Calculation of Reaction Rates for Hydrogen Abstraction by the Hydroperoxyl Radical from C1 through C4 Hydrocarbons

Transcription

1 Calculation of Reaction Rates for Hydrogen Abstraction by the Hydroperoxyl Radical from C1 through C4 Hydrocarbons Jorge Aguilera-Iparraguirre *,1, Henry J. Curran 2, Wim Klopper 1, John M. Simmie 2 1 Lehrstuhl für Theoretische Chemie, Institut für Physikalische Chemie, Universität Karlsruhe, D Karlsruhe, Germany 2 Combustion Chemistry Centre, National University of Ireland, Galway, University Road, Ireland Abstract A number of high-level calculations were carried out on the energetics and kinetics of H-abstraction by the hydroperoxyl radical, HO 2, from a series of hydrocarbons, to form hydrogen peroxide and methyl, ethyl, propyl, iso-propyl, primary and secondary butyl, isobutyl and tert-butyl radicals. The barrier height for the reaction of HO 2 with methane was computed at kj mol 1 from explicitly-correlated coupled-cluster calculations including singles, doubles and a perturbative correction for triples; this can be used as a benchmark for subsequent accurate estimates of the reaction barriers in those cases where such a procedure is presently computationally prohibitive. Introduction The H-atom abstraction from alkanes by the hydroperoxyl radical, HO 2, is an important reaction in combustion particularly so at temperatures between 600 and 1200 K. The subsequent fate of hydrogen peroxide plays a key role in degenerate branching or secondary initiation. Despite this there are very few reliable measurements of the rates of these reactions [1 4]; most values have been estimated [5 7]. In recent work Carstensen et al. [8] have used the CBS-QB3 formalism [9] to compute rate constants from transition state theory for a series of reactions of the HO 2 radical with methane, ethane, propane and isobutane. Specific Objectives Our aims were: (1) to compute the rate constant for H-abstraction by HO 2 from the simplest hydrocarbon, methane, at as high a level of theory as is currently feasible and (2) to use a number of less computationally expensive methods for the higher hydrocarbons. The results obtained were then to be compared with the benchmark result for methane in order to determine the most reliable, less demanding, model which could be employed subsequently in developing detailed chemical kinetic mechanisms to understand the combustion chemistry of both fossil and renewable fuels. Computational Details Geometry optimizations utilizing analytical nuclear gradients were carried out to locate minima and saddle points at the level of density functional theory (DFT), using the B3LYP exchange-correlation functional [10 12] in combination with the def2-tzvp basis [13] as implemented in the Turbomole application [14]. Moreover, single-point calculations were carried out in the same basis using the exchange-correlation functionals BP86 [15 16], TPSS [17], TPSSh [18], BMK [19] and B97K [19]. The BMK and B97K functionals have been designed specifically for accurate calculations of barrier heights [19]. Redundant internal coordinates were used for the geometry optimizations and the search for saddle points was performed using the trust radius image minimization approach (TRIM) [20]. Harmonic frequencies were calculated analytically for all species at the B3LYP/def2-TZVP level. The frequencies of the minima were all real, and the saddle points exhibited only one imaginary frequency. For all reactions, conventional spin-restricted coupled-cluster calculations were performed at the B3LYPoptimized geometries in the correlation-consistent triple-zeta basis (cc-pvtz) of Dunning [21]. These spinrestricted coupled-cluster calculations were based on a spin-restricted Hartree-Fock reference (restricted Hartree-Fock, RHF, or restricted open-shell Hartree-Fock, ROHF) and were carried out with Molpro [22]. The coupled-cluster calculations included singles and doubles (RCCSD) [23] as well as perturbative triples [RCCSD(T)] [24] and were performed in the frozencore approximation. A detailed study of the CH 4 + HO 2 system was performed using the conventional coupled-cluster method with the family of n-tuple-zeta basis sets (cc-pvnz) with n = 2, 3, 4 and 5 [21]. On this system, integraldirect explicitly-correlated CCSD-R12 calculations were performed with the DIRCCR12-OS program [25 26] using a spin-restricted Hartree-Fock reference wave function (RHF or ROHF). A spin-restricted coupled-cluster calculation was performed for the closed-shell systems while the open-shell systems were treated at the spin-unrestricted coupled-cluster ROHF- UCCSD-R12 level. For comparison, the conventional coupled-cluster calculations in the cc-pvnz basis sets on the CH 4 + HO 2 system were carried out similarly, that is, using the ROHF-UCCSD(T) method for the open shells. The CCSD-R12 calculations were carried out in the 19s14p8d6f4g3h basis (9s6p4d3f for H) sets [27]. * Corresponding author: aguilera@chem-bio.uni-karlsruhe.de Proceedings of the European Combustion Meeting 2007

2 Transition state theory was used to compute reaction rate constants. The expression for the reaction rate constant of a bimolecular reaction A + B AB is: kbt QTS( T ) k( T ) = κ ( T ) Vm ( T ) exp( EA h Q ( T ) Q ( T ) A B / RT ) where Q TS (T), Q A (T) and Q B (T), are the partition functions (including translation) of the transition state (AB or TS) and the reactants, respectively, calculated using the module Freeh of Turbomole. R is the gas constant, k B the Boltzmann constant, h the Planck constant and V m (T) the molar volume of an ideal gas at temperature T. The temperature T is varied from 600 to 1300 K in the present work. E A is the barrier height including zero-point vibrational energy (ZPVE). In the present work, the ZPVE was computed at the B3LYP/def2- TZVP level and all of the data presented in the present work include the B3LYP ZPVE. κ(t) is the transmission coefficient accounting for tunnelling effects, computed from the well-known Wigner formula: κ ( T ) = hν 1 + kbt RT E a Only the imaginary frequency, ν, associated with the reaction coordinate and the reaction barrier E a are required to calculate κ(t). Corrections to account for hindered rotations were included for all rotations about the C C and O O bonds as well as for rotations about the reaction coordinate C H O. Following Vansteenkiste et al. [28], instead of removing the harmonic vibrational modes from the partition function, we corrected the partition function by multiplying it, for each hindered rotation, with the ratio q hind-rot (T)/q vib-1d (T). To obtain q hind-rot (T), potential energy curves were computed for the rotations about the above mentioned bonds using discrete steps of five degrees, thereby allowing for a geometry relaxation of all of the other internal coordinates (for the transition state, the difference between the C H and H O distances was kept fixed since otherwise the geometry relaxation would have led to either the reactants or the products). A one-dimensional Schrödinger equation was solved to obtain the eigenstates needed to compute the partition function q hind-rot (T). Arrhenius-like expressions of the form: k (T) = A T n exp( E A / RT) were fitted to the computed rate constants at temperatures of 600, 700, 800,, 1300 K; A, n and E A were treated as fitting parameters. We fitted the above expression to rate constants computed from the B3LYP/def2-TZVP partition functions (corrected for hindered rotations) with the best estimates of the activation energies E A obtained at the coupled-cluster level of theory (vide infra). Note that these best estimates were used not only in the exponential but also in the Wigner formula for the transmission coefficient κ(t). Results and Discussion Before presenting and discussing the results for all of the reactions, let us first have a close look at the reaction: CH 4 + HO 2 TS CH 3 + H 2 O 2 For this reaction, we have calculated the electronic energies of the two reactants, the two products and the transition state (TS ) at the level of ROHF-UCCSD-R12 theory. This theory uses electronic wave functions that depend explicitly on all of the electron-electron distances and is capable of yielding results close to those that would be obtained in a complete basis set of atomic orbitals if it were possible to use such a basis. To the CCSD-R12 energies, we have added the perturbative (T) correction for connected triples computed in the ccpv5z basis, and we shall refer to the corresponding energies as CCSD(T)-R12 for short. Table 1 shows the electronic atomisation energies for the five species involved in the above reaction, computed at this CCSD(T)-R12 level. Table 1: Experimental and ROHF-UCCSD(T)-R12 calculated electronic atomisation energies in kj mol 1. Calculated values were obtained by adding the (T) triples correction from the cc-pv5z basis to the ROHF- UCCSD-R12 energies. System Experimental a Calculated CH CH HO H 2 O TS a Experimental atomisation enthalpy at 0 K and experimental zero-point vibrational energy from [29]. The CCSD(T)-R12 results agree to within 3 kj mol 1 with the known experimental data, and we expect that the atomisation energy of TS, and thus the barrier height, is similarly accurate. The agreement with experiment is very satisfactory. To obtain an even better agreement with experiment, it would be necessary to include higher excitations into the coupled-cluster treatment (full triples as well as quadruples and quintuples), to include core orbitals into the correlation treatment, and to correct for relativistic and non-born- Oppenheimer effects. The electronic reaction energy ΔE R,e and the electronic barrier height ΔE B,e can be calculated from the CCSD(T)-R12 data displayed in Table 1. In the following, we shall compare these very accurate values with the results obtained from more approximate (and computationally less involved) methods. On the one hand, we shall compare the CCSD(T)-R12 data with the results obtained from conventional CCSD(T) theory using the standard cc-pvnz basis sets. Also, we shall compare the coupled-cluster data with the results obtained from DFT using a variety of exchange-correlation functionals (in the def2-tzvp basis). Table 2 shows the computed data (ΔE R,e and ΔE B,e ) obtained with the conventional CCSD(T) method in the 2

3 cc-pvnz basis sets. The results show clearly that CCSD(T) calculations in small basis sets such as ccpvdz and cc-pvtz are not accurate enough for our purposes. The deviations from the CCSD(T)-R12 values are and 4 5 kj mol 1, respectively. Only in the case of the cc-pvqz basis is the error below 3 kj mol 1. Table 2: ROHF-UCCSD(T) calculated electronic reaction energy, ΔE R,e, and reaction barrier, ΔE B,e, for the reaction CH 4 + HO 2 CH 3 + H 2 O 2. Basis ΔE R,e / kj mol 1 ΔE B,e / kj mol 1 cc-pvdz cc-pvtz cc-pvqz cc-pv5z cc-pv(q5)z a CCSD(T)-R a Extrapolated from the cc-pvqz and cc-pv5z basis sets using the n 3 formula of Helgaker et al. [30]. Since the convergence of the computed data with the size of the atomic basis set is slow but systematic, it is possible to extrapolate the results to the limit of a complete basis (cc-pv Z). CCSD(T)-R12 values are using the cc-pvqz and cc-pv5z values, such an extrapolation yields energies that are only kj mol 1 away from the CCSD(T)-R12 benchmark data. This is a strong indication that they are accurate to within 1 kj mol 1 of the basis-set limit of CCSD(T) theory. At present, it is possible to perform frozen-core CCSD-R12 calculations on the transition state of the reaction of CH 4 with the HO 2 radical in the large 19s14p8d6f4g3h basis (9s6p4d3f for H), but similar calculations on the transition states for the reactions with the larger hydrocarbons are technically not feasible. Therefore, we have calculated the electronic barrier heights for all of the reactions in the def2-tzvp basis using various exchange-correlation functionals and in the cc-pvtz basis using frozen-core RCCSD(T) theory. For the reaction of the hydroperoxyl radical with methane, the B3LYP ZPVE correction to the barrier amounts to 9.8 kj mol 1. Adding this contribution to the ROHF-UCCSD(T)/cc-pVTZ and CCSD(T)-R12 electronic barriers (cf. Table 2) yields ΔE B,0 = kj mol 1 and ΔE B,0 = kj mol 1, respectively. Thus, the CCSD(T)/cc-pVTZ value appears to slightly overestimate the CH 4 + HO 2 barrier height. This can be easily corrected by scaling the electron-correlation contribution to the electronic barrier ΔE B,e by a factor of This factor is the ratio between the correlation contributions in the R12 and cc-pvtz basis sets. Note that the electron-correlation contribution to the barrier is negative. We have adopted this scaling factor of to obtain the best estimates of the barrier heights for all of the reactions under study. The results are presented in Table 3, with the best estimates given in the last row. Table 3: Reaction barriers (ΔE B,0 in kj mol 1 ) for the reactions of the hydrocarbons with the hydroperoxyl radical. All DFT data were obtained in the def2-tzvp basis, and include the B3LYP ZPVE. Method CH 3 C 2 H 5 n-c 3 H 7 i-c 3 H 7 p-c 4 H 9 s-c 4 H 9 i-c 4 H 9 t-c 4 H 9 B3LYP BP TPSS TPSSh BMK B97K RCCSD(T) a Best estimate b a Frozen-core CCSD(T)/cc-pVTZ value. b Best estimate of the barrier height ΔE B,0 obtained by scaling the frozen-core CCSD(T)/cc-pVTZ electron-correlation contributions to ΔE B,e by the factor (see text). Table 4: Calculated TST rate constants (in cm 3 molecule 1 s 1 ) at 600, 800, 1000 and 1200 K together with the fit parameters A / cm 3 molecule 1 s 1, n and E A / kj mol 1. Imaginary frequency ν / cm 1. Radical 600 K 800 K 1000 K 1200 K A n E A ν CH E E E E E i C 2 H E E E E E i n-c 3 H E E E E E i i-c 3 H E E E E E i p-c 4 H E E E E E i s-c 4 H E E E E E i i-c 4 H E E E E E i t-c 4 H E E E E E i In comparison with the DFT results obtained from calculations with various exchange-correlation functionals, we find that the B3LYP values are very close to the best estimates. On average, the B3LYP barriers are 3

4 only ca. 3 kj mol 1 below the best estimates. The TPSSh functional yields values that in turn are ca. 3 kj mol 1 below the B3LYP barriers. Furthermore, the BMK and B97K exchange-correlation functionals yield barriers that are too high in comparison with the best estimates derived from CCSD(T) calculations. These functionals overestimate the barriers by more than 10 kj mol 1. The results (B3LYP) show, not unexpectedly, that the reaction barriers for the abstraction of a primary hydrogen cluster around 79.0 ± 0.6 kj mol 1 (excluding methane which is a special case), abstraction of a secondary H is lower at about 64 kj mol and finally a tertiary hydrogen lower again at 54 kj mol 1 reflecting of course the decrease in the C H bond dissociation energies from primary to secondary to tertiary [31]. The final results are collected in Table 4 where we present the TST rate constants k(t) calculated from the B3LYP/def2-TZVP partition functions in conjunction with the best estimates of E A. The fit parameters A, n and E A needed to represent the rate constants by an Arrhenius-like expression are also given in Table 4. Note that the activation energy E A given in the table is the fit parameter, not the best estimate of E A. In fact, the fitted E A values are about 7 to 14 kj mol 1 lower than the best estimates. Comparison with previous work Methane: In the case of methane our computations are in good agreement with the calculations of Carstensen et al. [8] at 600 K but are almost a factor of two faster at 1500 K. A relative rate measurement (based on HO 2 + HO 2 H 2 O 2 + O 2 ) was reported by Baldwin et al. [2]. Both our calculations and those by Carstensen et al. are faster than those calculated by ; at 623 K our values are three times faster and almost an order of magnitude faster than at 1273 K. Figure 1: k (CH 4 + HO 2 CH 3 + H 2 O 2 ) 10 4 Baulch et al Tsang/Hampson 1986 Ethane: Our calculated values are in excellent agreement with the the relative rate measurements (based on the self-reaction of hydroperoxyl radicals) of [1]. Our rate expression shows a stronger temperature curvature compared to the previous recommendations of Scott and Walker [4] and Orme et al. [32] and the review of Tsang and Hampson [7] and thus is considerably faster at temperatures above 600 K. Our rate constant is in reasonable agreement with the calculations of Carstensen et al. [8] it is two times slower at 600 K, but both rate constants are almost identical at 1500 K), Figure 2. Figure 2: k (C 2 H 6 + HO 2 CH 2 CH 3 + H 2 O 2 ) Baldwin/Walker Tsang/Hampson Propane: In the case of propane there are two different abstractable H atoms leading to either n-propyl, Figure 3, or iso-propyl radicals, Figure 4. Figure 3: k(c 3 H 8 + HO 2 n-c 3 H 7 + H 2 O 2 ) Tsang 88 [6] performed a relative rate measurement at 773 K and found a relative rate of 0.03 for the production of n-propyl radical relative to the reaction CH 2 O + HO 2 HCO + H 2 O 2. All other rate constants presented in Figure 3 are either calculations or estimates. The rate constant calculated in this study is slower than all other calculations and/or estimations. The value calculated by Carstensen et al. [8] is seven times faster at 600 K, which reduces to a factor of two and a half times faster at 1500 K. [6] also carried out a relative rate measurement for the production of iso-propyl radical at 773 K and found a relative rate of relative to the reaction CH 2 O + HO 2 HCO + H 2 O 2. [8] is in error, private comm. H-H.Carstensen. 4

5 Figure 4: k(c 3 H 8 + HO 2 iso-c 3 H 7 + H 2 O 2 ) Figure 6: k(n-c 4 H 10 + HO 2 s-c 4 H 9 + H 2 O 2 ) Tsang 88 All other rate constants presented in Figure 4 are either calculations or estimates. The rate constant calculated in this study is approximately one and a half times faster than the recommendations of Scott and Walker [4], Orme et al. [32], [6] and Tsang [33]. However, our calculation is significantly slower than that computed by Carstensen et al. [8] our values are more than three times slower at 600 K reducing to one and a half times slower at 1500 K. n-butane: In the case of n-butane, both primary or secondary radicals can be generated, Figure 5 and Figure 6, respectively. There have been no measurements of the rate constant for hydrogen atom abstraction by the hydroperoxyl radical from n-butane. Figure 5: k(n-c 4 H 10 + HO 2 p-c 4 H 9 + H 2 O 2 ) The rate constant calculated in this study is slower than all other calculations/estimations. The rate constant calculated by Carstensen et al. [8] is seven times faster at 600 K and a factor of two and a half times faster at 1500 K. These relative disparities are identical to those in the case of primary hydrogen abstraction from propane, Figure 3. For secondary hydrogen atom abstraction from n- butane our calculations are in excellent agreement with the recommendations of Scott and Walker [4]. For secondary hydrogen atom abstraction from n- butane our calculations are consistently almost two times faster than the recommendations of Scott and Walker [4]. The rate constant calculated by Carstensen et al. [8] is between seven times (at 600 K) and four times (at 1500 K) faster than our calculations. iso-butane: For iso-butane, both iso-butyl, Figure 7, and tertiary-butyl radicals, Figure 8, are formed. Figure 7: k(i-c 4 H 10 + HO 2 i-c 4 H 9 + H 2 O 2 ) Tsang 90 [6] performed a relative rate measurement at 773 K and found a relative rate of for the production of iso-butyl radicals relative to the reaction CH 2 O + HO 2 products. Recommendations of Scott and Walker [4], Orme et al. [32], and Tsang [33] and the calculation of Carstensen et al. are in reasonable agreement with this rate constant. However, our calculation is almost four times slower than the value derived by [6] at 773 K. Figure 8 depicts rate constants for the abstraction of the tertiary hydrogen atom from isobutene. Baldwin et al. [6] performed a relative rate measurement at 773 K and found a relative rate of for the production of tert-butyl radicals relative to CH 2 O + HO 2 products. [8] is in error, private comm. H-H.Carstensen. 5

6 Figure 8: k(i-c 4 H 10 + HO 2 t-c 4 H 9 + H 2 O 2 ) Tsang 90 Recommendations of Scott and Walker [4], Orme et al. [32], and Tsang [33] are in reasonable agreement with this rate constant. However, our calculation and that of Carstensen et al. are both significantly faster than the value derived by [6] at 773 K, being approximately five and four and a half times faster, respectively. Conclusions We have shown that due to current limitations in compute power that it is not feasible to apply the highest levels of theory beyond the simplest case of methane + HO 2. But, perhaps surprisingly, the DFT method B3LYP allied to the def2-tzvp basis provides an inexpensive and reasonably accurate estimate of the reaction barrier height. Acknowledgements This work was conducted in the Sonderforschungsbereich (SFB) 551 Carbon from the Gas Phase: Elementary Reactions, Structures, Materials, which is funded by the Deutsche Forschungsgemeinschaft (DFG). This work was further supported by the European Commission through a Transfer-of-Knowledge grant, MKTD-CT We thank M. Olzmann and O. Welz for making available their program for computing corrections to the partition function due to hindered rotations following Vansteenkiste et al. References [1] R.E. Baldwin, C.E. Dean, M.R. Honeyman, R.W. Walker, J. Chem. Soc. Farad. Trans. 1, 82 (1986) [2] R.R. Baldwin, P.N. Jones, R.W. Walker, J. Chem. Soc. Farad. Trans. 2, 84 (1988) [3] S.M. Handford-Styring, R.W. Walker, Phys. Chem. Chem. Phys. 3 (2001) [4] M. Scott, R.W. Walker, Combust. Flame 129 (2002) [5] D.L. Baulch, C.J. Cobos, R.A. Cox, C. Esser, P. Frank, Th. Just, J.A. Kerr, M.J. Pilling, J. Troe, R.W. Walker, J. Warnatz, J. Phys. Chem. Ref. Data 21 (1992) [6] R.R. Baldwin, A.R. Fuller, D. Longthorn, R.W. Walker, Combust. Inst. European Symp. ed. F. J. Weinberg, Academic Press, London, 1 (1973) 70. [7] W. Tsang, R.F. Hampson, J. Phys. Chem. Ref. Data 15 (1986) [8] H.-H. Carstensen, A.M. Dean, O. Deutschmann, Proc. Combust. Inst. 31 (2007) [9] J.A. Montgomery, Jr., J.W. Ochterski, G.A. Petersson, J. Chem. Phys. 101 (1994) [10] A.D. Becke, J. Chem. Phys. 98 (1993) [11] P.J. Stephens, F.J. Devlin, C.F. Chabalowski, M.J. Frisch, J. Phys. Chem. 98 (1994) [12] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B, 37 (1988) [13] F. Weigend, R. Ahlrichs, Phys. Chem. Chem. Phys. 7 (2005) [14] R. Ahlrichs, M. Bär, M. Häser, H. Horn, Ch. Kölmel, Chem. Phys. Letters 162 (1989) ; Turbomole v5.9; [15] J.P. Perdew, Phys. Rev. B 33 (1986) [16] A.D. Becke, Phys. Rev. A 38 (1988) [17] J. Tao, J.P. Perdew, V.N. Staroverov, G.E. Scuseria, Phys. Rev. Lett. 91 (2003) [18] V.N. Staroverov, G.E. Scuseria, J. Tao, J.P. Perdew, J. Chem. Phys. 119 (2003) ; ibid. 121 (2004) (Erratum). [19] A.D. Boese, J.M.L. Martin, J. Chem. Phys. 121 (2004,) [20] T. Helgaker, Chem. Phys. Lett. 182 (1991) [21] T.H. Dunning, Jr., J. Chem. Phys. 90 (1989) [22] Molpro , H.-J. Werner, P.J. Knowles, R. Lindh, F.R. Manby, M. Schütz and others; [23] C. Hampel, K. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 (1992) [24] M.J.O. Deegan, P.J. Knowles, Chem. Phys. Lett., 227 (1994) [25] W. Klopper, J. Noga, ChemPhysChem 4 (2003) (2003). [26] J. Noga, W. Klopper, T. Helgaker, P. Valiron, see www-laog.obs.ujf-grenoble.fr/~valiron/ccr12/. [27] W. Klopper, C.C.M. Samson, J. Chem. Phys. 116 (2002) [28] P. Vansteenkiste, D. van Neck, V. Van Speybroeck, M. Waroquier, J. Chem. Phys. 124 (2006) ; ibid. 125 (2006) [29] Computational Chemistry Comparison and Benchmark DataBase, [30] T. Helgaker, W. Klopper H. Koch, J. Noga, J. Chem. Phys. 106 (1997) [31] Y.-R. Luo, Handbook of Bond Dissociation Energies, CRC Press, Boca Raton, [32] J.P. Orme, H.J. Curran, J.M. Simmie, J. Phys. Chem. A. 110 (2006) [33] W. Tsang, J. Phys. Chem. Ref. Data 17 (1988) [34] W. Tsang, J. Phys. Chem. Ref. Data 19 (1990)