J. Phys. Chem. A 2010, 114, 765–777
765
Roaming Radical Kinetics in the Decomposition of Acetaldehyde Lawrence B. Harding,* Yuri Georgievskii,* and Stephen J. Klippenstein* Chemical Sciences and Engineering DiVision, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: July 21, 2009; ReVised Manuscript ReceiVed: NoVember 16, 2009
A novel theoretical framework for predicting the branching between roaming and bond fission channels in molecular dissociations is described and applied to the decomposition of acetaldehyde. This reduced dimensional trajectory (RDT) approach, which is motivated by the long-range nature of the roaming, bond fission, and abstraction dynamical bottlenecks, involves the propagation of rigid-body trajectories on an analytic potential energy surface. The analytic potential is obtained from fits to large-scale multireference ab initio electronic structure calculations. The final potential includes one-dimensional corrections from higher-level electronic structure calculations and for the effect of conserved mode variations along both the addition and abstraction paths. The corrections along the abstraction path play a significant role in the predicted branching. Master equation simulations are used to transform the microcanonical branching ratios obtained from the RDT simulations to the temperature- and pressure-dependent branching ratios observed in thermal decomposition experiments. For completeness, a transition-state theory treatment of the contributions of the tight transition states for the molecular channels is included in the theoretical analyses. The theoretically predicted branching between molecules and radicals in the thermal decomposition of acetaldehyde is in reasonable agreement with the corresponding shock tube measurement described in the companion paper. The prediction for the ratio of the tight to roaming contributions to the molecular channel also agrees well with results extracted from recent experimental and experimental/theoretical photodissociation studies. 1. Introduction The dissociation of a closed-shell molecule by simple bond cleavage generally leads to two radical fragments. However, as the radicals separate they can roam,1 sampling large volumes of orientation space. This roaming occurs at radical-radical separations of about 3 Å (6 au). During this roaming, the radicals may sample orientations where there is a barrierless path leading to the abstraction of a radical from one radical to the other. Most commonly, this abstraction involves the transfer of an H atom. The importance of such roaming dynamics in the long-range region of the potential was noted some time ago in the ion-molecule literature.2-4 There, the increased strength of the long-range ion-molecule interactions greatly enhances the importance of such roaming mechanisms. The different orientational dependences of the long-range ion-molecule interactions and the short-range bonding interactions allows for the formation of long-range ion-molecule complexes. There are then two transition states involved in the formation of a covalently bound ionic complex. The long-range centrifugal barriers provide an “outer” transition state that separates the free ion and molecule from the ion-molecule complex. An “inner” transition state separates the ion-molecule complex from a chemically bonded ionic adduct. When roaming occurs there is a separate inner transition state that separates the longrange ion-molecule complex from a distinct set of products via H atom or proton transfer, for example. For example, in ref 2, Audier and Morton describe the roaming dynamics during the dissociation of protonated propylamines. The decomposition of the protonated n-propylamine initially leads to an [ipropyl+ · · · NH3] ion-molecule complex, * To whom correspondence should be addressed. E-mail:
[email protected] (L.B.H.),
[email protected] (Y.G.),
[email protected] (S.J.K.).
which can either collapse to chemically bound ipropylNH3+ or decompose to propene + NH4+. Lee and Grabowski review similar roaming dynamics between multiple channels from ion-molecule complexes in the reactions of O- with a great variety of neutrals.3 For the reactions with ethylene and benzene, they include potential energy surfaces illustrating the presence of multiple product channels from the long-range [O- · · · neutral] ion-molecule complex. Our own contribution to the treatment of roaming dynamics for ion-molecule reactions considered the multiple channels available from the [C2H2 · · · CH4+] ion molecule complex.4 To our knowledge, the first correctly interpreted evidence of a roaming radical mechanism for a neutral reaction can be traced to a remarkable paper by van Zee et al.5 They measured CO (ν,J) distributions resulting from the photolysis of formaldehyde at a number of different wavelengths. At energies above the threshold for formation of H + HCO they observed bimodal CO rotational distributions and took this to imply the existence of a second mechanism for the formation of CO, distinct from the standard mechanism involving the well studied, tight, threecenter, saddle point. They suggested the opening of “a second fragmentation path related to the exit channel of the H + HCO f H2 + CO abstraction reaction and accessed through the radical channel” as the most likely explanation. This is illustrated schematically in Figure 1. The hypothesis of van Zee et al. has now been confirmed by an extensive series of theoretical and experimental studies.1,6-11 In particular, it has been shown that the tight, three-center transition state yields rotationally hot CO, while the roaming radical mechanism yields rotationally cold CO. This work has been the subject of recent reviews.12,13 The implications of the roaming radical mechanism in the thermal dissociation of formaldehyde have also been discussed,14 and recently nonadiabatic mechanisms for the molecular dissociation of formaldehyde have been proposed.15,16
10.1021/jp906919w 2010 American Chemical Society Published on Web 12/28/2009
766
J. Phys. Chem. A, Vol. 114, No. 2, 2010
Figure 1. Schematic of the formaldehyde potential surface showing the tight molecular dissociation pathway in blue, the radical dissociation pathway in red, and the roaming radical pathway in purple.
Figure 2. Schematic of the acetaldehyde potential surface showing the tight molecular dissociation pathway in blue, the radical dissociation pathway in red, and the roaming radical pathway in purple.
A roaming mechanism was also suggested by Jackson and co-workers in their 1993 study of the CN + O2 reaction.17 In this case, the observation of CO + NO contradicted the presence of high barriers for the four-center tight transition state from the initially formed NCOO adduct. A roaming of the O atom about the NCO radical to abstract an N atom, may provide a much lower energy route for the formation of the CO + NO products. Unfortunately, no dynamical studies have yet been performed to verify this mechanism. An early theoretical indication of the presence of long-range roaming dynamics during the decomposition of a neutral molecular complex was provided in a direct dynamics study of the HCCO + O2 reaction.18 In particular, the O atom arising from OO fission in the initial HC(O2)CO complex was observed to roam around to abstract an H atom from the HC(O)CO complex. Similarly, CC fission in an HCOCO2 complex was observed to result in HOCO via a long-range abstraction by CO2 from HCO. More recently, it has been suggested that a roaming radical mechanism might also provide an explanation for the observed CO rovibrational distributions in the photodissociation of acetaldehyde.19,20 A schematic of the acetaldehyde potential surface (Figure 2) shows two tight molecular dissociations (leading to H2 + H2CCO or CH4 + CO). In acetaldehyde, the lowest threshold for radical formation is to CH3 + HCO (the H + CH3CO and H + CH2CHO asymptotes lie 5 and 12 kcal/
Harding et al. mol higher). It was suggested then that in acetaldehyde it is a CH3 fragment that roams and abstracts the hydrogen from HCO giving a second route to CH4 + CO. The barriers for both tight molecular dissociations are thought to be much closer to the threshold for radical formation than in the case of formaldehyde and for this reason dissociation over the tight transition states is not expected to compete as effectively with simple bond cleavage. The proposed roaming mechanism for acetaldehyde decomposition was quickly confirmed by a combined theoretical/ experimental study focusing on IR emission from the CH4 produced in the photodissociation.21 The conclusion from this study was that roaming is the dominant mechanism for formation of CH4 + CO. This conclusion was based on the comparison of experimental measurements of the CH4 vibrational distribution with predictions from full-dimensional quasiclassical trajectory simulations. The studies of refs 19 and 21 yielded two separate estimates for the branching between the roaming and tight transition state pathways for formation of CH4 + CO.22 That from ref 19 is obtained directly from experimental measurements of CO rovibrational distributions and has subsequently been significantly revised (from 15% roaming to 70% roaming) via consideration of the V/J correlations from each CO(J) state.22 Meanwhile, that from ref 21 (84% roaming) is obtained from a comparison of experiment and theory for the CH4 rovibrational distributions, which presume that (i) the fulldimensional quasiclassical trajectory simulations of the dynamics from the tight transition state on to CH4 + CO yield the correct CH4 product vibrational distribution from the tight transition state pathway and that (ii) there is essentially no contribution to the low energy tail of the distribution from the roaming pathway. However, it does not rely on any theoretical analysis for the rovibrational distribution arising from the roaming pathway. In general, the branching between radicals and molecules may be of some importance to combustion modeling. Unfortunately, in the acetaldehyde photodissociation experiments mentioned above, although the radical channel is observed, the branching between radicals (CH3 + HCO) and molecules (CH4 + CO) could not be determined. Gherman et al.23 report quantum yields for CH4 + CO of only 0.001 at 313.0 nm (91.3 kcal/mol) rising to 0.66 at 265.4 nm (107.7 kcal/mol). Shepler, Braams, and Bowman24,25 have suggested, on the basis of trajectory simulations, that at an excitation energy of 95 kcal/mol, branching to molecules (via roaming) is on the order of 80-90%, dropping to ∼60% at 120 kcal/mol. (The authors note, however, that up to 85% of the CH3 + HCO forming trajectories are discarded due to zero point violation.) In contrast, thermal experiments have been interpreted to imply much larger yields of radicals and consequently smaller yields of molecular products. For example, a shock tube study by Kiefer and co-workers was well described by a mechanism that presumes no CH4 + CO products.26 Inclusion of a dominant roaming channel would cause significant difficulties for the modeling of the laserschlieren gradient data.27 An ongoing point of discussion in the literature concerns appropriate theoretical frameworks for understanding and quantifying the importance of roaming radical mechanisms.28 For example, are these reactions inherently nonstatistical and hence inaccessible to transition state theory based treatments? Should these be considered new examples of “non-intrinsic reaction coordinate (IRC)” dynamics?29 Harding et al.30 have reported the existence of saddle points (and IRCs) for the roaming pathways in both formaldehyde and acetaldehyde. They
Decomposition of Acetaldehyde note, however, that a simple, rigid-rotor, harmonic-oscillator, transition-state theory calculation based on these saddle points yields unrealistically high rates for roaming. But this does not preclude the applicability of transition state theory when employing more accurate state counting procedures that properly treatment anharmonicities in the low frequency motions. For acetaldehyde, dynamical calculations have been reported using the full-dimensional, quasiclassical approximation.21,24,25 This approach introduces significant uncertainties with respect to zero-point conservation and usually requires the demanding task of fitting accurate, full-dimensional, analytic potential surfaces. Preliminary quantum dynamics calculations on H2CO have been reported by Yonehara et al.,31 but their results were of only qualitative significance due to computational limitations in basis set size and in propagation times. Another approach that has been used in modeling the dissociation of acetaldehyde is direct dynamics trajectories.32-34 Although this approach avoids the need to fit an analytic surface it is typically restricted to relatively inexpensive, singlereference, electronic structure methods. As noted above, roaming dynamics occurs at interfragment separations of ∼3 Å. A singlet wave function composed of two radical fragments separated by 3 Å (6 au) will typically have two dominant configurations. For example, for the roaming saddle point reported by Harding et al.,30 these two configurations have coefficients of 0.71 and -0.70 whereas, for comparison, the two dominant configurations for the tight transition state have coefficients of 0.98 and 0.18. For this reason, the single reference methods often used in direct dynamics calculations are not expected to be reliable for these roaming-type mechanisms. In this paper we report a novel theoretical framework for estimating the branching between roaming and radicals and demonstrate its application to the decomposition of acetaldehyde. This reduced dimensional trajectory (RDT) framework emphasizes the rigid-body dynamics of the two incipient radicals, which is appropriate because the roaming dynamics occurs at large separations where the interactions between the two radicals are relatively weak. In essence, this approach assumes that the bottlenecks to CC fission, to H abstraction, and for the motion between them all occur at separations where the interactions are relatively weak. Our prior and current variable reaction coordinate transition state theory analyses for this and related reactions suggest that these assumptions are met.26,35,36 This emphasis on rigid-body dynamics facilitates the development of analytic potentials and removes ambiguities related to the conservation of zero-point energy for the high frequency, conserved vibrational modes of the radicals. A key assumption in the RDT approach is that the dynamics of the conserved vibrational modes is essentially decoupled from that of the transitional modes within the roaming region. This may not always be the case. Importantly, the dominant effect of the coupling of the conserved and transitional modes may be readily accounted for within the RDT framework under the assumption of vibrationally adiabatic dynamics for the conserved modes. With this assumption, the proper effective potential for the RDT’s involves the sum of the potential and the adiabatic vibrational energies, which are readily calculated from a fulldimensional potential. This vibrationally adiabatic assumption is again motivated by the relatively weak interactions in the long-range region relevant to the roaming dynamics. The ratio of the conserved mode to transitional mode frequencies provides a measure of the appropriateness of this vibrational adiabaticity assumption.
J. Phys. Chem. A, Vol. 114, No. 2, 2010 767 Here, we simplify the calculation of such vibrationally adiabatic corrections by focusing on the two one-dimensional minimum energy paths for addition and for abstraction. Notably, the most important component of these corrections is generally just the relaxation energy arising from the change in the conserved mode minima. Note that the need to include such potential corrections does not by itself imply a departure from the reduced dimensional dynamics framework but rather signifies the need to employ the proper vibrationally adiabatic effective potentials. However, very large corrections arising from large internal geometry changes would correlate with a breakdown of the vibrationally adiabatic assumption. It should be clear that the RDT approach cannot be used to predict the product rovibrational state distributions. However, it does suggest a simpler approach to predicting the roaming contribution to such distributions. In particular, one could initiate trajectories directly from a statistical distribution in the abstraction region of the long-range potential. These trajectories would likely need to be propagated in all dimensions. But, such trajectories would proceed directly to products, thereby facilitating the prediction of the distributions close to the roaming threshold. The very short time for which such trajectories would need to be propagated implies that even multireference direct dynamics methods could be implemented. This approach would thereby bypass completely the challenging step of creating a global analytic potential energy surface. The overall branching ratio for this roaming contribution would be determined from the RDT approach. In the companion paper in this issue, Sivaramakrishnan et al.37 present new direct measurements of the branching between molecular and radical products in the thermal decomposition of acetaldehyde. The conclusion from this study is that the radical products, CH3 + HCO, dominate, making up 77 ( 9% of the products. This data provides an important test for our framework. To compare with their data, we perform a master equation analysis of the temperature and pressure dependence of the kinetics and include a rigid-rotor harmonic-oscillator transition state theory description of the contribution to molecular products from the tight transition states. The theoretical calculations indicate that there is little contribution from the tight molecular channels for the temperatures and pressures of relevance to the experiment. Thus, these experiments37 provide the first measurement of the branching between roaming and radical channels in a thermal decomposition. 2. Theoretical Methods In general, various low-energy pathways between distinct relative orientations open up at radical-radical separations on the order of 3-4 Å (6-8 au).30,38,39 Roaming dynamics involves the motion between these different orientations.1,6,7,21 At these separations, the interactions between the two incipient radicals are relatively weak, and their vibrational modes are expected to behave adiabatically, with conserved quantum numbers. In this instance, a separation of modes into the conserved vibrational motions of the radicals and the remaining “transitional” modes is appropriate, where the conserved modes are those vibrational modes that exist for the isolated radical fragments and the transitional modes are the six new modes that appear when the radical fragments are put at some relatively large, but finite separation. The transitional modes are generally very low frequency. In the case of the roaming saddle point for acetaldehyde reported by Harding et al.,30 these modes have frequencies ranging from 30 to 125 cm-1. For comparison, the two lowest-frequency modes of acetaldehyde are 160 (internal
768
J. Phys. Chem. A, Vol. 114, No. 2, 2010
Harding et al.
TABLE 1: Energies, Harmonic Frequencies, and Rotational Constants for the Stationary Points on the CH3CHO Potential Surfacea species CH3CHO
energy (kcal/mol)
frequencies (cm-1)
0.0 (0.0)
3144, 3092, 3019, 2922, 1750, 1452, 1441, 1403, 1360, 1121, 1107, 889, 756, 495, 140 4344 3305, 3190, 2151, 1400, 1134, 984, 582, 488, 433 2104 3144, 3144, 3144, 3016, 1535, 1535, 1320, 1320, 1320 2676, 1846, 1095 3290, 3290, 3101, 1406, 1406, 497 3131, 3125, 3020, 1858, 1439, 1435, 1333, 1035, 942, 860, 459, 96 3269, 3144, 2979, 1569, 1461, 1380, 1150, 958, 950, 737, 492, 429 3252, 3133, 2118, 1806, 1553, 1465, 1240, 1167, 1076, 926, 839, 663, 538, 351, 1502i 3175, 3155, 3131, 3005, 1768, 1406, 1405, 1036, 878, 727, 518, 505, 272, 142, 1712i
H2 + H2CCO
36.2 (27.7)
CO + CH4
-2.5 (-5.9)
HCO + CH3
90.8 (83.2)
CH3CO + H CH2CHO + H CH3CHO f H2 + H2CCO CH3CHO f CH4 + CO a
96.1 (88.7) 103.2 (95.5) 85.9 (80.4) 87.2 (82.9)
rotational constants (GHz) 8.92, 10.0, 55.6 1728.3, 1728.3 9.6, 10.0, 275.5 56.0, 56.0 154.6, 154.6, 154.6 41.0, 43.5, 699.4 139.9, 279.7, 279.7 9.23, 9.76, 81.6 9.54, 11.2, 65.5 8.86, 10.0, 62.6 7.02, 7.58, 45.7
All results are from CCSD(T)/CBS//CCSD(T)/aug-cc-pvdz calculations. Energies in parentheses include zero point.
rotation of the CC bond) and 500 (CCO bend) cm-1. Importantly, the roaming dynamics involves solely the low frequency transitional modes. This separation of modes provides the basis for the theoretical analysis of the roaming and radical channels presented here. It allows us to focus on the transitional mode dynamics of the two incipient radicals treated as rigid bodies. It has proven to be of great utility in our related transition state theory analyses of radical-radical recombination rates.35,36 In the present study, we simply replace the transition state theory analysis with rigid body classical trajectories. Related rigid body trajectory simulations have been employed in a number of studies of ion-molecule40-43 and radical-radical44-48 addition reactions. These RDT simulations require some description of the radical-radical interaction energies over a wide range of configuration space. Here, we develop a six-dimensional analytic potential energy surface to describe the interaction between the CH3 and HCO radicals. This potential is based on fits to a large number of wideranging multireference electronic structure calculations. It also incorporates corrections based on higher level evaluations and for the effect of conserved mode variations. These one-dimensional corrections are obtained from calculations along the addition and abstraction minimum energy paths. The RDT simulations employing the six-dimensional analytic potential energy surface yield microcanonical rates for the roaming and radical products. For the tight transition states leading to molecular products the microcanonical rates are obtained from rigid-rotor harmonic-oscillator transition state theory analyses employing ab initio determined energies and rovibrational properties. The ratios of these rates provide branching ratios for comparison with the ratios derived from photodissociation data either directly19,22 or from a comparison of experiment and theory.21 Alternatively, the incorporation of the microcanonical rates in master equation simulations of the reactive process yields temperature and pressure dependent rate coefficients for comparison with the experimental data of Sivaramakrishnan et al.37 The theoretical methods employed for each of these aspects of the calculation are described in more detail in the following subsections. 2.1. Electronic Structure Calculations. In this section we describe the coupled-cluster calculations used to characterize the minima and tight transition states and the multireference calculations used to characterize the roaming transition state and the CH3 · · · HCO interaction potential. All electronic structure calculations were done using the MOLPRO program package.49
TABLE 2: Energies (kcal/mol) of the Roaming Saddle Point Relative to CH3 + HCO basis set method
aug-cc-pvdz
aug-cc-pvtz
aug-cc-pvqz
(2E,2O)-CASPT2 (4E,4O)-CASPT2 (2E,2O)-CAS+1+2+QC (4E,4O)-CAS+1+2+QC
-1.17 -1.22 -0.96 -1.01
-1.06 -1.08 -0.86 -0.85
-1.03 -1.05 -0.80 -0.81
2.1.1. Coupled-Cluster Calculations. Geometries and frequencies for the minima and tight transition states were calculated using CCSD(T)/aug-cc-pvdz50-53 followed by singlepoint calculations using CCSD(T)/aug-cc-pvtz and CCSD(T)/ aug-cc-pvqz, which are extrapolated to the basis set limit54 to obtain the final CCSD(T)/CBS//CCSD(T)/aug-cc-pvdz relative energies. The results of these calculations are summarized in Table 1. The energies of CH4 + CO and CH3 + HCO relative to CH3CHO are in good agreement with the most recent active tables determinations of Ruscic.55 For CH4 + CO the comparison is: theory -5.88 vs active tables -6.06 ( 0.01 kcal/mol. For CH3 + HCO the comparison is: theory 83.19 vs active tables 82.80 ( 0.1 kcal/mol. The lowest pathway to decomposition is predicted to be the (1,2) elimination of H2. The calculated, zero-point-corrected barrier height of 80.4 kcal/mol is in good agreement with a previous G2 calculation56 (80.7 kcal/mol) and in nearly perfect agreement with the CCSD(T)/CBS+CV calculation of Shepler et al.24 The barrier for the higher molecular route (to CH4 + CO) is predicted to be 83.2 kcal/mol compared to 82.9 from G2 and 83.2 from Shepler et al. Acetaldehyde and the tight transition state leading to CH4 + CO are predicted to have internal rotor barriers of 1.1 and 0.8 kcal/mol respectively. 2.1.2. Roaming Saddle Point. The structure of the saddle point for roaming has been described previously.30 The coefficients of the two dominant configurations in the CASPT2 wave function for this structure are 0.709 and -0.705, clearly indicating the multireference character of the wave function. CCSD(T) calculations of this structure result in a T1 diagnostic of ∼0.05 again indicative of potential problems with standard single reference methods for this structure. The results of multireference CASPT257,58 and CAS+1 + 2+QC59,60 calculations of the energy of this saddle point relative to CH3 + HCO are summarized in Table 2. These calculations show that the energy is relatively insensitive to the basis set, the size of the active space, and CASPT2 vs MR-CI. All of the calculations lie in the range -1.0 ( 0.2 kcal/mol. Here and below, the two-
Decomposition of Acetaldehyde
J. Phys. Chem. A, Vol. 114, No. 2, 2010 769
TABLE 3: Harmonic Vibrational Frequencies (cm-1) for the Roaming Saddle Point and for CH3 + HCO from CASPT2/aug-cc-pvdz Calculations mode
TS
CH3
CHstr CHstr CHstr CHstr COstr HCHbend HCHbend HCObend CH3 umbrella transitional modes
3352 3351 3151 2754 1940 1421 1421 1077 566 125 85 67 55 33 262i
3356 3356 3156
reaction coordinate
HCO
2745 1944 1421 1421 1074 574
electron two- orbital active space simply includes the CH3 and HCO radical orbitals, while the four-electron four-orbital active space also includes the CO π and π* orbitals. Vibrational frequencies for the roaming saddle point structure are given in Table 3, where they are compared to the frequencies for the separated fragments, CH3 + HCO. The similarity in these two sets of frequencies provides some validation for the hypothesis that the branching between roaming and radicals occurs in regions of configuration space where the two fragments are only weakly interacting and the internal coordinates and frequencies of the two fragments are not significantly perturbed. 2.1.3. Analytic Six-Dimensional Surface. In this section we describe our fit of a six-dimensional, analytic surface for the interaction between a rigid CH3 radical and a rigid HCO radical. The primary focus here is to characterize that part of the potential relevant to the roaming reaction, i.e., where the two radical fragments are separated by a relatively large distance, 2.5-5 Å (5-10 au). The electronic wave functions in this region are inherently multireference in character, and the use of standard single reference methods such as CCSD(T) is inappropriate. The calculations used in this fit were done at the (2E,2O)-CASPT2/aug-cc-pvdz level. Approximately 100,000 points were calculated for C to CO bond midpoint distances spanning the range from 2.1 to 10 Å (4-20 au). Inclusion of permutation symmetry (among the three methyl hydrogens) expands this list to ∼300,000 points (We note here that it has been shown61 that permutation symmetry can be imposed on the fitting function which obviates the need to replicate points as done here). The fit employs direct product multinomials in Morse variables.6,62 For the present problem there are twelve Morse variables, one for each of the internuclear distances between the HCO and CH3 fragments. The multinomial expansions include all terms up to third order and a subset of the fourth order terms. Each multinomial contains a total of 533 terms. A total of eight multinomials were used for different (but overlapping) ranges of the distance between the two fragments. The eight individual fits are connected by switching functions to yield the final analytic potential used in the trajectory calculations. For the ∼50,000 points within (5 kcal/mol of the CH3 + HCO asymptote, the fit yields a root-mean-square (rms) error of
