Mechanism and Kinetics of the Reaction NO3 + C2H4 - The Journal

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Mechanism and Kinetics of the Reaction NO3 þ C2H4 Thanh Lam Nguyen,† Jaehee Park,‡ Kyungjun Lee,‡ Kihyung Song,*,‡ and John R. Barker*,† † ‡

Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, Ann Arbor, Michigan 48109, USA Department of Chemistry, Korea National University of Education, Chongwon, Chungbuk 363-791, Korea

bS Supporting Information ABSTRACT: The reaction of NO3 radical with C2H4 was characterized using the B3LYP, MP2, B97-1, CCSD(T), and CBS-QB3 methods in combination with various basis sets, followed by statistical kinetic analyses and direct dynamics trajectory calculations to predict product distributions and thermal rate constants. The results show that the first step of the reaction is electrophilic addition of an O atom from NO3 to an olefinic C atom from C2H4 to form an open-chain adduct. A concerted addition reaction mechanism forming a five-membered ring intermediate was investigated, but is not supported by the highly accurate CCSD(T) level of theory. Master-equation calculations for tropospheric conditions predict that the collisionally stabilized NO3C2H4 free-radical adduct constitutes 8090% of the reaction yield and the remaining products consist mostly of NO2 and oxirane; the other products are produced in very minor yields. By empirically reducing the barrier height for the initial addition step by 1 kcal mol1 from that predicted at the CBS-QB3 level of theory and treating the torsional modes explicitly as one-dimensional hindered internal rotations (instead of harmonic oscillators), the computed thermal rate constants (including quantum tunneling) can be brought into very good agreement with the experimental data for the overall reaction rate constant.

’ INTRODUCTION In daylight, isoprene (2-methyl-1,3-butadiene) and the monoterpenes (e.g., R- and β-pinene), which are the most important biogenic volatile organic compounds (VOCs) aside from methane,1,2 react rapidly with hydroxyl free radicals (OH) and more slowly with ozone (O3) in the atmosphere.3 The nitrate free radical, which is produced mostly by the reaction of NO2 with O3, is not abundant during daylight because it is photolyzed very rapidly; at night, however, NO3 is the dominant free radical.1,2 Similarly to the OH radical, NO3 attacks alkenes such as isoprene and the monoterpenes by initial electrophilic addition to the sp2 carbon atoms in the alkene double bond.49 The energy barriers associated with NO3 radical attack tend to be higher than those for OH radical attack, and hence, the reaction rate constants tend to be smaller. Both radicals can also abstract hydrogen atoms from alkyl groups, but the barriers are high, and therefore, abstraction is much slower than addition to the double bond. The free-radical adducts formed by OH addition to the olefinic double bond undergo rapid reaction with O2 to produce RO2 radicals. The reactions of the RO2 radicals that originate from the OH þ isoprene reaction produce copious yields of methacrolein and methyl vinyl ketone, which also contain olefinic double bonds.911 For accurate chemical simulations, it is important to understand the mechanisms and rates of these reactions. Empirical correlations have been used with reasonable success to estimate the rate constants at room temperature (see ref 12 and references therein). The reactions involving NO3 have received less attention than those involving OH, but the mechanism for the reaction of NO3 with alkenes is generally accepted to proceed by initial formation r 2011 American Chemical Society

of a vibrationally excited adduct, which can be stabilized by collisions or can form an epoxide by loss of NO2, as shown in Scheme 1.911 This general mechanism was first postulated by Bandow et al.4 and is supported by subsequent product yield measurements on larger alkenes. The reaction rate constants measured for the NO3 þ ethene reaction at a pressure of ∼5 Torr5,6,13,14 are in good agreement with those measured at a pressure of ∼1 atm,15,16 showing that the reverse of the primary adduct formation is unimportant. Pressure-dependent yields of NO2 and oxirane products can be explained by the competition between collisional stabilization of the excited adduct and epoxide formation. Olzmann et al. implemented a model based on this competition and demonstrated that it can explain quantitatively the pressure-dependent NO2 yields measured in several NO3 þ alkene reactions.7 Although the general mechanism for NO3 þ alkene reaction is thought to be understood, there is still uncertainty about the primary reaction step in which an adduct is formed. The mode of attack first postulated by Bandow et al. consists of initial formation of a single CO bond between the NO3 and one of the olefinic carbon atoms in the alkene to form an “open” adduct. The freeradical open adduct thus can may undergo collisional stabilization and subsequent secondary reactions, including loss of NO2 to produce an epoxide, as shown in Scheme 1. This scheme is supported by the theoretical calculations of Perez-Casany et al.17 Cartas-Rosado et al. reported theoretical results18,19 in agreement Received: January 16, 2011 Revised: March 21, 2011 Published: April 21, 2011 4894

dx.doi.org/10.1021/jp200460b | J. Phys. Chem. A 2011, 115, 4894–4901

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Scheme 2

with those of Perez-Casany et al., but also offered theoretical evidence supporting the possible existence of a direct cycloaddition to produce a cyclic adduct. This alternative pathway had been previously considered by Atkinson20 and by Canosa-Mas et al.13 The putative cycloaddition reaction pathway is shown in Scheme 2. This pathway is analogous to the addition of ozone to carboncarbon double bonds: two CO single bonds are formed simultaneously between the O3 and the two olefinic carbon atoms to form a highly vibrationally excited cyclic adduct, which subsequently undergoes rapid fragmentation (the Criegee mechanism). In the NO3 þ ethene reaction, the excited cyclic adduct is predicted to be unusually stable, but when excited, it can fragment into 2CH2O þ NO. This scheme is consistent with the aldehydic products identified by Barnes et al.16 in reactions involving alkenes larger than ethene. Most importantly, CartasRosado et al.18,19 reported on the basis of their calculations that the cycloaddition pathway is considerably lower in energy than the channel leading to formation of the open adduct. In the present work, we bring higher-level theory to bear on the NO3 þ ethene reaction system. The first goal of the present work was to determine the nature of the primary step. The second goal was to predict the product yields as functions of temperature and pressure. To examine the primary step, we employed several levels of quantum theory and large basis sets extrapolated to the complete-basis-set limit. For the kinetics calculations, we used a realistic multiwell, multichannel master-equation model for a range of realistic parameters. In the following sections, we show that the higher-level theory does not support the existence of a low-energy cycloaddition channel. We also show that a detailed treatment of hindered internal rotors in the transition state for forming the open adduct and a minor adjustment of the energy barrier produces essentially perfect agreement with the measured rate constant for the overall reaction. Finally, the product yields predicted by the master-equation calculations support the model of Olzmann et al.,7 which was based on competition between collisional stabilization and epoxide formation.

’ COMPUTATIONAL METHODS Electronic Structure Methods. The nitrate radical (NO3) has been studied extensively both experimentally and theoretically.21

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The experimental studies show that NO3 has D3h symmetry and that its ground electronic state is X2A0 2.21 Theoretical calculations using highly correlated levels of theory, such as multireference coupledcluster (MRCC)2225 or multireference configuration-interaction (MRCI),26 support the experimental measurements and reproduce the molecular properties very well. However, these methods are currently prohibitively expensive for investigating the chemical kinetics and reaction mechanisms of NO3 with C2H4. Unfortunately, single-reference HartreeFock (HF) and second-order MøllerPlesset perturbation theory (MP2), which are less expensive, are unsuitable for modeling the NO3 radical, because of the effects of symmetry breaking.21,27,28 Fortunately, several density-functional-theory- (DFT-) based methods, which are also less expensive, have been shown to predict geometry and molecular properties for NO3 fairly well. These DFT methods employ the hybrid density functionals B3P86, B3PW91, B3LYP, and B97-1, which have small HF exchange coefficients (also see Table S2, Supporting Information).2729 The DFT-based methods are computationally cost-efficient and less prone to problems with symmetry breaking and spin contamination. In the present work, the B3LYP method30 was used with the 6-311þG(d,p) basis set to optimize the geometries of all relevant species, and single-point calculations were then carried out at higher levels of theory to refine the energetics.31 A harmonic vibrational analysis was carried out to verify each stationary point on the potential energy surface (PES). Every minimum identified on the PES was found to have all positive harmonic vibrational frequencies, and each transition state structure was found to have only one imaginary frequency, corresponding to the reaction coordinate. The computed rovibrational parameters were used for the subsequent chemical kinetics simulations. To obtain more accurate energies at the stationary points on the PES, single-point energy calculations were carried out using the CBS-QB3 composite method.32 The CBS-QB3 approach uses the highly accurate CCSD(T) method33 to compute electron correlations and applies an extrapolation technique to estimate the energy at the complete-basis-set (CBS) limit.32 Note that, in the present work, the B3LYP/6-311þG(d,p) zero-point vibrational energies (ZPEs) were used in place of the ZPE computed using the smaller basis set [B3LYP/6-311G(d,p)], which was specified in the original description of the CBS-QB3 method.32 Table 1 shows that the reaction enthalpies computed using the CBS-QB3 method are within 12 kcal mol1 of the experimental values. It is thus anticipated that the barrier heights computed using the same method will be within about (2 kcal mol1 of the true values. In the earlier work discussed in the Introduction, CartasRosado et al.18,19 used MP2/6-31G(d,p) and B3LYP/6-31G(d,p) to characterize the addition reactions of NO3 with C2H4 and with other alkenes. They identified two parallel reaction mechanisms for the first step of NO3 addition to the olefinic double bond, namely, a concerted cyclic addition to produce a cyclic intermediate and a nonconcerted addition to form an open-chain intermediate, using the MP2 method. Surprisingly, the B3LYP method was unable to identify the concerted pathway, although it was successful in finding a transition structure for the nonconcerted pathway. As a result, CartasRosado et al. used MP2 and higher-level single-point calculations [including CCSD(T) with Dunning’s correlation-consistent triple-ζ basis set34,35] to obtain the energies of the two transition states. In the present work, all calculations were carried out using the Gaussian36 and CFOUR37 quantum chemistry software packages. Direct Dynamics Methods. Quasiclassical trajectory calculations were performed to determine the possible product channels. 4895

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Table 1. Calculated Relative Energies (kcal/mol) of Various Species in the C2H4 þ NO3 Reaction Obtained Using the CBS-QB3//B3LYP/6-311þG(d,p) and CBS-QB3//MP2/ 6-311þG(d,p) Levels of Theorya species R: C2H4 þ NO3

CBS-QB3//MP2 CBS-QB3//B3LYP exptl (0 K)b 0.00

0.00

0.00

D: oxirane þ NO2

37.55

34.13

34.11

E: CH3CHO þ NO2

64.91

61.28

62.81

F: 2CH2O þ NO

64.35

62.44

62.05

a

20.30

17.72 15.39

b

29.04 6.33

c TS-ar

2.76 c

TS-rc

3.53

7.10 d

TS-ab

15.58

13.73

TS-ac

0.46

2.40

TS-bd

0.43

0.62

TS-ce

21.80

20.71

TS-af

17.98

a

on effective mass and potential energy as functions of the dihedral angle.4851 The effective mass was obtained by using computed geometries (see above) with program Lamm (part of the MultiWell Program Suite). Energy transfer was treated using the conventional exponential-down model,52 which can be expressed as 1 ðE0 EÞ 0 exp PðE, E Þ ¼ for E0 g E ð1Þ NðE0 Þ R where P(E,E0 ) is the probability density that a molecule initially excited to energy E0 will be found in the range from E to E þ dE following a single collision. The probability density for collisions with E0 < E is computed using detailed balance. The energy-transfer parameter R, which is almost identical to ÆΔEædown, the average energy transferred in deactivating collisions, was varied empirically over a reasonable range based on experience with similar models.52 The collision frequency was computed for collision partners governed by a Lennard-Jones interaction potential. The Lennard-Jones parameters for pure N2 were obtained from the literature,53 and those for the excited adduct were assumed, based crudely on analogy with molecules of about the same mass (see the MultiWell User Manual for a handy list of Lennard-Jones parameters). All isomers of the excited adduct were assumed to have the same energy-transfer parameters. At low energies, the collision frequency is assumed to depend on the density of states, as described elsewhere,42 but this assumption has little effect at the energies of the present simulations. The vibrational frequencies and rotational constants for all relevant species are summarized in Tables S2 and S3 (Supporting Information), total relative energies are summarized in Table S4 (Supporting Information), and Lennard-Jones parameters are summarized in Table S5 (Supporting Information).

Experimental reaction enthalpies at 0 K given for the purpose of comparison. b “Experimental” reaction enthalpies at 0 K taken from the NIST-JANAF Thermochemical Tables,62 unless stated otherwise: ΔHf(C2H4) = 14.58 ( 07 kcal/mol, ΔHf(NO3) = 18.53 ( 5.00 kcal/mol, ΔHf(oxirane) = 9.59 ( 0.15 kcal/mol, ΔHf(NO2) = 8.59 ( 0.19 kcal/ mol, ΔHf(CH3CHO) = 38.29 ( 0.36 kcal/mol (from Wiberg et al.63), ΔHf(NO) = 21.46 ( 0.04 kcal/mol, and ΔHf(CH2O) = 25.20 ( 0.08 kcal/mol (from Czako et al.64). c Energy difference of 3.57 kcal/mol for TS-ar between the CBS-QB3//MP2 and CBS-QB3//B3LYP levels of theory is mainly due to a discrepancy of 4.83 kcal/mol in the MP2 ZPE for NO3 compared to the B3LYP ZPE (see Table S2 in the Supporting Information). d Computed as 6.33 þ (3.53 2.76) = 7.10 kcal/mol, (i.e., energy of TS-ar at the CBS-QB3//B3LYP level of theory plus an energy gap of (3.53 2.76) between TS-rc and TS-ar at the CBSQB3//MP2 level of theory).

’ RESULTS

The VENUS program38 coupled with the semiempirical quantum chemistry package MSINDO39 was used for this purpose. The relative translational energy was set to 20 kcal/mol, and other modes were sampled according to a 298 K Boltzmann distribution. Trajectories were run for 500 fs with a 0.05-fs time step using the velocity Verlet integration method. Because the purpose of the simulation was to determine possible product channels, only about 100 trajectory calculations were performed, and branching ratios to the channels were not determined precisely. Chemical Kinetics Simulations. Chemical kinetics simulations were carried out using the MultiWell Program Suite,4042 which includes computer codes for calculating thermochemistry (program Thermo), sums and densities of states (program DenSum), and canonical transition state theory rate constants (program Thermo). In addition, the MultiWell Program Suite includes program MultiWell, which is a multichannel, multiwell master-equation code suitable for simulating the atmospheric reactions of NO3 with olefins. The microcanonical rate constants needed for the simulations were computed using RiceRamspergerKasselMarcus (RRKM) theory,4345 based on sums and densities of states computed using the BeyerSwinehart algorithm46 as implemented by Stein and Rabinovitch.47 Hindered internal rotations were treated as separable one-dimensional degrees of freedom; the appropriate Schr€odinger equation was solved to obtain the eigenstates, based

Concerted Cyclic Addition Pathway? To determine which (B3LYP or MP2) optimized geometries are more reliable, singlepoint calculations were carried out at the CBS-QB3 level of theory to determine the relative energies of the transition structures for the initial addition reaction. Single-point calculations using CBS-QB3 at the MP2 and B3LYP geometries are presented in Table 1 for comparison. The CBS-QB3//B3LYP results are in better agreement with the experimental reaction enthalpies. Moreover, the MP2 method was tested and found to be unsuitable for describing the geometry of the NO3 radical (see Table S2, Supporting Information), suggesting that the MP2 method might not be reliable for investigating reactions involving NO3. A transition structure for the concerted cyclic addition (designated as TS-rc) was successfully located and optimized using the MP2 method in the present work, but our attempts to use the B3LYP method failed, just as reported previously by Cartas-Rosado et al.19 Because TS-rc might play an important role in the reaction of NO3 with alkenes, we investigated the reaction further by optimizing the geometry of the transition structure with other levels of theory. These included the CCSD(T)/ 6-311G(d,p) method, which is a much higher level of theory than MP2, and the B97-1/6-311þG(d,p) method,54 which describes NO3 radical fairly well (see Table S2, Supporting Information). Various geometries obtained by optimizing the geometry while imposing Cs symmetry for TS-rc are presented in Figure 1. The results in Figure 1 show that the optimized geometries obtained 4896



Figure 1. Optimized geometries for TS-rc of cyclic addition under imposed Cs symmetry obtained at various levels of theory. Note that the MP2 method gives a first-order saddle point, whereas the CCSD(T) and B97-1 methods generate a second-order saddle point.

with CCSD(T) and B97-1 are superficially in good agreement with that obtained using MP2. The transition state obtained using MP2 is a proper first-order saddle point (i.e., it has only one imaginary frequency), whereas the transition states obtained using CCSD(T) and B97-1 are second-order saddle points with two imaginary frequencies. Vibrational analyses of the two normal modes with imaginary frequencies reveal that the first imaginary frequency corresponds to the cyclic addition of NO3 to the double bond and the other imaginary frequency corresponds to a deformation of the open-chain adduct a. When the transition state is optimized without imposing a particular symmetry, the geometry converges on the open-ring transition state, TS-ac (Figure 2). Because the transition state for cyclic addition obtained at the CCSD(T) level of theory contains two imaginary frequencies and is therefore not a proper transition state and because CCSD(T) is more accurate than MP2, we conclude that the concerted mechanism to form a cyclic adduct is an artifact associated with use of the MP2 method. The results obtained using the B97-1 method are consistent with this conclusion, because the B97-1 method is more resistant than MP2 to the effects of symmetry breaking. The subsequent single-point energy calculations using the CBSQB3 level of theory based on the CCSD(T)- and B97-1-optimized geometries for TS-rc give similar results, 358.327208 and 358.327098 hartree, respectively. These values are slightly lower than the CBS-QB3//MP2 value of 358.323313 hartree. However, the CBS-QB3//CCSD(T) energy was not used for subsequent kinetics treatments because the CCSD(T)-optimized geometry is a second-order saddle point. Unless stated otherwise, the CBS-QB3//B3LYP energies and B3LYP rovibrational parameters were used to construct the PES and carry out the kinetics simulations, as described in the following sections. Potential Energy Surface and Reaction Mechanism. The critical points on the PES are shown schematically in Figure 2 and listed in Table 1. The first step in the reaction of NO3 with C2H4 is electrophilic addition of NO3 to an olefinic carbon atom to form the open-chain adduct a. This addition step takes place through transition structure TS-ar, which has a barrier height of 6.3 kcal/ mol (including ZPE corrections), as calculated using the CBSQB3//B3LYP/6-311þG(d,p) level of theory. The concerted

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cyclic addition pathway through TS-rc, which is ignored for the reasons given in the previous section, lies 0.8 kcal/mol higher in energy than TS-ar. Moreover, the entropy of TS-rc is much smaller than that of TS-ar, corresponding to a very “tight” transition structure. As a result, the rate of the cyclic addition would be about 2 orders of magnitude lower than that of the open-chain addition. Thus, even if the concerted addition mechanism took place through a proper transition state, the reaction would not be competitive with the open-chain addition step. The open-chain adduct a formed through TS-ar lies 17.7 kcal mol1 below the reactants NO3 þ C2H4. The internal energy of a must be at least 24 kcal mol1, because 6.3 kcal mol1 is required to overcome the barrier at TS-ar. The energized adduct a* can undergo reaction through any of four possible reaction pathways. Pathway I starts with rapid isomerization to b through the low-lying TS-ab (barrier height of 4.0 kcal mol1); isomer b then loses NO2 through TS-bd (barrier height of 17.0 kcal mol1) to produce ethylene oxide (oxirane). As discussed in the next paragraph, there is no evidence for direct NO2 loss from a. Pathway ii proceeds through TS-ac (barrier height of 20.1 kcal mol1) to form the five-membered ring intermediate c, which lies 29.0 kcal mol1 below the initial reactants. The internal energy of the nascent highly vibrationally excited intermediate c will be at least 25.3 kcal/mol; it will quickly decompose through TS-ce (barrier height of 8.3 kcal mol1) to produce two formaldehyde molecules plus nitric oxide. Pathway iii consists of redissociation back to the initial reactants through TS-ar. However, TS-ar lies about 46 kcal mol1 higher than TS-bd and TS-ac. As a result, the redissociation channel will be almost negligible under all atmospheric conditions. Finally, pathway iv proceeds through a simultaneous 1,2-H shift and NO2 loss (TS-af) to form acetaldehyde. However, this step is negligibly slow because it requires that an energy barrier of 35.7 kcal mol1 be overcome. A search was carried out for a direct NO2-loss channel from the vibrationally excited adduct a*, but no such transition state was found. Two different techniques were used to search for the transition state, namely, a full optimization and a constrained optimization, followed by reoptimization without freezing (e.g., using the keyword NOFREEZE in Gaussian). All of these calculations were done at the UB3LYP/6-311þG** level of theory. The full optimization always converged to TS-bd (see Figure 2), which connects b with the products oxirane þ NO2. In all of our tests, this behavior did not depend on whether the initial geometry more closely resembled the structure of a or b. On the other hand, the constrained optimization starting with the dihedral angle NOCC of a led to a nonstationary point (i.e., some of the gradients were not equal to zero). Further optimization using this geometry as an initial guess again generated TS-bd. Thus, we conclude that there is no direct NO2-loss pathway from a that does not pass through b on the way to forming oxirane þ NO2. In addition to the reaction paths described in the preceding paragraph, each intermediate can undergo collisional stabilization. Because all of the reaction rate constants depend on excitation energy, the resulting system can be described with reasonable accuracy only by a master-equation simulation. Direct Dynamics Calculations. To determine the possible reaction pathways, semiempirical direct dynamics simulations were performed as described above. Many trajectories were excluded because of poor energy conservation. However, it was possible to determine most of the product channels including oxirane þ NO2, open radical þ NO2, and NO þ 2CH2O. Snapshots of representative trajectories taken from the direct dynamics calculations are 4897



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Figure 2. Stationary points on the potential energy surface for the reaction NO3 þ C2H4 (based on the CBS-QB3//B3LYP energies and geometries; see Table 1 and the Supporting Information).

shown in Figure S5 (Supporting Information). Examination of the complete set of trajectories did not reveal any surprises, suggesting that all of the reaction channels that are important in the atmosphere have been included in Figure 2 and Table 1. Kinetics Simulations. The kinetics simulations were based on the PES shown in Figure 2, except that TS-rc was ignored for the reasons given above. The computed energies were used without adjustments, except for that of transition state TS-ar. For one set of simulations, the computed energy of TS-ar was used without adjustments, but a second set of simulations employed a barrier height for TS-ar reduced by 1.0 kcal/mol. The magnitude of this adjustment is within the range of error expected for the CBSQB3 method,32 especially when the enthalpy of formation of the NO3 free radical computed using CBS-QB3 is in error by ∼34 kcal/mol (see Table S1, Supporting Information). Experience shows that the CBS-QB3 method can predict “chemically accurate” energies (including barrier heights, heats of formation, bond dissociation energies, and others) within 12 kcal mol1 of experimental values.5559 The empirical adjustment (i.e., reduction of the energy of TS-ar by 1 kcal mol1) was made to obtain better agreement with the rate constants measured by Canosa-Mas,5 Atkinson et al.,15 Biggs et al.,14 and Boyd et al.,6 as described below. As described in the previous section, the primary step in the reaction of NO3 with ethene is formation of an open-chain adduct with two low-energy rotamers a and b separated by TS-ab. The species are the two minima of a hindered internal rotation, and TS-ab is the hindering barrier. For present purposes, the two rotamers (a and b) were treated as a single species ab containing a hindered internal rotation. To compute a reasonably accurate entropy for the open-chain adduct ab (including both rotamers), the torsional potential energy, V(χ), and effective reduced moment of inertia, Ir(χ), for all three hindered rotations (i.e., around the NO, OC, and CC bonds) in adduct ab were obtained by carrying out a sequence of geometry optimizations constrained at fixed dihedral angles, χ, corresponding to torsion around each bond

separately. The three torsions were then treated as separable, and the entropy and density of states were computed using the standard options available in programs Thermo and DenSum, respectively, which are modules in the MultiWell Program Suite.4042 The separable approximation neglects interactions among the three hindered rotors. The properties of TS-ar were obtained in a similar manner. The lowest two harmonic frequencies in TS-ar (i.e., 70 and 78 cm1) correspond to two torsions. When animations of the normal modes corresponding to these frequencies were viewed, it was noted that the motions are qualitatively very similar. It also was not readily apparent how to carry out scans to obtain the potential energy as a function of dihedral angles for the two torsions uniquely. Because the harmonic frequencies and the motions are so similar, we decided to carry out just one scan and then treat the two torsions as identical. The torsion potentials and reduced moments of inertia for the ON and OC torsions in TS-ar were obtained in the same way as for intermediate a (see above). Because of the double-bond character in TS-ar, the torsion around the CC bond has a high barrier to internal rotation and was treated accurately with the harmonic oscillator approximation. The simulations were initialized by using the chemical activation energy distribution function40,4345,60,61 for intermediate ab formed through TS-ar and continued in simulated time until no further reaction was observed. Reaction branching ratios were obtained from the ratios of simulated populations that reacted to form bimolecular products or were essentially thermalized by collisions. Simulations were carried out at 280 and 300 K and over the pressure range from 1 mbar to 10 bar of N2, which is assumed to be a good proxy for air. The branching ratios obtained using the computed energies (i.e., the ab initio PES) and a simulated temperature of 300 K are shown in Figure 3. The results obtained for a simulated temperature of 280 K are very similar (see Figure S2, Supporting Information). In general, the branching ratios are not very sensitive to temperature or to the details of TS-ar. 4898



Figure 3. Simulations at 300 K using the computed PES without adjustments. The solid curves show the fractional yields of collisionally thermalized adduct ab (the combined concentrations of rotamers a and b). The dashed curves show the fractional yields of oxirane þ NO2. Simulations were carried out for energy-transfer parameter values of R = 250, 500, and 1000 cm1.

The results in Figure 3 show that, at low pressures, the products consist almost entirely of NO2 þ oxirane, whereas at high pressures, where collisional stabilization is very important, the thermalized NO3C2H4 adduct ab is dominant. The other possible products amount to only a few percent at all pressures. In particular, the back-reaction to regenerate NO3 þ C2H4 amounts to only ∼0.2% at the lowest simulated pressure and vanishes at the highest pressure. At atmospheric pressure, the stabilized free-radical adduct ab accounts for 8090% of the yield and most likely subsequently reacts with O2 to form the nitratoethylperoxyl radical (O2NOCH2CH2OO•), as is generally assumed in tropospheric chemistry models.911 In this chemical activation system, the reaction rate constant for formation of the ith reaction product is given by ki = fik¥, where fi is the fractional yield of the ith reaction product and k¥ is the high-pressure rate constant for formation of the nascent intermediate ab through the reaction between NO3 and C2H4. The high-pressure rate constant k¥ is identical to the rate constant computed using canonical transition state theory (CTST).4345 Because the fractional yield corresponding to the back-reaction to form NO3 þ C2H4 was found in the simulations to be essentially negligible, the rate constant k¥ can be identified with the experimental rate constant. The CTST rate constant, including quantum mechanical tunneling through an unsymmetrical Eckart potential barrier, was computed for temperatures from 100 to 2000 K using the program Thermo41 and is shown along with the experimental data5,6,1315 in Figure 4. The critical energy for reaction, E0, computed using CBSQB3, as described above, is 6.33 kcal mol1. Although tunneling was included in the calculations, its effect was negligible, as the imaginary frequency was found to be only ∼135i cm1. The rate constant was first calculated using the harmonic oscillator approximation for the torsional degrees of freedom, with the result being a factor of 30 or more smaller than the experimental data, as shown in the figure. It is well known that the harmonic oscillator approximation performs poorly for hindered internal rotations

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Figure 4. Experimental rate constants and computed models. The experimental error bars are (1σ (from the original publications).

with harmonic frequencies on the order of kBT/hc or less. Thus, the more accurate treatment of the torsions described above is needed for treating transition state TS-ar accurately. When the torsions in transition state TS-ar were treated explicitly as hindered internal rotations as described above, the computed CTST rate constants increased by about an order of magnitude (the middle curve in Figure 4). Lowering E0 to 5.33 kcal mol1 brings the computed rate constant into very good agreement with the room-temperature experimental data. Moreover, the computed rate constant is also in very good agreement with the temperature-dependent data of Canosa-Mas et al.5 Branching ratios computed as described above are hardly affected at all by this adjustment (see Figures S2 and S3, Supporting Information). Thus, the effective rate constants for production of the major products can be obtained by combining the computed k¥(T) value with the branching ratios in Figure 4 (or Figures S1S3, Supporting Information), which are not very sensitive to the details of TS-ar. For convenience, the results for the CTST rate constant were fitted to the following empirical expression by nonlinear least-squares, weighted by assuming constant relative errors k¥ ðTÞ ¼ 6:375 1013 ð300=TÞ2:148 expð2:383=TÞ for 100 K e T e 2000 K

ð2Þ (four significant digits were retained in order to accurately reproduce the calculated results).

’ SUMMARY AND CONCLUSIONS The calculations reported herein are in good agreement with the experimental kinetics data on the reactions of NO3 with olefins. Not only are the rate constants in good quantitative agreement (after a small empirical adjustment to the energy of TS-ar that is acceptably within the known error limits of the calculation method), but the reaction mechanism is predicted to proceed by first forming an open-chain adduct (ab) and not by concerted formation of a cyclic intermediate. The latter mechanism is ruled out because the transition state to form the cyclic intermediate contains two imaginary frequencies when computed using 4899


The Journal of Physical Chemistry A high levels of theory and large basis sets. The kinetics simulations illustrated once again that hindered internal rotations play a major role in determining the rate constants and must be treated accurately: the harmonic approximation is simply not acceptable for the reactions considered here. When the torsions are treated explicitly as separable hindered internal rotations, the computed rate constants for the total reaction are in very good agreement with the experimental data. By using the computed total rate constant and the computed branching ratios, rate constants for formation of the major products can be predicted as functions of temperature and pressure for conditions relevant to the atmosphere.

’ ASSOCIATED CONTENT

bS

Supporting Information. Optimized geometries, calculated electronic energies, harmonic zero-point energies, rovibrational parameters, and the results of kinetics simulations. This information is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (K.S.), [email protected] (J.R.B.).

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Mechanism and Kinetics of the Reaction NO3 + C2H4 - The Journal

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