < 205 cm−1.3 The most ready experimental observable is the fraction of roaming to TS production of CO (reactions R1 and R2) because the different product state distributions for each mechanism make it easy to distinguish and measure CO produced via each pathway. The PST-roaming theory, however, predicts the relative yield of reactions R2 and R3. The experiment and theory can be connected by measurement or calculation of the yield of reaction R1 to R3, which has been carried out previously.3,4 Experimental data for photolysis in the range of 30150−32000 cm−1, combined with QCT calculations,3 show that the overall branching fraction for reaction R1, F(R1), decreases with energy from 0.82 to 0.15, while the yield of reaction R3, F(R3), increases from 0.0 to 0.75. The total branching fraction for reaction R2, F(R2), decreases from 0.18 to 0.10. These data are reproduced in Table 1, which has been adapted from refs 3 and 4. In all cases through this paper, we reserve lower case f for the roaming to dissociation fraction as defined in eqs 3 and 4 while using upper case F for other branching fractions that we adapt from the literature. The radical dissociation and roaming branching fractions, f, are also shown in the table, defined as the ratio of the individual branching fractions, F 7634



Article

Figure 3. The H2CO roaming and radical dissociation sums of states, wroam (red solid line) and wdiss (blue dashed line), as a function of available energy for the radical dissociation channel, for ΔEroam = 146 cm−1. (Inset) wroam and wdiss near the radical dissociation threshold (30327.6 cm−1).

Figure 2. Relative roaming (red line) and radical dissociation (blue line) branching fractions for H2CO predicted by the PST-roaming theory, with ΔEroam = 146 cm−1 and Proam = 1. The literature radical dissociation and roaming branching fractions (blue and red circles, respectively, see Table 1) are also shown. The energies at which the HCO vibrational levels become accessible for radical dissociation and roaming states are indicated by the blue and red combs, respectively.

3. The roaming sum of states, wroam, however, rises more slowly with energy, and wdiss increasingly dominates with increasing energy. This is further illustrated in Figure 4, where the state count is plotted schematically for a number of available energies as a function of translational energy, Etrans. The roaming branching fractions, f roam and fdiss, are proportional, respectively, to the ratio of the red and blue shaded areas shown in Figure 4. To illustrate the effect of increasing the available energy on f roam, the y-axis has been normalized for Etrans = 0. We have shown schematic data to avoid any confusion arising from the discrete nature of the actual state counts. In panel (a) of Figure 4, for diss Eroam avail = ΔEroam (that is, Eavail = 0), only potential roaming states are present. As the available energy increases above ΔEroam, panels (b) and (c), wdiss increases rapidly. There is a more modest increase in wroam, and therefore, f roam decreases with increasing available energy. Imaging experiments, together with QCT simulations, have also been used to explore roaming in deuterated formaldehyde, D2CO.34 A lower bound for the total roaming branching ratio of 15% in D2CO was determined at a total energy of 31 348 cm−1 (an energy of 328 cm−1 relative to the dissociation threshold). The uncertainty arises from background interference that limited the range of CO rotational levels that could be experimentally probed for in D2CO.34 This value is similar to the 18% observed in H2CO at a comparable energy,2 and the authors concluded that the increased mass of the “roamer” in D2CO does not inhibit roaming. The roaming to dissociation relative branching fractions predicted by the PST-roaming theory for D2CO are shown overlapping the H2CO data in Figure 5. The energetic and spectroscopic parameters used are given in the Supporting Information, while the roaming parameters were kept at the best-fit parameters for H2CO, Proam = 0.99 and ΔEroam = 161 cm−1. The results for H2CO and D2CO are very similar; there are slight differences in structure because all three vibrational levels become accessible at lower energy in D2CO than those in

Figure 4. Schematic representation of the PST state count as a function of translational energy, Etrans, for three representative available diss roam energies, (a) Eroam avail = ΔEroam (that is, Eavail = 0), (b) Eavail = 2ΔEroam, = 4ΔE . Potential roaming states are indicated by the and (c) Eroam avail roam fine hatched area, while the dissociative states are more coarsely hatched in blue. The state counts at zero translational energy have been normalized.

H2CO. Overall, however, the PST-roaming theory predicts that once the difference in radical dissociation energies has been accounted for, the relative roaming to dissociation branching ratios will be similar in D2CO and H2CO, which is consistent with the experimental observation. 7635



Article

the relevant thermal distribution of the parent proves to be an essential addition to the PST-roaming theory. The method by which we implement a thermal distribution favors simplicity over spectroscopic accuracy. The code for the PST-roaming theory loops over the thermally accessible rotational and vibrational states of the parent and evaluates wdiss and wroam at a total energy, Etot, that is Etot = E(λ) + Evib + Erot

where the vibrational and rotational energies, Evib and Erot, are calculated from the NO3 harmonic oscillator and rigid rotor energy levels using the spectroscopic constants given in the Supporting Information. We recognize that this is an expedient approximation in that a spectroscopically accurate treatment would include the appropriate P, Q, and R rotational transitions (we are effectively treating only Q-type transitions) as well as anharmonic corrections to the vibrational hot band transitions. At each total energy, Etot, wroam and wdiss are weighted by the relevant thermal population of the NO3 parent. In the current calculations, we use a Boltzmann distribution but allow different rotational and vibrational temperatures in order to model the experimental conditions. More details of the Boltzmann averaging and its implementation, including conversion between quantum yield fractions and branching fractions, can be found in the Supporting Information, as well as the appropriate constants for the NO2 + O dissociation. Photofragmentation translational spectroscopy experiments show a roaming fraction (that is, the NO + O2 channel) of 0.7 ± 0.1 at 17007 cm−1, dropping rapidly to less than 0.05 at 17123 cm−1 and less than 0.01 by 17153 cm−1.61 These data are the discrete points plotted in Figure 6. We have performed PST-roaming calculations to compare with these data, as shown in the figure. In each case, we have set ΔEroam = 258 cm−1 as determined experimentally. The roaming probability, Proam, has been treated as an adjustable parameter. The PST-roaming calculation shown as the blue dotted line in Figure 6 is the pure PST-roaming calculation for a parent molecule in its ground rovibrational state. To fit the rapid drop in roaming fraction exhibited by the experimental data, we need to use a very low value of Proam. The best fit to the data provides Proam = 0.0075 (vide inf ra for discussion of the magnitude of Proam). However, the PST-roaming theory must also predict f roam =1 for all energies between the roaming and bond dissociation thresholds, where dissociation is not energetically possible. The solid red line in Figure 6 shows the PST-roaming calculation for the reported experimental conditions of Tvib = 300 K and Trot ≈ 0K.61 Although Proam was refitted to the data, its magnitude did not change. The only effect on the f roam distribution is to lower f roam to about 0.67 in the region between the dissociation and roaming thresholds, in agreement with the experimental result. The PST-roaming result might appear unusual in that a 300 K vibrational temperature does not affect the roaming fraction above the bond dissociation threshold and that it produces only a constant reduction in f roam below this threshold. However, this is a result of the unusual experimental conditions, a room-temperature vibrational distribution but an effectively 0 K rotational distribution. The lowest vibrational frequency in NO3 is the umbrella vibration with ν = 363 cm−1. At 300 K, this is the only excited state with any significant population, and the ground state retains 67% of the total vibrational population. Molecules in excited states are excited to a total energy of the photon energy

Figure 5. Relative roaming branching fractions for H2CO (red line) and D2CO (blue dashed line) predicted by the PST-roaming theory as a function of energy above the radical dissociation threshold (30328.5 cm−1 for H2CO and 31020 cm−1 for D2CO). ΔEroam = 161 cm−1 and Proam = 0.99 were used in both cases.

B. NO3. The second benchmark system that we have chosen is the photolysis of NO3 to yield NO2 + O(3P) (the dissociation channel) or NO + O2 (the roaming channel), with energy thresholds as indicated NO3 + hν → NO + O2 (roaming channel, 16821 ± 14 cm−1, ref 60) → NO2 + O(3P) (dissociation channel, 17079 ± 14 cm−1, ref 60)

(10)

(R6) (R7)

Because of its atmospheric relevance, there is a wealth of experimental data for NO3 photodissociation. The thresholds above come from a review of previous data61−65 by Johnstone, Davis, and Lee (JDL).60 Recently, the NO + O2 products have been determined to arise solely from roaming;22,23,27,28 theoretical calculations agree that there is no accessible TS at the energy of the experiments.26,27 This provides a significant simplification for our analysis because reactions R6 and R7 become the only two chemical pathways in this energy window. The experimentally determined thresholds above therefore provide ΔEroam = 258 ± 20 cm−1. In modeling the yields of reactions R6 and R7, there are three different types of experiments to which we can benchmark our PST-roaming theory, (i) collision-free, molecular beam measurements of the roaming fraction via photofragment translational spectroscopy,61 (ii) room-temperature quantum yields of both channels,66,67 and (iii) a photofragment (NO) yield spectrum of jet-cooled NO3 through this wavelength region.68 Unlike the formaldehyde experiments described above, where the rotational and vibrational energy of the parent is precisely (spectroscopically) defined, all of the NO3 experiments are carried out from parent molecules with a range of internal energies. Even the molecular beam experiments suffer from incomplete cooling in the expansion, characterized by the authors as Tvib = 300 K and Trot ≈ 0 K,61 and the inclusion of 7636



Article

Figure 6. Roaming branching fraction for NO3 predicted by the PSTroaming theory, with ΔEroam = 258 cm−1 and Proam = 0.0075. The calculations were carried out at 0 K and at the experimental temperature, Tvib = 300 K.61

Figure 7. Quantum yields for room-temperature NO3 photolysis. The symbols are the average and standard deviation of data by Magnotta and Johnston. (MJ, ref 66). The dashed lines are recommended by Orlando et al. from their own experimental data. (OTMC, ref 67). The bold lines are the predictions of PST-roaming theory using ΔEroam = 258 cm−1 and Proam = 0.0075 as determined previously. The data are plotted using the lower abscissa. The wavelength scale on the upper axis is nonlinear and included for reference only.

plus its initial internal energy. In NO3, the roaming fraction drops to zero within ∼50 cm−1 of the dissociation threshold; therefore, excited vibrational states do not produce roaming at this excitation energy. The effect of introducing a nonzero vibrational temperature is simply to reduce the roaming fraction for excitation between the roaming and bond dissociation thresholds. Quantum yields for reactions R6 and R7 at room temperature have been reported by many groups due to the atmospheric importance of this reaction. The two most extensive sets of data are due to the work of Magnotta and Johnston (MJ, ref 66) and Orlando et al. (OTMC, ref 67), which are in reasonable agreement, as summarized in Figure 7. Here, we have shown the average and range of the MJ data as discrete points with error bars. OTMC provided a recommended set of quantum yields, and these recommendations are shown as dashed lines in the figure. Figure 7 also shows the results of the PST-roaming calculations for the bond dissociation (R6) and roaming (R7) channels for Tvib = Trot = 298 K and using the same values of ΔEroam = 258 cm−1 and Proam = 0.0075 as derived from the molecular beam data above. The calculations represent the experimental data well. They fall generally within the error bars of the MJ data but somewhat overpredict the roaming quantum yield, ϕroam, and underpredict the dissociation quantum yield, ϕdiss, compared with the recommendations of OTMC. We stress, however, that the calculations for Figure 7 had no adjustable parameters. The final NO3 experiments that we can compare to the PSTroaming theory are NO yield (photofragment excitation) spectra from Wittig and co-workers.68 The intensity in a phofex spectrum arises from the product of the absorption coefficient and product yield at the relevant wavelength. Wittig and coworkers measured phofex spectra under several sets of molecular beam conditions and pump−probe delay times. The spectrum reproduced in Figure 8 corresponds to their Figure 7 (bottom) at “long delay” to allow complete reaction and to molecular beam conditions of Trot = 25 K and Tvib = 0 K. Overlaid on the experimental spectrum are three simulations based on the PST-roaming theory using Proam = 0.0075 as

Figure 8. NO yield spectrum of NO3 at Trot = 25 K, Tvib = 0 K (black line, adapted from Figure 7 of ref 68), overlaid with NO yield predictions obtained by multiplying the 230 K NO3 absorption spectrum (obtained from Figure 11 of ref 61) with scaled ϕroam predictions from the PST-roaming theory, with Proam = 0.0075 and ΔEroam = 238, 258, and 278 cm−1.

before and using three values of ΔEroam = 258 ± 20 cm−1, corresponding to the extent of experimental uncertainty. To calculate these simulations, we evaluated ϕroam for the reported experimental conditions and multiplied this by the reported NO3 absorption strength at 230 K.69 Although this does not correspond to the experimental conditions, it is the coldest reported NO3 absorption spectrum and the same one that Wittig and co-workers used to compare with their spectra.68 The model spectra are in very good agreement with the 7637



Article

experiment. At all but the lowest energy, the agreement is quantitative. At the lower energy, the simulation dies faster than the experiment for the ΔEroam = 258 cm−1 but is in reasonable agreement with ΔEroam = 238 cm−1, within experimental uncertainty. It is worth noting that NO3 is an extremely complicated system. At energies close to the bond dissociation threshold, vibronic levels of NO3 are strong mixtures of the electronically excited B̃ (2E′) state and high rovibrational states on the electronic ground X̃ (2A2′) state.26−28 Moreover, roaming occurs on both the X̃ state and the optically-dark first excited Ã (2E″) state.26−28 In the roaming region, the X̃ and Ã state PESs converge, with a conical intersection just before oxygen abstraction.26 Our theory does not distinguish between roaming mechanisms on the two electronic states. As an effectively asymptotic model, the PST-roaming theory considers NO2 + O rovibrational states at quasi-infinite separation, where energy cannot be transferred from NO2 to O. The theory therefore predicts the total roaming fraction from both electronic states, and a single ΔEroam is used to model both roaming mechanisms. In summary, the PST-roaming calculations are successful in reproducing three completely different types of NO3 photodissociation experiments. ΔEroam = 258 cm−1 was fixed throughout at the experimental value, while Proam was modeled in the first set of experiments and left unchanged to model the other two sets. C. CH3CHO. Finally, we apply the PST-roaming theory to the photolysis of acetaldehyde, CH3CHO. Irradiated in the near-ultraviolet region of the spectrum (λ < 300 nm), CH3CHO is known to have several photochemical and photophysical pathways, which are shown below in order of increasing threshold energy CH3CHO + hν → CH4 + CO (roaming channel, 28770 cm−1) → CH4 + CO (TS channel, 29120 cm−1)

Figure 9. Roaming branching fractions for CH3CHO predicted by the PST-roaming theory as a function of energy. ΔEroam was set to 385 cm−1 as determined by previous ab initio calculations.15,36 The PSTroaming result was fit to RDT results of HGK,36 yielding Proam = 0.21. The solid red curve shows the PST-roaming theory results, for the same parameters, at Trot = Tvib = 298 K. In comparison, the roomtemperature results of Horowitz and Calvert (HC, red squares), which are quantum yields for molecular products, Fmolec, represent an experimental upper limit for f roam (see text).80

In the absence of directly comparable experimental data, the RDT results of HGK provide a theoretical benchmark with which we can compare the PST-roaming theory. Harding and co-workers conducted an extensive search for a “roaming saddle point” and concluded that such a saddle point exists with an energy that lies 1.0 ± 0.2 kcal/mol below the radical dissociation limit (the error range represents the range of values for different theoretical approaches).15 In the absence of experimental data about the roaming threshold, we use ΔEroam = 385 cm−1 (1.1 kcal/mol), as used by HGK. Figure 9 shows the results of the PST-roaming calculation fit to the HGK data, yielding Proam = 0.21 as the best fit. The PSTroaming theory results drop more rapidly than those of HGK once the radical dissociation threshold is attained but are more persistent at higher energy. Nonetheless, the dependence of f roam in the PST-roam calculation is quite similar to that of the RDT calculations. The method of deriving a roaming rate coefficient from full-dimensional QCT or RDT methods remains challenging; therefore, it is difficult to comment on the correct functional form of f roam for CH3CHO other than to say that the two methods provide a similar result, recognizing that the PST-roaming theory has one adjustable parameter while the more time-consuming RDT or QCT methods are parameter-free. Although there are no directly comparable experimental data for the radical dissociation/roaming branching ratio, there are several experimental measurements of room-temperature acetaldehyde quantum yields that can be usefully compared with theory.79−81 In comparing with room-temperature data, we restrict our comparison to wavelengths λ > 320 nm (E < 31 090 cm−1) so that the production of radical products on the triplet surface (reaction R12) is excluded and the radical dissociation yield can be assumed to arise from reaction on S0. In this range, there are two experimental measurements of the radical and molecular quantum yields, at 320 and 334 nm,

(R8) (R9)

→ CH3 + HCO (radical dissociation channel, 29120 cm−1)

(R10)

→ CH3CO + H (acetyl channel, 301710 cm−1)

(R11)

→ CH3 + HCO (triplet channel, 31090 cm−1)

(R12)

where the threshold energies were evaluated in two previous papers70,71 from a variety of original references.72−78 Although roaming has been long-established in this system,10−15 branching ratios have only been published for roaming versus TS production of CH4 + CO (reaction R8 versus R9), rather than roaming versus bond dissociation products (reaction R8 versus R10), and even these experimental ratios show large variation. It is therefore difficult to compare our results directly with these molecular beam experiments. Roaming in CH3CHO photolysis has also been investigated by quasi-classical trajectories by Bowman and co-workers11,12,35 and using RDTs by Harding et al. (HGK).36 The HGK paper reports f roam, defined in the same way as here, as a function of energy above the radical dissociation threshold. Their values are reproduced in Figure 9. 7638



Article

data with the PST-roaming theory may enable an estimation of the roaming threshold. E. Meaning and Interpretation of Proam. The underlying assumption of the PST-roaming theory is that the roaming states resemble those of the infinitely separated bond dissociation products, and the sum of roaming states can be calculated using the rotational constants and vibrational frequencies of these products. This assumption provides an upper limit to the roaming sum of states; if roaming is restricted in any way, the sum of states will be lower, for example, if rotation of the roaming fragments is hindered. Similarly, if roaming is sterically limited to a volume smaller than 4π stearadians, unsuccessful roaming states will evolve back to parent states, reducing the effective roaming sum of states. All of these effects act to lower the roaming fraction relative to that calculated using PST for the free products. Hence, the unadulterated PST-roaming rate, eq 5, will be an upper limit. These issues are commonplace in various kinetic theories. For example, RRKM theory can employ free or hindered rotors, and minimization of the recrossing of the TS dividing surface is implicit in VTST.42,43 In trajectory calculations on CH3CHO, both Klippenstein and co-workers36 and Bowman and coworkers29 identify many putative roaming trajectories that return to reactants. For the present, we combine all of these effects in a single, energy-independent parameter, Proam, which serves to reduce the roaming sum of states. We treat Proam as an empirical fitting parameter. In the examples above, Proam was found to vary from 0.0075 for NO + O2 to 0.21 for CH4 + CO to 1.0 for H2 + CO. In H2CO, a single value, Proam = 1, proved sufficient to fit the experimental data across the entire photolysis energy range, 3000 cm−1. This implies that the sum of roaming states, wroam, is at its maximum value. In our interpretation, we would reason that because both the roaming fragment, H•, and the “target” formyl H atom are spherically symmetric, there is no geometrical constraint on the intramolecular abstraction. Proam = 1 also implies that the HCO radical behaves essentially as a free particle. This interpretation is supported by Bowman and co-workers’ QCT calculations of the H2 + CO roaming channel.37 The QCTs reveal that there is no correlation between the CO angular momentum vector and the recoil velocity vector. Also, the HCO was found to rotate about any of its inertial axes during the roaming event. These observations imply that the H atom can roam anywhere in 4π steradians about the HCO core, approaching the target formyl H atom in any orientation. Intuitively, one would expect Proam to be relatively high; indeed, the data for H2CO are fit well with Proam = 1. In the acetaldehyde roaming reaction, Proam = 0.21 was obtained as a fit to the HGK theoretical results. This reaction has also been studied by both full-dimensional and reduceddimensional trajectories.11,12,35−37 Both experiment10 and theory37 reveal no vector correlation between the CO rotational vector and the recoil velocity vector of the departing CO and CH4 fragments. This implies that the CH3 moiety roams about an essentially free rotor HCO core. The RDT calculations show that there is a well-defined orientation required by the CH3 moiety to abstract the formyl H atom; the effectively planar CH3 group must orient such that its pz orbital can interact with the formyl H atom.36 This steric inhibition of the roaming reaction will reduce the roaming sum of states, and hence increase the number of potentially roaming states reforming reactants (recrossing). This is consistent with our

Table 2. CH3CHO Experimental Quantum Yields of Horowitz and Calvert79 and Derived Branching Fractions (at energies below the threshold for triplet reaction R12) energy (cm−1)

ϕR10(CH3 + HCO)a

ϕR8 + ϕR9(CH4 + CO)a

Fmolecb

30193 31250

0.057 0.47

0.035 0.079

0.38 0.14

a

Extrapolated to zero pressure. bFmolec is an upper limit to f roam (see text).

which were extrapolated to zero pressure in the original paper (see Table 2). The two molecular CH4 + CO mechanisms (reactions R8 and R9) were not distinguished at that time. The data points on Figure 9 are the yields for molecular products, Fmolec = (ϕR8 + ϕR9)/(ϕR8 + ϕR9 + ϕR10), and therefore represent an upper limit of the roaming yields plotted in the figure, which are f roam = ϕR8/(ϕR8 + ϕR10). The red line in Figure 9 shows results from the PST-roaming theory (omitting the TS channel, reaction R9) using a parent CH3CHO temperature of 298 K and the same values of ΔEroam and Proam used to fit the RDT data. Similarly to the NO3 modeling, the addition of a Boltzmann distribution of energy to the parent molecule results in the roaming branching fraction being less than 1 below Ediss and allows it to fall off more slowly than the 0 K results. The PST-roaming theory passes close to the 320 nm datum and below the 334 nm datum, which is consistent with the experimental values being upper limits. In H2CO, roaming increases in importance, compared to the TS mechanism, for increasing energy (see Table 1). This is the same trend as that observed in Figure 9 where TS production of CO would seem to be more important at 334 nm but roaming more important at 320 nm. Indeed, it has been argued previously that roaming production of molecular products (reaction R8) should exceed the production of molecular products via the TS pathway (reaction R9) by up to an order of magnitude at 308 nm,36,71,82 and therefore, the close agreement between the PST-roaming theory and the experimental upper limits also seems reasonable. D. Meaning and Interpretation of ΔEroam. There are two extrinsic parameters in the PST-roaming theory, and we interpret these in a similar fashion to many other kinetic models. ΔEroam is the critical energy for the appearance of roaming, measured relative to the critical energy for the commensurate bond dissociation channel. This can be determined experimentally as an “appearance threshold”, such as the case for H2CO and NO3 above. Where there is no experimental measurement of ΔEroam, such as in CH3CHO and several other roaming systems [acetone, CH 3 NO 2 , CH3OCHO], calculation of ΔEroam is still problematic at the present time, although there are active theoretical efforts toward this goal,15,36 and we used one such calculation in modeling roaming in CH3CHO. Where there are neither experimental nor theoretical values to employ, ΔEroam can be treated as an adjustable parameter in the PST-roaming theory. We have used this approach in modeling H2CO and NO3, fitting ΔEroam and Proam to the experimental data. For H2CO, this gave ΔEroam = 161 cm−1, still within the experimental range of 87−205 cm−1. When allowed to float in the NO3 calculations, the global best fit yielded ΔEroam = 238 cm−1, again within the experimental range of 258 ± 20 cm−1. Indeed, our results for H2CO and NO3, where there are good-quality experimental data, suggest that modeling the 7639



Article

interpretation of Proam and a value of Proam significantly less than 1. The lowest value of Proam obtained was for roaming in NO3 photolysis to yield NO + O2. Here, Proam is very small compared to the other two systems. Roaming in NO3, however, is much more unusual than that in the two prototypical aldehydes. First, roaming occurs in two electronic states.22,23,26−28 Initially NO3 is excited to the B̃ electronic state before relaxing to the excited Ã state. Roaming can occur on the Ã state, but a second electronic curve crossing, in the bond dissociation channel, from the Ã state to the to the electronic ground state, X̃ , is also possible. Roaming, reformation of NO3, and bond dissociation can then all occur on the X̃ state. Therefore, recrossing is a recognized feature of roaming in this system. Additionally, there is a strong Lambdadoublet propensity observed for roaming on both the X̃ and Ã states.27,28 A preference for one Lambda doublet over the other can only occur if the intramolecular abstraction is restricted to in-plane, an interpretation that is supported by theory.26−28 Both of these effects, limiting roaming to in-plane and significant recrossing into the reactant well, will reduce Proam. Our interpretation of the fitted Proam value of 0.0075 is that the reduction in available roaming states in NO3 is very significant; less than 1% of the possible free rotor NO2/isotropic O atom roaming states actually lead to roaming reactions.

and NO3) or to the theoretical value (CH3CHO). When ΔEroam and Proam were allowed to float freely, the global best fit returned values of ΔEroam that remained inside the experimental range. This suggests that the PST-roaming theory could be used to estimate the roaming threshold from experimental roaming fractions, even where the experiments lie above the roaming threshold. The fitted values of Proam ranged from 1 in H2CO, to 0.21 for CH3CHO, to 0.0075 for NO3. Proam = 1 in H2CO suggests that there is little or no hindrance of the H atom in roaming about the HCO core and that the HCO core behaves as a free rotor. The drop in Proam in CH3CHO can be understood as a reduction in the number of states of the CH3 moiety roaming about the HCO core that have the correct steric orientation to abstract the formyl H atom. The very small value of Proam in NO3 is interpreted in terms of the constraint for in-plane roaming and significant recrossing into the NO3 reactant well. The PST-roaming theory was developed as a way to calculate branching ratios without the need for extensive calculations of the PES. When combined with conventional kinetic theories, such as RRKM, TST, VTST, or VRC-TST, the PST-roaming theory will provide rate coefficients for roaming as a function of energy or temperature.

4. SUMMARY AND CONCLUSIONS We have described a simple theory of roaming reactions. The theory assumes that roaming branches from a barrierless bond dissociation channel at large interfragment separations, af ter the variational transition state for the dissociation process. We assume that all incipient bond dissociation products that have enough translational energy to dissociate do dissociate. Those that do not have enough translational energy may form roaming products or return to the reactant well. The relative sum of states of the bond dissociation and roaming pathways was calculated by PST. PST requires rotational constants and vibrational frequencies for the bond dissociation products, which are generally available from spectroscopic data, and these are used to calculate the bond dissociation and roaming sum of states. The PST-roaming theory requires two further parameters be determined or fit. ΔEroam is the energetic threshold for roaming, measured as the energy below the bond dissociation threshold. This parameter can be measured, or at least bracketed, experimentally, and progress is being made toward being able to calculate it theoretically. Proam is a scaling factor that encapsulates the limitations in the theory. First, the theory calculates the rovibrational states of two free particles, and this state count will be an overestimation if the fragments are hindered or if there is a steric restriction to the roaming reaction. Second, the theory assumes that all potential roaming states do roam rather than “recross” back to the reactant well. Both effects reduce the roaming sum of states and are combined into a single, energyindependent, scaling factor that can vary between 0 and 1. The PST-roaming theory was applied to the roaming reactions in H2CO, CH3CHO, and NO3. The theory quantitatively reproduced the branching between bond dissociation and roaming products for all reaction energies that have been experimentally measured for H2CO and NO3. It also gave reasonable agreement with the energy dependence of roaming in CH3CHO, as calculated using RDTs. ΔEroam was initially fixed to the center of the experimental range (H2CO

The parameters used in the PST-roaming theory as well as a detailed description of the PST-roaming theory and the Boltzmann calculations used to incorporate the parent internal energy. This material is available free of charge via the Internet at http://pubs.acs.org.

■

ASSOCIATED CONTENT

S Supporting Information *

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.J.T.J.); scott.kable@ sydney.edu.au (S.H.K.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was funded by the Australian Research Council (Grant DP1094559). Parts of this research were undertaken on the NCI National Computational Infrastructure Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government.

■

REFERENCES

(1) Townsend, D.; Lahankar, S. A.; Lee, S. K.; Chambreau, S. D.; Suits, A. G.; Zhang, X.; Rheinecker, J.; Harding, L. B.; Bowman, J. M. The Roaming Atom: Straying from the Reaction Path in Formaldehyde Decomposition. Science 2004, 306, 1158−1161. (2) Lahankar, S. A.; Chambreau, S. D.; Zhang, X.; Bowman, J. M.; Suits, A. G. Energy Dependence of the Roaming Atom Pathway in Formaldehyde Decomposition. J. Chem. Phys. 2007, 126, 044314. (3) Lahankar, S. A.; Goncharov, V.; Suits, F.; Farnum, J.; Bowman, J. M.; Suits, A. G. Further Aspects of the Roaming Mechanism in Formaldehyde Dissociation. Chem. Phys. 2008, 347, 288−299. (4) Suits, A. G. Roaming Atoms and Radicals: A New Mechanism in Molecular Dissociation. Acc. Chem. Res. 2008, 41, 873−881. (5) van Zee, R. D.; Foltz, M. F.; Moore, C. B. Evidence for a Second Molecular Channel in the Fragmentation of Formaldehyde. J. Chem. Phys. 1993, 99, 1664−1673. (6) Bamford, D. J.; Filseth, S. V.; Foltz, M. F.; Hepburn, J. W.; Moore, C. B Photofragmentation Dynamics of Formaldehyde: 7640



Article

CO(v,J) Distributions as a Function of Initial Rovibronic State and Isotopic Substitution. J. Chem. Phys. 1985, 82, 3032−3041. (7) Busch, G. E.; Wilson, K. R. Triatomic Photofragment Spectra. I. Energy Partitioning in NO2 Photodissociation. J. Chem. Phys. 1972, 56, 3626−3638. (8) Korsch, H. Vibrational State Distributions for Impulsive Triatomic Reactions. Chem. Phys. 1978, 33, 313−318. (9) Dugan, C. H. The Impulse Approximation in Photofragmentation: How Accurate an Estimate of Fragment Rotation? J. Phys. Chem. 1991, 95, 3477−3480. (10) Houston, P. L.; Kable, S. H. Photodissociation of Acetaldehyde as a Second Example of the Roaming Mechanism. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 16079−16082. (11) Heazlewood, B. R.; Jordan, M. J. T.; Kable, S. H.; Selby, T. M.; Osborn, D. L.; Shepler, B. C.; Braams, B. J.; Bowman, J. M. Roaming is the Dominant Mechanism for Molecular Products in Acetaldehyde Photodissociation. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 12719− 12724. (12) Shepler, B. C.; Braams, B. J.; Bowman, J. M. Quasiclassical Trajectory Calculations of Acetaldehyde Dissociation on a Global Potential Energy Surface Indicate Significant Non-Transition State Dynamics. J. Phys. Chem. A 2007, 111, 8282−8285. (13) Amaral, G. A.; Arregui, A.; Rubio-Lago, L.; Rodríguez, J. D.; Bañares, L. Imaging the Radical Channel in Acetaldehyde Photodissociation: Competing Mechanisms at Energies Close to the Triplet Exit Barrier. J. Chem. Phys. 2010, 133, 064303. (14) Rubio-Lago, L.; Amaral, G. A.; Arregui, A.; González-Vázquez, J.; Bañares, L. Imaging the Molecular Channel in Acetaldehyde Photodissociation: Roaming and Transition State Mechanisms. Phys. Chem. Chem. Phys. 2012, 14, 6067−6078. (15) Harding, L. B.; Klippenstein, S. J.; Jasper, A. W. Synergies between Experimental and Theoretical Studies of Gas Phase Reactions. Phys. Chem. Chem. Phys. 2007, 9, 4055−4070. (16) Goncharov, V.; Herath, N.; Suits, A. G. Roaming Dynamics in Acetone Dissociation. J. Phys. Chem. A 2008, 112, 9423−9428. (17) Chao, M. H.; Tsai, P. Y.; Lin, K. C. Molecular Elimination of Methyl Formate in Photolysis at 234 nm: Roaming vs. Transition State-Type Mechanism. Phys. Chem. Chem. Phys. 2011, 13, 7154− 7161. (18) Hause, M. L.; Herath, N.; Zhu, R.; Lin, M. C.; Suits, A. G. Roaming-Mediated Isomerization in the Photodissociation of Nitrobenzene. Nat. Chem. 2011, 3, 932−937. (19) Sivaramakrishnan, R.; Michael, J. V.; Klippenstein, S. J. Direct Observation of Roaming Radicals in the Thermal Decomposition of Acetaldehyde. J. Phys. Chem. A 2010, 114, 755−764. (20) Sivaramakrishnan, R.; Su, M. C.; Michael, J. V.; Klippenstein, S. J.; Harding, L. B.; Ruscic, B. Shock Tube and Theoretical Studies on the Thermal Decomposition of Propane: Evidence for a Roaming Radical Channel. J. Phys. Chem. A 2011, 115, 3366−3379. (21) Harding, L. B.; Klippenstein, S. J. Roaming Radical Pathways for the Decomposition of Alkanes. J. Phys. Chem. Lett. 2010, 1, 3016− 3020. (22) Grubb, M. P.; Warter, M. L.; Suits, A. G.; North, S. W. Evidence of Roaming Dynamics and Multiple Channels for Molecular Elimination in NO3 Photolysis. J. Phys. Chem. Lett. 2010, 1, 2455− 2458. (23) Grubb, M. P.; Warter, M. L.; Johnson, K. M.; North, S. W. Ion Imaging Study of NO3 Radical Photodissociation Dynamics: Characterization of Multiple Reaction Pathways. J. Phys. Chem. A 2011, 115, 3218−3226. (24) Kamarchik, E.; Koziol, L.; Reisler, H.; Bowman, J. M.; Krylov, A. I. Roaming Pathway Leading to Unexpected Water + Vinyl Products in C2H4OH Dissociation. J. Phys. Chem. Lett. 2010, 1, 3058−3065. (25) Zhu, R. S.; Lin, M. C. Decomposition/Isomerization Mechanism and Product Branching Ratios: An Ab Initio Chemical Kinetic Study. Chem. Phys. Lett. 2009, 478, 11−16. (26) Xiao, H.; Maeda, S.; Morokuma, K. Excited-State Roaming Dynamics in Photolysis of a Nitrate Radical. J. Phys. Chem. Lett. 2011, 2, 934−938.

(27) Grubb, M. P.; Warter, M. L.; Xiao, H.; Maeda, S.; Morokuma, K.; North, S. W. No Straight Path: Roaming in Both Ground- and Excited-State Photolytic Channels of NO3 → NO + O2. Science 2012, 335, 1075−1078. (28) Grubb, M. P.; Warter, M. L.; North, S. W. Stereodynamics of Multistate Roaming. Phys. Chem. Chem. Phys. 2012, 14, 6733−6740. (29) Zhang, X.; Zou, S.; Harding, L. B.; Bowman, J. M. A Global Ab Initio Potential Energy Surface for Formaldehyde. J. Phys. Chem. A 2004, 108, 8980−8986. (30) Bowman, J. M.; Zhang, X. New Insights on Reaction Dynamics from Formaldehyde Photodissociation. Phys. Chem. Chem. Phys. 2006, 8, 321−332. (31) Rheinecker, J. L.; Bowman, J. M Quasiclassical Trajectory Studies of the Dynamics of H2CO on a Global Ab Initio-Based Potential Energy Surface. Mol. Phys. 2005, 103, 1067−1074. (32) Zhang, X. B.; Rheinecker, J. L.; Bowman, J. M. Quasiclassical Trajectory Study of Formaldehyde Unimolecular Dissociation: H2CO → H2 + CO, H + HCO. J. Chem. Phys. 2005, 122, 114313. (33) Farnum, J. D.; Zhang, X.-B.; Bowman, J. M. Formaldehyde Photodissociation: Dependence on Total Angular Momentum and Rotational Alignment of the CO Product. J. Chem. Phys. 2007, 126, 134305. (34) Goncharov, V.; Lahankar, S. A.; Farnum, J. D.; Bowman, J. M.; Suits, A. G. Roaming Dynamics in Formaldehyde-d2 Dissociation. J. Phys. Chem. A 2009, 113, 15315−15319. (35) Shepler, B. C.; Braams, B. J.; Bowman, J. M. “Roaming” Dynamics in CH3CHO Photodissociation Revealed on a Global Potential Energy Surface. J. Phys. Chem. A 2008, 112, 9344−9351. (36) Harding, L. B.; Georgievskii, Y.; Klippenstein, S. J. Roaming Radical Kinetics in the Decomposition of Acetaldehyde. J. Phys. Chem. A 2010, 114, 765−777. (37) Bowman, J. M.; Shepler, B. C. Roaming Radicals. Annu. Rev. Phys. Chem. 2011, 62, 531−553. (38) Herath, N.; Suits, A. G. Roaming Radical Reactions. J. Phys. Chem. Lett. 2011, 2, 642−647. (39) Troe, J.; Ushakov, V. Classical Trajectory Study of the Reaction between H and HCO. J. Phys. Chem. A 2007, 111, 6610−6614. (40) Troe, J. Refined Analysis of the Thermal Dissociation of Formaldehyde. J. Phys. Chem. A 2007, 111, 3862−3867. (41) Troe, J. Analysis of Quantum Yields for the Photolysis of Formaldehyde at λ > 310 nm. J. Phys. Chem. A 2007, 111, 3868−3874. (42) Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell Scientific Publications: Oxford, U.K., 1990. (43) Pilling, M. J.; Seakins, P. W. Reaction Kinetics; Oxford University Press: New York, 1995. (44) Klippenstein, S. J.; Georgievskii, Y.; Harding, L. B. Statistical Theory for the Kinetics and Dynamics of Roaming Reactions. J. Phys. Chem. A. 2011, 115, 14370−14381. (45) Klippenstein, S. J. Variational Optimizations in the Rice− Ramsperger−Kassel−Marcus Theory Calculations for Unimolecular Dissociations with No Reverse Barrier. J. Chem. Phys. 1992, 96, 367− 371. (46) Klippenstein, S. J. An Efficient Procedure for Evaluating the Number of Available States within a Variably Defined Reaction Coordinate Framework. J. Phys. Chem. 1994, 98, 11459−11464. (47) Georgievskii, Y.; Klippenstein, S. J. Variable Reaction Coordinate Transition State Theory: Analytic Results and Application to the C2H3 + H → C2H4. Reaction. J. Chem. Phys. 2003, 118, 5442− 5455. (48) Chesnavich, W. J.; Bowers, M. T. Statistical Phase Space Theory of Polyatomic Systems: Rigorous Energy and Angular Momentum Conservation in Reactions Involving Symmetric Polyatomic Species. J. Chem. Phys. 1977, 66, 2306−2315. (49) Pechukas, P.; Light, J. C.; Rankin, C. Statistical Theory of Chemical Kinetics: Application to Neutral AtomMolecule Reactions. J. Chem. Phys. 1966, 44, 794−805. (50) Hunter, M.; Reid, S. A.; Robie, D. C.; Reisler, H. The Monoenergetic Unimolecular Reaction of Expansion-Cooled NO2: 7641



Article

NO Product State Distributions at Excess Energies 0−3000 cm−1. J. Chem. Phys. 1993, 99, 1093−1108. (51) Shepler, B. C.; Han, Y.; Bowman, J. M. Are Roaming and Conventional Saddle Points for H2CO and CH3CHO Dissociation to Molecular Products Isolated from Each Other? J. Phys. Chem. Lett. 2011, 2, 834−838. (52) North, S. W.; Hall, G. E. Quantum Phase Space Theory for the Calculation of v·j Vector Correlations. J. Chem. Phys. 1996, 104, 1864− 1874. (53) Bonnet, L.; Larregaray, P.; Rayez, J. C. On the Statisticodynamical Approach of Final State Distributions in Simple Bond Fissions. Phys. Chem. Chem. Phys. 2005, 7, 3540−3544. (54) Yin, H.-M.; Nauta, K.; Kable, S. H. Fully State-Resolved Photodissociation of Formaldehyde, H2CO → H + HCO: K Conservation and a Rigorous Test of Statistical Theories. J. Chem. Phys. 2005, 122, 194312. (55) Yin, H.-M.; Rowling, S. J.; Büll, A.; Kable, S. H. Photodissociation Dynamics of the Reaction H2CO → H + HCO via the Singlet (S0) and Triplet (T1) Surfaces. J. Chem. Phys. 2007, 127, 064302. (56) Heazlewood, B. R.; Rowling, S. J.; Maccarone, A. T.; Jordan, M. J. T.; Kable, S. H. Photochemical Formation of HCO and CH3 on the Ground S0 (1A′) State of CH3CHO. J. Chem. Phys. 2009, 130, 054310. (57) Guyer, D. Y.; Polik, W. F.; Moore, C. B. Dissociation Rates for Individual Eigenstates of S0 Formaldehyde: Fluctuations and Barrier Height. J. Chem. Phys. 1986, 84, 6519−6521. (58) Terentis, A. C.; Waugh, S. E.; Metha, G. F.; Kable, S. H. HCO (N,Ka,Kc,J) Distributions from Near-Threshold Photolysis of H2CO (J,Ka,Kc). J. Chem. Phys. 1998, 108, 3187−3198. (59) Wang, H. Y.; Eyre, J. A.; Dorfman, L. M. Activation Energy for the Gas Phase Reaction of Hydrogen Atoms with Carbon Monoxide. J. Chem. Phys. 1973, 59, 5199−5200. (60) Johnston, H. S.; Davis, H. F.; Lee, Y. T. NO3 Photolysis Product Channels: Quantum Yields from Observed Energy Thresholds. J. Phys. Chem. 1996, 100, 4713−4723. (61) Davis, H. F.; Kim, B.; Johnston, H. S.; Lee, Y. T. Dissociation Energy and Photochemistry of NO3. J. Phys. Chem. 1993, 97, 2172− 2180. (62) Davis, H. F.; Ionov, P. I.; Ionov, S. I.; Wittig, C. Picosecond Resolution Measurements of NO3 Unimolecular Decomposition. The NO + O2 Channel. Chem. Phys. Lett. 1993, 215, 214−220. (63) Nelson, H. H.; Pasternack, L.; McDonald, J. R. Laser-Induced Excitation and Emission Spectra of NO3. J. Phys. Chem. 1983, 87, 1286−1288. (64) Nelson, H. H.; Pasternack, L.; McDonald, J. R. Excited State Dynamics of NO3. J. Chem. Phys. 1983, 79, 4279−4284. (65) Ishiwata, T.; Fugiwara, I.; Naruge, Y.; Obi, K.; Tanaka, I. Study of NO3 by Laser-Induced Fluorescence. J. Phys. Chem. 1983, 87, 1349−1352. (66) Magnotta, F.; Johnston, H. S. Photodissociation Quantum Yields for the NO3 Free Radical. Geophys. Res. Lett. 1980, 7, 769−772. (67) Orlando, J. J.; Tyndall, G. S.; Moortgat, G. K.; Calvert, J. G. Quantum Yields for NO3 Photolysis between 570 and 635 nm. J. Phys. Chem. 1993, 97, 10996−11000. (68) Mikhaylichenko, K.; Riehn, C.; Valachovic, L.; Sanov, A.; Wittig, C. Unimolecular Decomposition of NO3: The NO + O2 Threshold Regime. J. Chem. Phys. 1996, 105, 6807−6817. (69) Yokelson, R. J.; Burkholder, J. B.; Fox, R. W.; Talukdar, R. K.; Ravishankara, A. R. Temperature Dependence of the NO3 Absorption Spectrum. J. Phys. Chem. 1994, 98, 13144−13150. (70) Thompson, K. C.; Crittenden, D. L.; Kable, S. H.; Jordan, M. J. T. A Classical Trajectory Study of the Photodissociation of T1 Acetaldehyde: The Transition from Impulsive to Statistical Dynamics. J. Chem. Phys. 2006, 124, 044302. (71) Heazlewood, B. R.; Maccarone, A. T.; Andrews, D. U.; Osborn, D. L.; Harding, L. B.; Klippenstein, S. J.; Jordan, M. J. T.; Kable, S. H. Near-Threshold H/D Exchange in CD3CHO Photodissociation. Nat. Chem. 2011, 3, 443−448.

(72) Smith, B. J.; Nguyen, M. T.; Bouma, W. J.; Radom, L. Unimolecular Rearrangements Connecting Hydroxyethylidene (CH3− C−OH), Acetaldehyde (CH3−CHO), and Vinyl Alcohol (CH2 CH−OH). J. Am. Chem. Soc. 1991, 113, 6452−6458. (73) Ruscic, B.; Boggs, J. E.; Burcat, A.; Csaszar, A. G.; Demaison, J.; Janoschek, R.; Martin, J. M. L.; Morton, M. L.; Rossi, M. J.; Stanton, J. F.; et al. Critical Evaluation of Thermochemical Properties of Selected Radicals. Part I. J. Phys. Chem. Ref. Data 2005, 34, 573−656. (74) Yang, X.; Maeda, S.; Ohno, K. Insight into Global Reaction Mechanism of [C2, H4, O] System from Ab Initio Calculations by the Scaled Hypersphere Search Method. J. Phys. Chem. A 2007, 111, 5099−5110. (75) Lifschitz, A.; Ben-Hamou, H. Thermal Reactions of Cyclic Ethers at High Temperatures. 1. Pyrolysis of Ethylene Oxide Behind Reflected Shocks. J. Phys. Chem. 1983, 87, 1782−1787. (76) Joshi, A.; You, X.; Barckholtz, T. A.; Wang, H. Thermal Decomposition of Ethylene Oxide: Potential Energy Surface, Master Equation Analysis, and Detailed Kinetic Modeling. J. Phys. Chem. A 2005, 109, 8016−8027. (77) Terentis, A. C.; Kable, S. H. Near Threshold Dynamics and Dissociation Energy of the Reaction H2CO → HCO + H. Chem. Phys. Lett. 1996, 258, 626−632. (78) Nguyen, T. L.; Vereecken, L.; Hou, X. J.; Nguyen, M. T.; Peeters, J. Potential Energy Surfaces, Product Distributions and Thermal Rate Coefficients of the Reaction of O(3P) with C2H4(X1Ag): A Comprehensive Theoretical Study. J. Phys. Chem. A 2005, 109, 7489−7499. (79) Horowitz, A.; Kershner, C. J.; Calvert, J. G. Primary Processes in the Photolysis of Acetaldehyde at 3000 Å and 25°C. J. Phys. Chem. 1982, 86, 3094−3105. (80) Horowitz, A.; Calvert, J. Wavelength Dependence of the Primary Processes in Acetaldehyde Photolysis. J. Phys. Chem. 1982, 86, 3105−3114. (81) Moortgat, G. K.; Meyrahn, H.; Warneck, P. Photolysis of Acetaldehyde in Air: CH4, CO and CO2 Quantum Yields. Chem. Phys. Chem. 2010, 11, 3896−3908. (82) Andrews, D. U.; Heazlewood, B. R.; Maccarone, A. T.; Conroy, T.; Payne, R. J.; Jordan, M. J. T.; Kable, S. H. Photo-Tautomerization of Acetaldehyde to Vinyl Alcohol: A Potential Route to Tropospheric Acids. Science 2012, 337, 1203−1206.

7642


A Phase Space Theory for Roaming Reactions - The Journal of

Recommend Documents