Predictive Theory for the Addition and Insertion Kinetics of 1 CH 2 Reacting with Unsaturated Hydrocarbons

Transcription

1 Paper # 070RK-0281 Topic: Reaction Kinetics 8 th U. S. National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University of Utah May 19-22, 2013 Predictive Theory for the Addition and Insertion Kinetics of 1 CH 2 Reacting with Unsaturated Hydrocarbons Daniela Polino, 1,2 Stephen J. Klippenstein, 1,* Lawrence B. Harding, 1 and Yuri Georgievskii 1 1 Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, IL, Dipartimento di Chimica, Materiali e Ingegneria chimica G. Natta, Politecnico di Milano, via Mancinelli 7, 20131, Milano, Italy Abstract The reaction of singlet methylene, 1 CH 2, with unsaturated hydrocarbons is of considerable significance to the formation and growth of polycyclic aromatic hydrocarbons (PAHs). In this work, we employ high level ab initio transition state theory to predict the high pressure rate coefficient for singlet methylene reacting with acetylene (C 2 H 2 ), ethylene (C 2 H 4 ), propyne (CH 3 CCH), allene (CH 2 CCH 2 ), 1,3-butadiene (CH 2 CHCHCH 2 ), and benzene (C 6 H 6 ). Both addition and insertion channels are found to contribute significantly to the kinetics, with the insertion kinetics of increasing importance for larger hydrocarbons due to the increasing number of CH bonds. We treat the addition kinetics with direct CASPT2 based variable-reaction-coordinate transition-state-theory (VRC-TST). One dimensional corrections to the CASPT2 interaction energies are obtained from geometry relaxation calculations and CCSD(T)/CBS evaluations. The insertion kinetics are treated with traditional TST methods employing CCSD(T)/CBS energies obtained along CASPT2/cc-pVTZ distinguished reaction coordinate paths. The overall rate constant and branching fractions are obtained from a multiple transition state model that accounts for the physical distinction between tight inner and loose outer transition states. Our predicted rate constants, which cover the range from 200 to 2000 K, are found to be in excellent agreement with the available experimental data, with maximum observed discrepancies of 20-30%.

2 1. Introduction The insertion of singlet methylene, 1 CH 2, into unsaturated hydrocarbons yields resonantly stabilized radicals, which play important roles in hydrocarbon growth through their recombination reactions. For instance, the addition of 1 CH 2 to acetylene leads to propargyl radical, whose self-recombination plays a significant role in the formation of the first aromatic ring under most conditions [Pope 2000, Miller 2010, Georgievskii 2007, Hansen 2010, Hansen 2009, McEnally 2006]. Similarly, addition to benzene yields benzyl radical and/or toluene with toluene readily converted to benzyl radicals via H abstractions. Notably, just as with propargyl, benzyl radical is resistant to further decomposition [Derudi 2011, Cavallotti 2009, da Silva 2009, Braun-Unkhoff 1990, Hippler 1990, Oehlschlaeger 2006, Sivaramakrishnan 2006, Sivaramakrishnan 2011] and so its recombination reactions may be key steps in the formation of larger PAHs [Matsugi 2012]. Analogous reactions of 1 CH 2 with larger PAHs would similarly contribute to further growth in PAH size [Slavinskaya 2009]. Hence, information about the rates of 1 CH 2 addition to a variety of unsaturated hydrocarbon compounds is useful for the development of combustion models for soot formation. The reaction of 1 CH 2 with acetylene has served as a benchmark reaction for the study of the reactivity of carbenes with unsaturated species. Its key role in the formation of propargyl radical has led to a large number of theoretical [Guadagnini 1998, Frankcombe 2001, Yu 2005] and experimental [Canosamas 1985, Hack 1988, Hayes 1995, Adamson 1996, Hayes 1996, Blitz 2000, Davis 2004, Gannon 2010] studies aimed at delineating the temperature and pressure dependence of its kinetics. Its overall kinetics has been experimentally investigated by several authors at room temperature [Canosamas 195, Hack 1988, Hayes 1995, Adamson 1996], with reported values ranging from 2.5 to 3.7 x cm 3 molecule -1 s -1. However, only a few studies have examined the temperature-dependence of the kinetics ( K) [Blitz 2000, Gannon 2010]. Notably, the theoretical predictions of a nearly temperature independent rate constant are in sharp contrast with the experimental results, which show a significant decay with increasing temperature. Furthermore, this discrepancy cannot be explained by a temperature dependent contribution from deactivation to 3 CH 2. In the case of ethylene, a number of authors [Canosama 1985, Hayes 1985, Langford 1983, Hack 1989, Staker 1991] measured the room temperature rate coefficient for its addition to singlet methylene, obtaining results ranging from 1.5 to 2.5 x cm 3 molecule -1 s -1. Temperature dependent studies were carried out by Wagener et al. [Wagener 1990] and Hayes et al. [Hayes 1996] over the range K, and more recently by Gannon et al. [2010] over the range K. Hack and coworkers [Hack 1989] found that the relaxation contribution to the overall 1 CH 2 removal rate at room temperature is 0.20±0.04, which is very similar to the value they determined for acetylene (0.22±0.07). There have been only a few studies of the kinetics of singlet methylene reacting with larger hydrocarbons. Wagner s group used laser induced fluorescence (LIF) to measure the removal rate of 1 CH 2 for a large set of olefins and aromatic species at room temperature [Hack 1988, Wagener 1990, Hack 1989, Koch 1990]. In these studies, they also evaluated the branching ratios for physical deactivation at room temperature and the temperature dependence for the overall removal rate with methane, ethane, ethylene, and benzene. Hayes et al. [Hayes 1996] also determined the room temperature removal rate constant for a large number of saturated and unsaturated hydrocarbons, with laser flash photolysis/laser absorption experiments. Finally, Gannon et al. [Gannon 2010] have recently reported a temperature dependent rate coefficient for the reaction of methylene with propene. 2

3 Unfortunately, there have been no experimental studies of 1 CH 2 addition reactions for the temperature range of relevance to combustion modeling. Furthermore, the difficulty of deconvolving the relaxation and reaction contributions combined with the considerable discrepancy between the available theoretical and experimental studies implies that there is considerable uncertainty in the proper temperature dependence of the chemical reaction rate even over the studied range of 200 to 700 K. Here, we apply high level theoretical methods to the study of the high pressure rate constant for the reaction of 1 CH 2 with a series of unsaturated hydrocarbons. Both addition and insertion channels are found to contribute significantly to the kinetics. The overall rate constant and branching fractions are obtained from a multiple transition state model that accounts for the physical distinction between these two channels and between tight inner and loose outer transition states. We employ the direct variable reaction coordinate transition state theory (VRC-TST) approach [Georgievskii 2003a, Georgievskii 2003b, Klippenstein 1992, Klippenstein 1997] to treat the formation of a long-range complex and for the addition across the π- bond to form a three membered ring. This approach has been shown to yield accurate predictions for the kinetics of barrierless radical-radical recombination reactions [Georgievskii 2007, Harding 2005, Klippenstein 2002, Klippenstein 2006]. Meanwhile, we employ reaction path variational transition state theory to treat the inner transition state for the insertion process. All singlet methylene addition reactions are highly exothermic because 1 CH 2 is a very unstable species. Thus, for each reaction considered here, low energy bimolecular channels are accessible and the overall rate coefficients are expected to be pressure independent. Complete kinetic analyses of the temperature and pressure dependent branching between the addition, insertion, and subsequent isomerization channels [Frankcombe 2001] is important, but is beyond the scope of this work. The outline of this paper is as follows: in section 2 we provide the details of the electronic structure and TST methods used here. Then, in section 3, the predictions for the high pressure addition rate coefficients are presented, discussed, and compared in detail with the available experimental data. Finally, some concluding remarks are given in section Methods In this section we first review the kinetic theory employed here and then discuss the electronic structure methods used to generate the interaction energies that are needed in the kinetic theory. 2.1 Kinetic Theory Multiple Transition State Model The reactive flux for the addition process shows two bottlenecks. One correlates with the formation of a longrange/van der Waals complex, while the other correlates with formation of either the 3-membered ring or the insertion into one of the CH bonds. In our recent discussion of roaming radical reactions [Klippenstein 2012], we described a statistical framework for treating the coupled kinetics for just such situations. For the present reactions, the effective flux for formation of the 3-membered ring may be expressed as 3

4 !!"!#$!!""!,! =!!"#$%!,!!!"!#$!!""#$!,! /!!"#$%!,! +!!"!#$!!""#$!,! +!!"#$%&!'"!""#$!,! while the effective flux for formation of the insertion products is given as!!"#$%&!'"!"!!,! =!!"#$%!,!!!"#$%&!'"!""#$!,! /!!"#$%!,! +!!"!#$!!""#$!,! +!!"#$%&!'"!""#$!,! The individual rate constants are obtained from thermal averages of these channel dependent effective fluxes. Obvious generalizations apply to the cases with more than one possible insertion cite. Variable Reaction Coordinate Transition State Theory The present calculations implement the direct variable reaction coordinate transition state theory (VRC-TST) approach to treat both!!"#$%!,! and!!"!#$!!""#$!,!. The direct VRC-TST approach has been described in detail in previous investigations on hydrogen atom addition to alkyl [Harding 2005] and alkenyl radicals [Georgievskii 2003b] and on alkyl + alkyl combination reactions [Georgievskii 2007, Klippenstein 2006]. Thus, we provide only a brief review of the methodology focusing primarily on the details relevant to its application to the addition of singlet methylene to unsaturated hydrocarbons. A separation into conserved and transitional modes is a key simplifying assumption within the VRC-TST approach. The conserved modes are taken to be the vibrational modes of the reacting fragments, while the remaining transitional modes describe their rotational and translational motions. The contribution to the transition state partition function from the conserved modes is readily evaluated via a direct sum over the vibrational levels, which are typically treated as harmonic oscillators. The VRC-TST approach then focuses on an accurate evaluation of the transitional mode contribution to the transition state partition function via Monte Carlo integration of classical phase space integral representations. An important feature of the VRC-TST protocol for the transitional modes is the definition of the transition state dividing surface in terms of a set of pivot point locations and separations. These pivot points define the location of the origin for the effective fragment rotations within the transition state. A fixed distance between a pivot point on one fragment and one on the other fragment defines a dividing surface separating reactants from products. Hence, in accord with the variational principle of TST, the rate coefficient is minimized with respect to both the pivot point locations and separations. For the present study of 1 CH 2 + alkene/alkyne reactions we employ the multi-faceted dividing surface approach [Georgievskii 2003], where multiple pivot points are specified for each fragment. The overall dividing surface is then composed of the set of faces corresponding to fixed distances between each pair of pivot points. Previous studies have demonstrated that at large separations the optimum position for the pivot point is at the center-of-mass (CoM). Meanwhile, at short separations orbital centered pivot points are preferred. For the singlet methylene fragment we used two pivot points, positioning them along the direction perpendicular to the plane formed by the three atoms and passing from the CoM of the fragment (c.f., Fig. 1a). We located one point on each side of the plane in order to take into account the possibility that the 1 CH 2 can attack the unsaturated bond from either of its sides. The distance of these two pivot points from the CoM was varied from 0.01 to 0.5 and 1.0 Bohr in the optimizations. 4

5 Figure 1. Number of pivot points and locations for a) methylene, b) acetylene, c) ethylene, d) propyne, e) propene, f) allene, g) butadiene, and h) benzene. Distances are in Bohr. For all other reactive fragments the pivot point positions are summarized in Figure 1(b-h). Importantly, within the multifaceted dividing surface VRC-TST approach, it is possible to take advantage of the symmetry properties of the fragments. For example, in the case of acetylene plus methylene, two pivot points were employed for each fragment and the pair of pivot points for each fragment are identical. Thus, by symmetry, the dividing surfaces generated by their combination are also identical. Hence, we calculate the flux corresponding to one dividing surface and then simply multiply it by a factor of 4. For the pivot point to pivot point distance, a grid spacing of 0.26 Å was employed for short separations, while a grid spacing ranging from 0.26 to 1.1 Å was used for larger separations. As a consequence, on the order of 100 to 500 dividing surfaces have been considered for each addition reaction investigated here. The evaluation of the transitional mode contribution to the partition function is then accomplished via Monte Carlo integration over the transitional mode orientations, with up to a few thousand samplings per dividing surface. The Monte Carlo simulations were terminated when the estimated integration error bars were below the threshold of 10%. Finally, based on the results of previous studies [Harding 2005, Klippenstein 2006], a temperature independent dynamical correction factor of 0.85 was applied. This correction factor accounts for the limited recrossing of the VRC transition state dividing surface observed in trajectory studies for radical-radical reactions. 5

6 Variational Transition State Theory The insertion channels present an interesting challenge for TST methods. They are too tight, with too much structural rearrangement, to allow for an accurate treatment with our variable reaction coordinate approach. However, they are also too loose for a single fixed transition state to be appropriate. Indeed, for many of the insertions there is no clear saddle point in the insertion region. Nevertheless, we are able to obtain distinguished coordinate reaction paths via constrained optimizations with fixed CH distances. Following these paths out to large CH distances generally yields a long-range saddlepoint that correlates with the transition from the cyclic adduct to the insertion products. Rovibrational analyses along the distinguished coordinate path provide the data for variational rigid-rotor harmonic oscillator based treatments of the flux. We also incorporate hindered rotor treatments for the appropriate low frequency modes. 2.2 Electronic structure methods Cyclic Addition The Monte Carlo integration over the transitional mode coordinates requires accurate estimates of the interfragment interaction energies for on the order of ten thousand geometries. The development of analytic representations for the, generally 6 dimensional, transitional mode potential energy surface is difficult and timeconsuming. Instead, we evaluate the transitional mode interaction energies on the fly using an approach that is similar to the one employed in our previous studies for reactions such as the addition of atomic hydrogen to alkyl radicals and the recombination of two alkyl radicals [Harding 2005, Klippenstein 2006]. In particular, we employ a CASPT2 approach, which yields fairly accurate energies very efficiently, to directly evaluate the orientation dependence of the interaction energies. Additional calculations are performed along the minimum energy path to yield onedimensional corrections based on higher level evaluations and also to account for the effects of the relaxation of the fragment geometries at shorter separations. To begin, the reactant geometries were optimized at the CASPT2/cc-pVDZ level [Knowles 1985, Werner 1985, Werner 1996, Celani 2000, Dunning 1989]. For singlet methylene a two electron two orbital (2e,2o) active space was used. For the unsaturated hydrocarbons the π bonding and π * anti-bonding orbitals are strongly correlated and so we included each of these orbitals in the active space for their geometry optimization. Thus, for example a four electron four orbital (4e,4o) active space was considered for acetylene. The orientation dependent interfragment interaction potentials were generally determined with a CASPT2(2e,2o)/cc-pVDZ approach, where the (2e,2o) active space correlates with the singlet methylene orbitals. For computational expediency, the π and π * orbitals of the unsaturated hydrocarbons were generally not included in the active space for this orientational sampling. As described below, test calculations for acetylene demonstrate that their exclusion is acceptable. However, for ethylene and propylene, certain wavefunction convergence problems were encountered with the (2e,2o) active space. For these cases, we instead employ a (4e,4o) active space that includes the π and π * orbitals of ethylene/propene because it yields a more consistent set of interaction energies. The one-dimensional, orientation-independent correction (ΔV corr ) to the CASPT2/cc-pVDZ potential is obtained from the expression: 6

7 ( ( )/ ) ( 2/ ) Δ Vcorr = E CCSD T CBS E CASPT cc pvdz + fixed ( 2/ ) ( 2/ ) E CASPT cc pvdz E CASPT cc pvdz relaxed fixed (1) In this expression, the subscript fixed means that the calculations were carried out while keeping the geometries fixed at their asymptotic equilibrium geometries, while the subscript relaxed corresponds to energy evaluations allowing for the relaxation of the internal degrees of freedom of the two fragments. The term in the first brackets accounts for limitations in the accuracy of the CASPT2/cc-pVDZ method, and from here on we will call it the method correction. This correction is based on CCSD(T)/CBS(cc-pVTZ;cc-pVQZ) [Dunning 1989, Raghavachari 1989] one-dimensional orientation independent evaluations along the minimum energy path. The extrapolation to the complete basis set limit was obtained using the scaling coefficient proposed by Martin [Martin 1998]. The term in the second brackets accounts for the contribution due to the relaxation of the conserved modes, and from now on it will be referred to as the relaxation correction. To obtain this correction we optimized the internal fragment geometries (at the CASPT2/ccpVDZ level) at various points along the reaction coordinate, while also constraining the relative orientation of the two fragments. All the calculations carried out in this work were performed with the MOLPRO package [Werner]. Insertion The distinguished coordinate insertion paths were determined at the CCSD(T)/CBS(cc-pVTZ;ccpVQZ)//CASPT2/cc-pVTZ level. Rovibrational analyses were performed with the CASPT2/cc-pVTZ method. The active spaces again consisted of the (2e,2o) actives spaces for the 1 CH 2 component and (2e,2o) or (4e,4o) actives spaces for the component correlating with the double and triple bond unsaturated hydrocarbons, respectively. 3. Results and Discussion The present dynamically corrected VRC-TST predictions for the temperature dependent rate coefficient, based on the corrected CASPT2/cc-pVTZ energies, are plotted in Fig. 2 for the addition reaction of 1 CH 2 with acetylene. Variations in the various correction terms discussed suggest that the overall uncertainty in the rate predictions is on the order of 20-30%. The experimental data and theoretical results available in the literature are also compared with these predictions. The calculated VRC-TST rate coefficient is in very good agreement with the numerous experimental rates measured at room temperature. Moreover, our predictions quantitatively reproduce the temperature dependence found in the experimental work of Blitz et al. [Blitz 2000] and Gannon et al. [Gannon 2010]. Notably, for this reaction the contribution from insertion is predicted to be negligible, being 15% or less throughout the whole 200 to 2000 K temperature range. 7

8 Figure 2. VRC-TST high pressure limit rate coefficient for the 1 CH 2 + C 2 H 2 reaction (solid red line) compared with experimental measurements (symbols) and other theoretical predictions (lines). It is important to point out that all the experimental data are based on removal measurements of 1 CH 2 and thereby contain information about both reaction with acetylene and physical deactivation to the triplet ground state. As mentioned in the introduction, Hack et al. [Hack 1988] measured the contribution from relaxation to the total removal rate of singlet methylene in the presence of acetylene to be 0.22±0.07 at room temperature. Further insight on this aspect has been given by Gannon et al. [Gannon 2010]. Briefly, they monitored directly the loss of 1 CH 2 and the production of H atoms. Thus, they deduced a branching ratio for the reaction of 1 CH 2 with C 2 H 2 to yield propargyl and an H atom, which has been proposed as the main reactive path [Adamson 1996, Davis 2004]. Interestingly, at room temperature they obtained a value of 0.88±0.09, which is coherent with the fraction of electronic relaxation measured by Hack et al. [Hack 1988]. 29 In addition, Gannon and coworkers investigated the temperature dependence of H atom yields. In particular, they observed that the branching ratio between reaction and electronic relaxation increases slightly between 298 and 398 K and becomes equal to 1 above 398 K. This temperature dependence is similar to the dependence observed for the 1 CH 2 + H 2 /D 2 system, but is in contrast with measurements made with inert gases. Hence, all experimental results shown in Figure 2 were plotted after subtracting this contribution, which was maintained constant for all temperatures. This choice is motivated by the lack of experimental data on the temperature dependence of physical deactivation. Previous theoretical predictions made by Guadagnini et al. [Guadagnini 1998], and more recently by Yu and Muckerman [Yu 2005], reproduce the room temperature experimental data, but fail to predict the negative temperature dependence found by Blitz et al. [Blitz 2000] and Gannon et al. [Gannon 2010]. By necessity, the early study of Guadagnini et al. was based on relatively low-level electronic structure evaluations for the vibrational frequencies (CASSCF/DZ), and furthermore presumed rigid rotor harmonic oscillator behavior in the transition state region. Thus, it is perhaps not surprising that the present predictions yield quite different predictions for the temperature dependence. Yu and Muckerman, on the other hand, carried out ab initio trajectory simulations using forces calculated at the UCCSD/cc- 8

9 pvdz level and then extrapolated to the UCCSD(T)/CBS level with the SAC approach of Truhlar et al. [Gordon 1989, Gordon 1986, Gordon 1987]. However, they employ a 300 K canonical ensemble for their rotational sampling, apparently even for their high temperature evaluations of the rate coefficient. This inappropriate restriction in rotational states is likely responsible for their discordant prediction for the temperature dependence of the addition kinetics. Figure 3. VRC-TST high pressure limit rate coefficient for the 1 CH 2 + C 2 H 4 reaction (solid red line) compared with experimental measurements (symbols). The present dynamically corrected VRC-TST predictions for the temperature dependent rate coefficient in the reaction of singlet methylene with ethylene is plotted in Figure 3 together with the available experimental data. These calculations also employed a corrected CASPT2/cc-pVTZ potential. However, in this case it was necessary to introduce a 4 electron 4 orbital CAS active space in order to correctly follow the potential energy surface. The present a priori predictions again show good agreement with both the rate values measured at room temperature, and also the temperature dependence found by Gannon et al. [Gannon 2010]. The plotted experimental values were again reduced by a temperature independent factor of 20%, in order to account for the contribution from deactivation to the measured total removal rates. The contribution from insertion is slightly larger in this case, with a typical contribution of 20%. 3.2 Predictions for 1 CH 2 + pc 3 H 4, ac 3 H 4,, C 3 H 6, C 4 H 6, and C 6 H 6 The multiple TS model, incorporating VRC-TST calculations for the key channels, was implemented for the addition of singlet methylene to a series of unsaturated hydrocarbons that are commonly present in combustion environments: propyne, propene (pc 3 H 4 ), allene (ac 3 H 4 ), butadiene and benzene. For each of these reactions the orientation dependent interaction potentials were evaluated with the CASPT2/cc-pVDZ method and one dimensional corrections based on CCSD(T)/CBS and relaxed CASPT2/cc-pVDZ calculations were employed. As already discussed in the methods section a (2e,2o) CAS active space was adopted for most of the calculations. Unfortunately this choice could not be applied to all of the reactions that were investigated. For propene, as for ethylene, a (4e,4o) active space was 9

10 required in order to correctly estimate the interaction energies at shorter distances. Meanwhile, for the allene and butadiene cases, it was necessary to employ a (6e,6o) active space, which includes the π spaces for both of the double bonds. Table 1. Contributions of physical deactivation to the removal of 1 CH 2 by the reactants considered in this study. Reactant C 2 H 2 C 2 H 4 pc 3 H 4 C 3 H 6 ac 3 H 4 C 4 H 6 C 6 H 6 k d /k tot 0.22 a 0.20 b 0.24 c 0.24 b 0.30 b 0.20 d 0.30 a a [Hack 1988], b [Hack 1989], c [Koch 1990], d Assumed equal to the value corresponding to ethylene. For these reactions only limited comparisons with experimental data are possible. The complete series of hydrocarbons has only been studied by the Wagner group [Hack 1988, Hack 1989, Koch 1990] and only at room temperature. All experimental data were again scaled by a constant correction factor that approximately accounts for the contribution to the singlet methylene removal rate from the physical deactivation of singlet to triplet methylene. Correction factors were mainly obtained from the experimental values measured by Wagner and coworkers and are presented in Table 1. 1 CH 2 + allene and butadiene In Figure 4a and 4b the results of our VRC-TST simulations are plotted for the reaction of singlet methylene with allene and butadiene, respectively. These predicted rate coefficients show a slightly larger discrepancy with experiment as compared to the reactions previously analyzed. Nevertheless, our predictions are still fairly consistent with the experimental measurements as we overestimate the values determined by Hack et al. [Hack 1989] and Hayes et al. [Hayes 1995] by only about a factor of 1.3. Notably, the presence of additional H atoms yields an increased contribution from the insertion channels at lower temperatures. 10

11 Figure 4. VRC-TST high pressure limit rate coefficient (solid red line) for (a) 1 CH 2 + ac 3 H 4 and (b) 1 CH 2 + C 4 H 6 compared with experimental measurements (symbols). 1 CH 2 + benzene In our analysis of the potential energy surface for the cyclic addition of singlet methylene to benzene we found that the magnitude of the interaction energy does not increase smoothly along the reaction coordinate. Instead it shows a slope change at a distance of Å, suggesting the presence of a saddle point, as illustrated in Figure 5. This behavior can be ascribed to the necessity to break the resonance within the aromatic ring in order to form a new bond with the methylene carbon atom. Furthermore, the difficulty of forming bicyclic compounds may also contribute to this change in the form of the potential energy surface. Saddle point searches at the CASPT2/cc-pVDZ level were unsuccessful, while CCSD(T)/CBS calculations along the minimum energy path showed a reduced variation of the slope, confirming the absence of a saddle point. Thus, the variable reaction coordinate TST approach was again used to predict the high pressure addition rate coefficient as described in section 2. 11

12 Figure 5. Potential energy surface for the 1 CH 2 + C 6 H 6 reaction evaluated at the CASPT2/cc-pVDZ level (blue fine dashed) and CCSD(T)/CBS level (red dashed) keeping the geometries of the approaching fragments fixed, and at the CASPT2/cc-pVDZ level (green solid) allowing the relaxation of the internal degrees of freedom of the two reactants. The predicted addition rate constant is plotted in Figure 6 along with the experimental data from the works of Hack et al. [Hack 1989], Wagener [Wagener 1990], and Hayes et al. [1996]. Again the agreement with their measurements is excellent, as the predictions fall within the experimental error bars. Notably, in this case the cyclic addition and insertion contributions are of comparable magnitudes. Figure 6. VRC-TST predicted high pressure limit rate coefficient (solid red line) for 1 CH 2 + C 6 H 6 compared with experimental measurements (symbols). 12

13 4. Conclusions The present theoretical approach has proven to be an effective strategy for the estimation of the high pressure rate coefficients for the addition of singlet methylene to unsaturated hydrocarbons. Specifically, the CASPT2 method was demonstrated to yield a valid procedure for the direct evaluation of the interaction energies within variable reaction coordinate transition state theory. The introduction of the one-dimensional correction potential yields predicted rate coefficients that are in good agreement with the available experimental data. The effects due to the relaxation of the conserved modes are particularly significant. Overall, we estimate the present rate predictions of the total rate coefficient to be accurate to within about +/- 25%. The present dynamically corrected VRC-TST rate coefficients are reproduced reasonably well (errors <10-15%) by the modified Arrhenius expressions reported in Table 2. The present study has focused on the addition of the simplest carbene ( 1 CH 2 ), but the reliability of the present approach opens the possibility to extend the same procedure to the addition of other important carbenes to unsaturated hydrocarbons, such as ethylidene (CH 3 CH) and vinylidene (CH 2 C). Barrierless reactions of such carbenes with other unsaturated hydrocarbons are also expected to be of importance in combustion and pyrolysis environments. 13

14 Table 2: Fitted rate expressions; k = AT n exp(-e/rt) a Reactant Products A n E C 2 H 2 CH 3 CCH 4.19x CHCH 2 CH- 4.03x total 1.57x C 2 H 4 CH 3 CHCH x CH 2 CH 2 CH x total 2.97x CH 2 CCH 2 CH 3 CHCCH x C(CH 2 )CH 2 CH x Total 6.76x CH 3 CCH CH 3 CCCH x CH 3 CH 2 CCH 9.18x C(CH 3 )CH 2 CH- 9.00x total 2.56x CH 3 CHCH 2 CH 3 CHCHCH x (CH 3 ) 2 CCH x CH 3 CH 2 CHCH x CH(CH 3 )CH 2 CH x total 2.33x CH 2 CHCHCH 2 CH 3 CHCHCHCH x CH 2 C(CH 3 )CHCH x CH(CHCH 2 )CH 2 CH x total 1.76x C 6 H 6 C 6 H 5 CH x bicyclic 3.94x total 1.56x a In these expressions, T is the temperature in K, and the rate coefficients are in cm 3 mole -1 s -1. These expressions are valid over the K temperature range. 14

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