Nonmonotonic Temperature Dependence of the Pressure-Dependent

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Nonmonotonic Temperature Dependence of the Pressure-Dependent Reaction Rate Constant and Kinetic Isotope Effect of Hydrogen Radical Reaction with Benzene Calculated by Variational Transition State Theory Hui Zhang, Xin Zhang, Donald G. Truhlar, and Xuefei Xu J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b09374 • Publication Date (Web): 02 Nov 2017 Downloaded from http://pubs.acs.org on November 7, 2017

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The Journal of Physical Chemistry

11/02/2017

Nonmonotonic Temperature Dependence of the Pressure-Dependent Reaction Rate Constant and Kinetic Isotope Effect of Hydrogen Radical Reaction with Benzene Calculated by Variational Transition State Theory Hui Zhang,1,2 Xin Zhang,1,* Donald G. Truhlar,3 and Xuefei Xu2,* 1 State Key Laboratory of Chemical Resource Engineering, Institute of Materia Medica, College of Science, Beijing University of Chemical Technology, Beijing, 100029, P.R. China 2 Center for Combustion Energy and Department of Thermal Engineering, Tsinghua University, Beijing 100084, China 3 Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States Abstract: The reaction between H and benzene is a prototype for reactions of radicals with aromatic hydrocarbons. Here we report calculations of the reaction rate constants and the branching ratios of the two channels of the reaction (H addition and H abstraction) over a wide temperature and pressure range. Our calculations, obtained with an accurate potential energy surface, are based on variational transition state theory for the high-pressure limit of the addition reaction and for the abstraction reaction and on system-specific quantum RRK theory calibrated by variational transition state theory for pressure effects on the addition reaction. The latter is a very convenient way to include variational effects, corner-cutting tunneling, and anharmonicity in falloff calculations. Our results are in very good agreement with the limited experimental data and show the importance of including pressure effects in the temperature interval where the mechanism changes from addition to abstraction. We found that a negative temperature effect of the total reaction rate constants at 1 atm pressure in the temperature region where experimental data are missing and accurate theoretical data were previously missing as well. We also calculated the H + C6H6/C6D6 and D + C6H6/C6D6 kinetic isotope effects, and we compared our H + C6H6 results to previous theoretical data for H plus toluene. We report a very novel nonmonotonic dependence of the kinetic isotope effect on temperature. A particularly striking effect is the prediction of a negative temperature dependence of the total rate constant over 300–500 K wide temperature ranges, depending on the pressure but generally in the range from 600 to 1700 K, which includes the temperature range of ignition in gasoline engines, which is important because aromatics are important components of common fuels.

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1. INTRODUCTION Aromatic hydrocarbons are important components of commercial fuel because of their higher octane numbers and the resulting good antiknock properties.1 The reactions of radicals with aromatic hydrocarbons play a key role in combustion chemistry; and it is of special concern that aromatics in fuels promote the emission of pollutants, such as the formation of soot or black carbon, which affect the atmospheric environment and human health.2,3,4 The kinetics of radical–aromatic reactions is essential for understanding the safety, efficiency, and air pollution implications of fuel mixtures, but experimental data on radical reactions with aromatics is incomplete, and even when overall reaction rates have been measured, the branching ratios (ratios of production of various products) are usually unavailable. As the simplest aromatic hydrocarbon, benzene is a prototype molecule for study of the kinetics of aromatic hydrocarbons, and its reactions with various small radicals have been extensively investigated both experimentally and theoretically. The reaction of hydrogen radical with benzene has two possible channels, the addition channel C6H6 + H → C6H7

(R1)

and the hydrogen abstraction channel, C6H6 + H → C6H5 + H2

(R2)

The experimental studies are usually limited to a small temperature region and to low pressure. The pressure in some compression-ignition engines exceeds 200 bar, but unfortunately most available combustion data were measured at pressures below 1 bar. As an example of the scarcity of data, the total rate constant for reaction of hydrogen radical with benzene has never been reported in the temperature region of 600-1100 K, which is the region most important for ignition. At these temperatures and even more so at higher combustion temperatures, the addition reaction can be strongly pressure-dependent due to the competition between bimolecular stabilization of the adduct and unimolecular redissociation. However, most theoretical studies of the addition reaction have been 2


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limited to the high-pressure limit so they do not reveal the pressure dependence. The only detailed theoretical study on the pressure effect for hydrogen addition reaction of benzene was reported by Mebel et al. in 19975, using conventional Rice-Ramsperger-Kassel-Marcus (RRKM) theory by solving the master equation. As shown in Figure 1, combined with their calculations for the R2 channel using conventional transition state theory, Mebel et al. predicted the overall reaction rate constant ktotal of H + C6H6 at 1 atm pressure with He as bath gas having a minimum around 1300 K as a result of a negative temperature dependence of rate constant for R1 but a positive temperature dependence for R2. However, Giri et al.’s shock-tube study in 1200-1350 K showed a different trend of ktotal.6 We also noticed that the high-pressure limit of R1 rate constant given by Mebel et al. is ~3-10 times larger than most of experimental rate constants in the temperature range 300-500 K (see Figure 1). The lack of experimental data in a key temperature range and the inconsistencies of current experimental and theoretical studies motivated us to use theory to investigate this prototype reaction. The pressure-dependence of reaction rate constants (i.e., falloff effects) may be calculated by solving a master equation with energy-dependent microcanonical rate constants k(E) or k(E,J) as input, where E is total energy and J is total angular momentum.7,8,9,10,11,12,13,14,15 Mebel et al. calculated k(E) for H + benzene by RRKM theory, which is the same as conventional transition state theory (TST) and hence does not include a variational transmission coefficient (which is the ratio of the transition state flux through a variationally optimized dividing surface to the transition state flux through a conventional dividing surface at the saddle point), which is often significant. In addition, it is also difficult to consistently include the multidimensional tunneling effect and/or anharmonicity effects on rate constants in conventional RRKM calculations. A simpler method is the quantum Rice-Ramsperger-Kassel (QRRK) approximation combined with chemical activation theory, as developed in modern form by Dean16 and improved 3



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further by Dean and Bozzelli and their coworkers,17 which uses kinetic parameters estimated empirically or obtained from the high-pressure results of canonical-ensemble transition state theory calculations combined with electronic structure calculations to predict the pressure effect on the dissociation rate. These workers usually used CBS-QB3 theory18 for the electronic structure calculations. Recently, Bao, Zheng, and one of us proposed a new calibration procedure for the QRRK unimolecular rate theory to study the pressure effect; the new procedure is called the system-specific (SS) QRRK method.19 The SS-QRRK method reduces the requirement for empirical parameters by calibrating the rate constants through full high-pressure-limit canonical variational transition state theory (VTST) calculations for the specific reaction and temperature of interest. This method can include variational effects and anharmonicity (including multistructural effects, torsional potential anharmonicity, high-frequency vibrational anharmonicity), as well as multidimensional tunneling in the direct dynamics calculations based on accurate potential energy surfaces along a reaction path, tunneling path, and/or variable reaction coordinate. It has shown excellent agreement with the experimental results in several cases,19,20,21,22,23 including a study of the kinetics of hydrogen radical reactions with toluene, which is similar to the prototype aromatic reaction studied here. In the present work, we calculate the reaction rate constants for reactions R1 and R2 and their sum as functions of temperature and pressure; we also calculate the kinetic isotope effect, and we calculate the branching ratio between the two reaction channels, R1 and R2. We used the SS-QRRK method for pressure-dependent R1 reaction, and we employed canonical variational transition state theory24,25,26 (CVT) including a multidimensional treatment of tunneling with the small-curvature tunneling (SCT) approximation27 (CVT/SCT) for direct dynamics calculations on the pressure-independent R2. More details are given in Section 2, which also contains a brief introduction to the theory, and in Section 3. Sections 4 and 5 contain our results, 4


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discussion, and a summary.

2. THEORY 2.1 SS-QRRK theory The SS-QRRK method has been described in detail in Refs. 19 and 20, based on the earlier work of Ref. 16. Since full details and derivations are given in those references, here we just summarize the application to calculate pressure-dependent rate constants for reaction R1. The following thermal activation mechanism is considered for R1:

(M1) where k1 is the high-pressure-limit rate constant for the formation of energized adduct C6H7* with energy E, k-1(E) is the energy-dependent dissociation rate constant, kc is the collisional deactivation rate constant, M is the bath gas, and brackets denote a concentration in molecules per unit volume. We will calculate [M] as p/RT using the ideal gas law, so p is a function of [M], which we denote as p([M]). The rate constant

for formation of the stabilized adduct C6H7 is defined as

. In the steady-state approximation,

(1) where E0 is the critical energy (also called threshold energy) of the unimolecular redissociation reaction, and f(E,T) is the fraction of C6H7* at energy E and is given by (2) where K(E,T) is the ratio of excitation to de-excitation rate constants of C6H7*. In the single mean-frequency version of the theory, the molecule is considered to have s 5



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identical oscillators (s = 3N – 6 = 33, where N is the number of atoms in C6H7*) with vibrational frequency

(in cm-1), and K(E,T) is then given by (3)

In these equations, the energy is measured in units of the average vibrational quantum , where

is Planck's constant, and

is the speed of light. Then E = n

the number of quanta of excitation. We model

, where n is

, which is the geometric

mean frequency of stabilized adduct C6H7. In order to calculate

, we need to know k1, k-1, and kc.

The rate constant k1 is treated as the high-pressure-limit rate constant for addition, which is calculated using CVT/SCT theory. The pressure-dependent rate constant k-1 of the reverse dissociation reaction is treated as function of total energy E using the QRRK assumption that k-1(E) is the rate constant for a quantum mechanical oscillator with an energy E and with a frequency factor AQRRK of intramolecular energy exchange to accumulate the threshold energy E0 =

: (4)

In the SS-QRRK treatment, the Arrhenius pre-exponential factor

and the Arrhenius

activation energy Ea (both of which are temperature-dependent) of the high-pressure-limit unimolecular dissociation reaction rate constant

are used as AQRRK and E0. Thus (5)

where

is the slope of a plot of ln

The collisional deactivation rate constant kc is obtained using Troe’s energy transfer model,28,29,30 6


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kc=βc kLJ

(6)

where kLJ is the Lennard-Jones collision rate constant, and βc is the collision efficiency that is computed by (7) where is the average vibrational energy transferred per collision during both energization and de-energization processes; FE0 is the thermal population of unimolecular states above the threshold energy of the reactant normalized by a density of states factor at the threshold energy29,30 (following Troe,29 we label it as the energy dependence factor of the density of states). We calculate FE0 by using the Whitten-Rabinovitch method31 for the harmonic density of states, which is suitable for the present case because multi-structural torsional anharmonicity is not important. In the present study of benzene plus H, we used experimental values32,33 for , which yields 35, 83, and 33 cm−1 for bath gases Ar, H2, and He respectively. We also used half of the experimental for all three bath gases (i.e., 18, 42, and 17 cm−1 for Ar, H2, and He, respectively) as a sensitivity check of how the rate constants depend on values. When not specified otherwise, the results are based on the experimental values. In eq 6, the Lennard-Jones collision rate constant is calculated using the empirical Lennard-Jones parameters ε/kB and σ as described in Ref. 19. We obtained these parameters from Ref. 32, and in particular, ε/kB and σ are 440 K and 5.27 Å for C6H7, 93.3 K and 3.542 Å for Ar, 59.7 K and 2.827 Å for H2, and 10.22 K and 2.551 Å for He. We note that the only empirical or experimental parameters used in SS-QRRK method are , ε, and σ. All other quantities are calculated or calibrated by direct high-pressure-limit dynamics calculations on the accurate potential energy surfaces.

2.2 High-pressure-limit rate constants The high-pressure-limit rate constants needed in the present work are the 7



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pressure-independent rate constant of reaction R2 and the high-pressure-limit rate constants k1 and

of the SS-QRRK calculations for the pressure-dependent reaction

R1. These were calculated using reaction-path VTST, in particular canonical variational transition state theory including small-curvature tunneling (CVT/SCT) using standard methods described in detail elsewhere.34 The reaction coordinate is the isoinertial minimum energy path (MEP).35,36 There is no torsional degree of freedom, and there is only one structure for each of the reactants, products and transition states in this study, so that we did not need to consider torsional and multi-structural anharmonicity in the present applications. We included the vibrational anharmonicity by applying a universal scaling factor to the calculated frequencies (see Section 3.2). The calculated high-pressure-limit rate constants were fitted by the following equations:37,38 •

for endothermic reaction: (8)

•

for exothermic reaction: (9)

The corresponding temperature-dependent Arrhenius activation energies ,39,40 which yields •

for endothermic reaction: (10)

•

for exothermic reaction: (11)

8


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In equations (8-11), A, E, T0 and n are fitting parameters, and R is the ideal gas constant equal to 1.9872 cal mol-1 K-1.

3. COMPUTATIONAL DETAILS 3.1 Electronic structure calculations All the kinetics calculations are carried out by direct dynamics41,42 using Kohn-Sham density functional theory. In order to choose an exchange-correlation functional and basis set for the direct dynamics calculations, we first performed benchmark calculations of the energy of reaction and the forward and reverse barrier heights, and we used them to test the accuracy of a variety of exchange-correlation functionals and basis sets. First we checked whether the systems studied here are single-reference systems, and the tests in Supporting Information (SI) show that they are. Therefore the method chosen for the benchmark calculations is CCSD(T)-F12b43,44/jul-cc-pVTZ,45 which denotes coupled-cluster theory with single and double excitations and a quasiperturbative treatment of connected triple excitations with the F12b method to include explicitly correlated basis functions along with a basis set of Gaussian one-electron functions large enough to be expected to be close to converged (on average, within about 0.4 kcal/mol of the correct result46). We compared the benchmark results to the results of 25 theoretical model chemistries for R1 and to the results of 33 theoretical model chemistries for R2 (a “theoretical model chemistry,”47 in these cases, is a combination of an exchange-correlation functional and a one-electron basis set). For each reaction, we selected the model chemistry with the best agreement with the benchmark results for the relative energies of reactants, transition structure, and products; this procedure selects MN12-L48/jul-cc-pVDZ45 for the hydrogen addition reaction R1 and M06-2X49/jun-cc-pVDZ45 for hydrogen abstraction reaction R2. Details of the comparisons are given in the Supporting Information. This procedure of selecting an exchange-correlation functional may be viewed as selecting a density functional method 9



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to interpolate the high-level ab initio benchmark results from the stationary points to the entire reaction valley, which is done because applying the high-level method directly for the dynamics is impractical, but density functional theory is affordable. Gaussian 09,50 a locally modified version of Gaussian,51 and Molpro 2012.152 were the software packages used for electronic structure calculations.

3.2 Rate constant calculations Direct CVT/SCT dynamics calculations were carried out by using the Polyrate 2010-A53 and Gaussrate 2009-A54 programs, using the implicit potential energy surfaces defined by the selected model chemistries specified above. The SS-QRRK calculations were carried out by using Polyrate 2016.55 For the VTST calculations, the Page-Mclver algorithm56 was used with a step size of 0.005 Å to follow MEP, with the reorientation of the dividing surface (RODS) algorithm57,58 turned on to refine the energies and frequencies along the reaction path. The vibrational frequencies along the reaction path are required to calculate the free energy of activation profiles to find the variational transition state and to calculate the vibrationally adiabatic ground-state potential curve used in the tunneling calculations; these frequencies were calculated using redundant curvilinear internal coordinates.59 In the calculations of the partition functions and zero-point energies for dynamics study, the harmonic frequencies were scaled by a universal empirical factor60 of 0.974 for MN12-L/jul-cc-pVDZ method and 0.976 for M06-2X/jul-cc-pVDZ method to take account of the vibrational anharmonicity and systematic errors in the model chemistries. By comparing to the vibrational anharmonicity of transition state structures calculated directly using the hybrid61 degeneracy-corrected62 second-order perturbation theory (HDCPT2),63,64 we verified that using these universal scaling factors on harmonic frequencies can represent well the vibrational anharmonicity of the transition states for the present system, although they may not be reliable for the transition-state structures in 10


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some other systems.65

4. RESULTS AND DISCUSSION 4.1 Potential energy surfaces The experimental enthalpies of the reactions at 298 K were obtained from experimental heats of formation66,67 of reactants and products, and these numbers are listed in Table 1. The calculated potential energy profiles (excluding zero-point energies) of the R1 and R2 reaction channels are given in Figure 2; these results were calculated by the selected model chemistries, in particular, MN12-L/jul-cc-pVDZ for R1 and M06-2X/jul-cc-pVDZ for R2, and the numbers in parentheses are the CCSD(T)-F12b/jul-cc-pVTZ//M06-2X/ma-TZVP68 benchmark results. For comparison, the experimental classical reaction energies are given in square brackets, estimated by using experimental enthalpy of reaction at 298 K and the difference between the calculated classical reaction energy and the calculated reaction enthalpy at 298 K (the difference is an averaged result for several model chemistries; full details are in the Supporting Information). The figure shows that the CCSD(T)-F12b calculations agree very well with the experimental estimations for classical reaction energies (deviations are 0.3 kcal/mol or less, much less than the experimental uncertainties). The calculated classical reaction energies using the selected DFT model chemistries also agree well with experiment (average deviation 0.5 kcal/mol, compared to an average experimental uncertainty of 0.8 kcal/mol.) By using the selected model chemistries, the calculated barrier heights including the zero-point energies of the reactants and transition structures are 5.74 and 16.18 kcal/mol for the forward reactions of R1 and R2, respectively, and these barrier heights are lower than those (8.9 and 19.9 kcal/mol, respectively) obtained by G2M(rcc,MP2) method in Mebel et al.’s work5. This level of accuracy for G2M(rcc,MP2) is consistent with our finding that G2M(rcc,MP2) results have an error of ~3 kcal/mol in the calculations of 11



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enthalpy of reactions for both R1 and R2 as compared to the experimental data. The relatively more accurate G3 method gave a forward barrier height (including zero-point energies) of 16.5 kcal/mol for the H abstraction reaction R2,3 which is similar to our M06-2X/jul-cc-pVDZ result (16.18 kcal/mol). We conclude that our potential energy surfaces are accurate enough for direct dynamics calculations.

4.2 Kinetics 4.2.1 High-pressure-limit rate constants First we calculated the high-pressure-limit rate constants for both the forward and reverse reactions of the two channels by CVT/SCT direct dynamics in the temperature range of 180-3000 K. In order to ensure the accuracy of the fitted rate constants, we only fitted the temperature range of 298-3000 K. These rate constants were fitted by using eq 8 for the reverse reaction of R1 and the forward reaction of R2 (endothermic reactions) and eq 9 for the forward reaction of R1 and the reverse reaction of R2 (exothermic reactions), and the fitting parameters are shown in Table 2. (The directly calculated rate constants, i.e., before fitting, at selected temperatures including those below 298 K, are listed in Table S6.) The corresponding temperature-dependent Arrhenius activation energies are computed with equations (10) and (11). We list the calculated rate constants and

at selected temperatures for the reverse reaction of R1 in Table S7 of the

Supporting Information. Figure 3 shows the common logarithm of variational transmission coefficients (Γ, calculated as the ratio of CVT rate constants to conventional TST rate constants) and the tunneling transmission coefficients (κ, calculated as the ratio of CVT/SCT rate constants to CVT rate constants) as functions of 1000/T, for both the R1 and R2 reactions. We see that the variational effect is most important for R2 reaction in the low temperature range below 400 K, and we get Γ = 0.91, 0.82, and 0.60 for R2 at 400, 298, and 180 K, 12


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respectively. Less variational effect was observed for high-pressure-limit kinetics of R1. The tunneling effect becomes more and more significant as the temperature decreases for both the R1 and R2 reactions; the calculated tunneling coefficients (κ) are 3.8 and 6.2 at 298 K, 15 and 42 at 220 K, and 76 and 425 at 180 K for R1 and R2, respectively. The total high-pressure-limit rate constants ktotal (sum of the rate constants for the R1 and R2 reactions) are plotted in Figure 1, where they are compared to experiment, which in some cases only measures the sum, and the previous theoretical results5,69. The figure shows that in the low-temperature range, ktotal agrees well with the experimental data, but at high temperatures, the high-pressure-limit ktotal rate constants are remarkably overestimated as compared to those obtained experimentally at a lower pressure (Kiefer et al.: 0.2-1.0 atm;70 Giri et al.: ~1.3 or 4.3 bar6) because the rate constant of the addition reaction R1 is strongly pressure-dependent in the high-temperature range. We will show below that when the falloff effect is considered, the calculations agree well with experiment at both high and low temperature.

4.2.2 Intermediate quantities in calculating the pressure dependence It has sometimes been assumed16 in the literature that one can use a generic value of

FE0 equal to 1.15,30 but we find that this is appropriate only for small molecules at low temperature; hence we do not make that assumption here. Rather we calculate FE0 by using the Whitten-Rabinovitch method. The resulting FE0 values at selected temperatures are shown in Table 3. We can see that the FE0 value increases rapidly as the temperature increases in the present case; it is 1.31 at room temperature, 3.53 at 1000 K, and 169 at 2000 K. In Table 4, we tabulate the collision efficiency βc and the collisional deactivation rate constant kc as calculated using the energy dependence factor of the density of states obtained by Whitten-Rabinovitch method with Ar as bath gas. We also compare these 13



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values to those obtained by using FE0 = 1.15; this clearly shows that using empirical value (1.15) of the energy dependence factor of the density of states for all temperatures will overestimate βc and kc values by up to several orders of magnitude at high temperatures, and consequently it will strongly affect the calculated addition reaction rate constants. Hence, we warn future researchers not to simply assume FE0 is 1.15. The Supporting Information contains more extensive tables of βc and kc values for the three bath gases we considered.

4.2.3 Pressure-dependent rate constants We use the high-pressure limit of the temperature-dependent rate constant k1 (T), the energy-dependent k-1(E), and the collisional deactivation rate constant kc to obtain the pressure-dependent rate constant for R1 by eq 1. We already obtained the high-pressure-limit k1(T) for the forward reaction of R1, and the high-pressure-limit k-1(T) for the reverse reaction of R1 and the corresponding Arrhenius activation energies as mentioned in Section 4.2.1. Inserting k-1(T) and the corresponding the Arrhenius pre-exponential factor

in eq 5 gives

, which is taken as frequency factor AQRRK in

eq 4 for the calculation of k-1(E). The resulting

are listed in Table S7 of

Supporting Information. Using kc calculated by using eqs 6 and 7 and the empirical parameters , ε/kB and σ, we then obtained the pressure-dependent rate constants for R1 with H2 , Ar, or He as bath gas, and these are shown as functions of 1000/T in Figure 4. Figure 4 shows that the pressure dependence of the R1 addition reaction rate constants is much stronger at high temperatures than at low temperatures, as expected and as was found using RRKM theory by Mebel et al.5 The calculated pressure dependence is different for the three bath gases (because they have different , ε/kB, and σ), and it 14


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is found that the falloff effect is more obvious with Ar as bath gas than with H2 as bath gas; He is between Ar and H2 and is closer to Ar. Based on our results obtained using the best estimate of value (35, 83, and 33 cm−1 for colliding with bath gases Ar, H2, and He, as obtained from experiments) at T = 500 and 298 K, the high-pressure limit is almost reached at 100 torr for H2 at 500 K and at 10 torr for H2 at 298 K, and it is almost reached at 1 atm (500 K) and 10 torr (298 K) for Ar and He. From Figure 4, we can see that if is (arbitrarily – as a sensitivity test) decreased by a factor of two, at relative low pressure, for example, the calculated addition rate constants at intermediate temperatures and p ≤ 1 atm decrease by a factor of 2–3 for all three bath gases. As pressure increases, the calculated rate constant gradually becomes less affected by the value of . In Figure 5, we compare the calculated R1 rate constants to experimental values from the literature as functions of 1000/T. Our results calculated with H2 as bath gas at 1 atm are about 3.5 times larger than those used for estimating equilibrium constants by Berho et al.71 at T = 628 and 670 K, which were estimated as averages of the values listed in the NIST kinetics database, but our results agree very well with Yang’s result72 at room temperature with H2 as bath gas. At T = 300–357 K, our results at 1 atm are basically same as those at the high-pressure limit, and they also fall within the error range of the rate constants fitted by using the measured values at 0.9 atm of Ar, 6 atm of Ar, and 54 atm of Ar by Sauer and Ward.73 As compared to Mebel et al.’s RRKM results5 at 1 atm He for T > 700 K, our results exhibit relatively weaker pressure dependence. The high-pressure-limit R1 rate constants by Mebel et al. at T < 500 K are much larger than ours and than the experimental results. Because only a small variational effect was found for the R1 reaction (as we mentioned in the section 4.2.1), and Mebel et al.’s G2M(rcc,MP2) calculations gave a higher barrier for the R1 forward reaction, it is not clear what causes their notably larger high-pressure-limit rate constants. The total reaction rate constant for H plus benzene at a given pressure is the sum of 15



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R1 rate constant at that pressure and the high-pressure-limit of the rate constant of R2. Figure 6 shows these calculated total rate constants at several pressures with Ar as bath gas. A negative temperature dependence of the total rate constants is observed even for a pressure of 500 atm, and it becomes more significant with the decreasing pressure. Furthermore, the range of temperatures over which the temperature dependence is negative lowers from about 1200–1700 K at 500 atm to about 600–900 K at 10 torr. Figure 1 compares our results at 1 atm Ar and at 10 torr He to the data from the literature, and it shows that our calculations reproduce most of the experimental values well. In the temperature region of 296-493 K, our total rate constants at 10 torr He are in good agreement with those obtained in Hoyerman et al.’s study74 at a pressure of 1.9-14.7 torr with He as bath gas in an isothermal flow reactor with ESR, mass spectrometry, and gas-chromatography detection techniques. The results at 1 atm Ar also agree with Sauer and Ward’s experimental values73, which were measured directly by the pulsed radiolysis at ~1 atm Ar in the temperature range of 300–357 K. At T < 500 K, the calculated total rate constants at 1 atm Ar are close to the data at 100 torr Ar obtained by Nicovich et al., using pulsed photolysis–resonance fluorescence. Although Nicovich et al.’s experiments were performed at a lower pressure, they observed a negligible pressure effect in this temperature range.85 Unlike Mebel et al.’s results, the present work confirms the positive temperature dependence of the total rate constants in the 1200-1350 K temperature range as observed in Giri et al.’s shock-tube study6 at 1.3 bar, although the calculated total rate constants at ~1200 K are slightly higher than those obtained by Giri et al. In the region missing experimental data, our results show a negative temperature dependence of the total rate constants at 1 atm Ar and 10 torr He as a result of the competition between the hydrogen addition and abstraction reactions. We find good agreement with experiment at high and low temperature where pressure effects are unimportant. This is encouraging for the validity of our potential energy surface. At 16


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intermediate T, we have limited experimental data to evaluate the predicted pressure falloff by the SS-QRRK method, and therefore our results constitute a prediction. To test the sensitivity of our calculated pressure dependence to the value of , we have recalculated the rate constants with a lower value of equal to half the best experimental estimate of . These calculations show that the agreement with experiment is improved. For example, as we mentioned, at 1 atm Ar and around 1200 K, the predicted total rate constants using experimental are overestimated as compared to Giri et al.’s experiment6; but when half value is used, the calculated values are within the uncertainty of the observed values. Similarly, the predicted values at 10 torr He with half also show better agreement with Hoyermann et al.’s measurement74 (Figure 1). The dependence of calculated rate constants on value is larger at intermediate T than high and low T, and becomes more significant with decreasing pressure. Even with the experimental value of , the agreement with experiment is reasonably good, and the agreement with the varied is slightly better. But it is still valuable to try to identify the aspect of our calculation that most limits its accuracy. If one uses the energy transfer model used here, namely that the energy transfer is described by the simple exponential model with a single , then our treatment should be reasonable because we have now shown in previous papers19,20,22 that SS-QRRK agrees well with full microcanonical calculations and with Troe's approximation method, which also employs this energy transfer model, and previous workers75,76,77,78 have shown that strong and weak exponential collision models can agree well with solution of the master equation if the master equation is used with this energy transfer model. Therefore, the largest uncertainty is probably not due to the use of SS-QRRK and the use of a collision model rather than a numerical solution of the master equation, but rather due to the use of a single mean-frequency in the QRRK representation or the one-parameter exponential model used here and in most of the literature. It would be interesting to develop a 17



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three-frequency version77 of the SS-QRRK method. It would also be interesting to know how the SS-QRRK method performs if a more accurate model of the energy transfer is used, as has been done in recent work that explicitly includes the dependence of the energy transfer on rotational angular momentum or on excitation energy.12,13,79,80,81,82

4.2.4. Branching ratios of R1 and R2 We plotted the temperature-dependent branching ratios of the R1 and R2 reaction channels at high-pressure limit and at 500 atm, 100 atm, 20 atm, 10 atm, 1 atm, 100 torr and 10 torr with H2 or Ar as bath gas in Figure S1. (The prefix S indicates that the figure is in Supporting Information.) We found that the addition reaction R1 dominates the total reaction up to 3000 K in the high-pressure limit, and as the pressure decreases, the crossover temperature where the dominance passes from R1 to R2 decreases due to the falloff effect of the R1 rate constant. At 1 atm with Ar as bath gas, the crossover in the branching ratio occurs around 1150 K. At 1 atm with H2 as bath gas, crossover is around 1300 K. Figure 7 shows the relationship between the pressure and the crossover temperature. In the temperature range around 2000 K, the reaction is dominated by the H abstraction reaction at most pressures. The RRKM model of Kiefer et al.,70 fitted to their rate constants for the high-temperature pyrolysis of benzene as measured in a shock tube, provided R2 rate constants at T = 1900-2200 K, and these values are in reasonable agreement (with a factor of ~3.5) with our results and those of Mebel and coworkers.5,70

4.2.5 Kinetic isotope effects Using the same methods, we also calculated the rate constants for the following isotopolog reactions: H + C6D6 → products; D + C6H6 → products; and D + C6D6 → products. The kH/kD ratios calculated for C6H6/C6D6 at selected pressures with Ar as bath gas are plotted as functions of temperature in Figure 8. 18


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Similar kinetic isotope effect studies have been performed both experimentally and theoretically at low temperatures, where the addition channel dominates the total reaction rate constants. However, due to the complexity introduced by the isotopic exchange reaction of the adduct product, for example, C6D6H → C6D5H + D, there are large discrepancies among the kH/kD ratios reported in the literatures. At a pressure of ~1 torr and 300 K, by using ESR and mass spectrometric techniques, Kim et al.83 obtained kH/kD equal to 0.54 and 1.44 for H + C6H6/C6D6 and D + C6H6/C6D6 reaction pairs, respectively; Knutti and Buhler84 reported similar kH/kD ratios to those of Kim et al., although they realized that the subsequent chemistry of the adduct could complicate the experimental observations. Nicovich and Ravishankara’s pulsed photolysis-resonance fluorescence study85 did not observe any kinetic isotope effect for H + C6H6/C6D6 reaction pairs, and they obtained kH/kD ≈1 for T < 500 K at the pressure of 100 torr with Ar as bath gas. The theoretical calculations are not affected by the isotopic exchange reactions of adducts. We found that there is basically no kinetic isotope effect for the addition channel at low T, so we obtain kH/kD of 1.04 at 298 K, 0.99 at 400 K, and 0.89 at 500 K for the H + C6H6/C6D6 reaction pair at 100 torr Ar, which agrees well with Nicovich and Ravishankara’s observation.85 For the D + C6H6/C6D6 kinetic isotope effect, we also obtain kH/kD of ~1 for T < 500 K, which is smaller than the experimental value of ~1.4. Mebel et al.’s RRKM calculations5 overestimated the kinetic isotope effect (kH/kD =1.74) for H + C6H6/C6D6 reaction pair at 300 K but notably underestimated the kinetic isotope effect (kH/kD =0.116) for D + C6H6/C6D6 reaction pair as compared to our results and those from the experiments. As temperature rises, the H/D abstraction reaction has a growing contribution when the pressure is far from the high-pressure limit, and the kinetic isotope effect gradually becomes governed by the abstraction reactions. Consequently, kH/kD at 100 torr has a peak near 1400 K for H + C6H6/C6D6 and near 1100 K for D + C6H6/C6D6 (see Figure 8); and correspondingly, a minimum of kH/kD is found near 800 K and 500 K for H + 19



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C6H6/C6D6 and D + C6H6/C6D6, respectively. At 1000 and 2000 K, the calculated kH/kD ratios are 0.74 and 1.70 for H + C6H6/C6D6 and 1.44 and 1.42 for D + C6H6/C6D6.

4.2.6 Comparison between the rate constants for the reactions of H with benzene and toluene The reactions of H with toluene have been investigated in Ref. 19 by using the same method as used here, so it is interesting to compare the kinetics of the H reaction with benzene and to the kinetics for reaction with toluene. Figure 9 shows the total rate constants for benzene and toluene calculated at 1 atm with Ar as bath gas. For toluene this plot only shows the hydrogen addition and hydrogen abstraction reactions in calculations for the total rate constant, and the less important two-step substitution reaction channel to produce benzene and methyl radical is not included. As shown in Figure 9, based on our results and those in Ref. 19, the total reaction of H with toluene is faster than that with benzene at high and intermediate T because of the substituent effect of the methyl group, and in the intermediate temperature range, the difference in rate constants is more than one order of magnitude. At low T, where addition reactions dominate, toluene and benzene have similar reaction rate constants. Figure 10 shows that for both addition and abstraction channels, except for low-temperature addition reactions, toluene always has a higher total rate constant for a specific channel than does benzene at the same T. As compared to those of benzene, the hydrogen abstraction rate constants of toluene are much higher at low temperatures, and the addition rate constant is higher at high temperatures at the same pressure. The crossover temperature of toluene, at which the dominance of addition and abstraction channel in the total reaction changes (see discussion above) is also around 1150 K, very similar to that of benzene in 1 atm Ar, although the falloff of the addition channel of toluene starts at a relatively higher temperature.

20


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5. SUMMARY AND CONCLUSIONS We have calculated the total reaction rate constants of H with benzene and the branching ratio of the two reaction channels: H addition and H abstraction. We used the system-specific QRRK method for the association channel and reaction-path variational transition state theory for the abstraction channel and the high-pressure limit of association; all calculations are performed by direct dynamics using MN12-L/jul-cc-pVDZ and M06-2X/jun-cc-pVDZ for the potential energy surfaces of the hydrogen addition reaction R1 and hydrogen abstraction reaction R2, respectively. These model electronic structure theories used for the dynamics were chosen based on high-level calibrations. We report calculations over a wide temperature range (180–3000 K) and at pressures including the high-pressure-limit and several pressures corresponding to possible combustion conditions. Our results are in very good agreement with the limited experimental data at high and low temperatures where pressure effects are less important. If the limited experimental rate constants are correct at near intermediate temperatures, using the energy transfer parameter determined experimentally slightly underestimates the falloff effect at intermediate temperature, but better agreement with experiments is achieved when half of the experimental is used. We conclude that the current simple model for energy transfer is fine for this system, although it may be the factor that most limits the accuracy, and the results have some sensitivity to at intermediate T. We find that due to the significantly increasing falloff effect of the H addition reaction rate constant as temperature rises, the total reaction rate constant displays a negative dependence in the temperature range where experimental data are missing, for example at temperatures in the 800-1200 K range at 1 atm with Ar as bath gas. The falloff effect depends on the bath gas, and it is stronger with Ar than with H2 as bath gas. Our study clearly shows how the dependence of H addition reaction rate constant on pressure and on the bath gas at low pressures affects the branching ratios of H addition and H abstraction reactions, as well as the kinetic isotope effects for both H + 21



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C6H6/C6D6 and D + C6H6/C6D6, which are found to be non-monotonic functions of temperature. We also compared the reaction rate constants of H with benzene and H with toluene based on the previously reported study of the toluene reactions using the same SS-QRRK method. The total addition and abstraction reaction rate constants of toluene at intermediate and high T are larger than those of benzene due to the substituent effect of the methyl radical, and the difference is especially significant at intermediate temperatures around 1000 K (for 1 atm pressure with Ar as bath gas), which is also near the region of the crossover between dominance of H addition and dominance of H abstraction. ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ Selection of appropriate DFT model chemistry, multireference diagnostics, classical reaction energies and barrier heights, frequency scaling factors, enthalpies of reaction, rate constants, collision rates and other intermediate results, reaction rates, and Cartesian coordinates of optimized geometries. (PDF) AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] (XZ), [email protected] (XX) ORCID Hui Zhang: 0000-0001-5541-4663 Xin Zhang: 0000-0003-3559-2096 Xuefei Xu: 0000-0002-2009-0483 Donald G. Truhlar: 0000-0002-7742-7294 Notes The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Junwei Lucas Bao and Steven L. Mielke for helpful assistance and to Ahren Jasper for helpful discussions. This work was supported in part by the National Natural Science Foundation of China (91641127) and by the U.S. Department of Energy under Award Number DE-SC0015997.

Figure 1. The total rate constants ktotal for the reaction of H with benzene in He and Ar bath gases. HPL denotes the high-pressure limit.

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Figure 2. Potential energy profiles for H addition to benzene (R1) and for H abstraction from benzene by H (R2); the potential energy for nuclear motion is the Born-Oppenheimer electronic energy plus nuclear repulsion, and it does not include zero point energy. The zero of energy for this plot is the reactant potential energy. Therefore the conventional transition state theory classical barrier heights are the potential energies of the transion structures (saddle points, labeled as TSabs and TSadd)), and the classical reaction energies are equal to the potential energies of the products. The numbers in bold are calculated by MN12-L/jul-cc-pVDZ for H addition and by M06-2X/jun-cc-pVDZ for H abstraction. The numbers in parentheses and brackets are respectively the CCSD(T)-F12b/jul-cc-pVTZ//M06-2X/ma-TZVP benchmark results and the classical reaction energies calculated from the experimental data.

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Figure 3. Calculated variational transmission coefficients (Γ) and SCT tunneling transmission coefficients (κ) for H addition (reaction R1) and H abstraction (reaction R2).

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(a)

(b)

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(c)

Figure 4. Computed addition reaction (R1) rate constants at various pressures including the high-pressure-limit (HPL), as functions of temperature. The solid lines are calculated with the from the experimental work; the dotted lines are calculated (for comparison) with reduced to half the experimental value. (a) H2 as bath gas; (b) Ar as bath gas; (c) He as bath gas.

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Figure 5. Computed and experimental rate constants for the H addition reaction R1. The calculated rate constants are for 1 atm with Ar, H2, or He as the bath gas and also at the high-pressure-limit (HPL, which is independent of bath gas). The shaded part represents experimental uncertainty.

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Figure 6. Computed total rate constants for reactions of H with benzene at various pressures, including (from top to bottom) high-pressure-limit, 500 atm, 100 atm, 20 atm, 10 atm, 1 atm, 100 torr and 10 torr, as functions of temperatures, with Ar as bath gas.

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Figure 7. The crossover temperature (defined such that R1 gives the dominant product at lower temperatures, and R2 gives the dominant product at higher temperatures). The green line is for H2 bath gas, and the purple line is for Ar bath gas.

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(a)

(b)

Figure 8. The calculated kH/kD kinetic isotope effects as functions of temperature. (a) H + C6H6/C6D6. (b) D + C6H6/C6D6.

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Figure 9. Total rate constants for benzene and toluene at 1 atm Ar; for toluene, only H addition and abstraction reactions are considered.

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Figure 10. Comparison of the rate constants of H addition and abstraction reactions of benzene and toluene at 1atm Ar.

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Table 1. The experimental heats of formation (kJ/mol) of the species of interest and the calculated enthalpies of reactions for both R1 and R2 reaction using the heats of formation Molecule

C6H6

∆Hf (298K)

82.93

H a

218

Reaction

C6H6 + H → C6H7 (R1) C6H6 + H → C6H5 + H2 (R2) a

ref. 66

b

C6H5 a

338±3

a

H2

C6H7

0

208.0±3.9b

Reaction enthalpies -92.93±3.9 (-22.21±0.93 kcal/mol) 37.07±3 (8.86±0.72 kcal/mol)

ref. 67

Table 2. Fitting parameters for high-pressure-limit rate constantsa Reaction

Method

A

n

E

T0

H addition (R1) C6H6 + H →C6H7

MN12-L/jul-cc-pVDZ

3.552E-11

1.359

4.863

-45.092

C6H7 → C6H6 + H

MN12-L/jul-cc-pVDZ

5.980E+13

0.251

29.053

-15.940

H abstraction (R2) C6H6 + H → C6H5 + H2

M06-2X/jun-cc-pVDZ

3.832E-11

1.752

15.547

-18.015

C6H5 + H2 → C6H6 + H

M06-2X/jun-cc-pVDZ

2.335E-14

2.999

3.004

127.632

a

The units of parameter A are cm3 molecule-1 s-1 for bimolecular reactions and s-1 for unimolecular reactions. The parameters T0 and E are in units of K and kcal/mol, respectively. The value of R is 0.001987204 kcal/mol.

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Table 3. Energy dependence factor of the density of states as a function of temperature as calculated by the Whitten-Rabinovitch method

T/K

FE 0

298 400 500 600 800 1000 1200 1600 2000 2400 3000

1.31 1.45 1.62 1.83 2.44 3.53 5.69 23.07 169.40 1.74×103 6.51×104

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Table 4. Collision efficiency βc and collisional deactivation rate constant kc (in cm3 molecule-1 s-1) calculated with two choices for the energy dependence factor of the density of states for bath gas Ar ( = 35 cm-1) T/K

FE0 calculated by Whitten-Rabinovitch method

βc 298 1000 2000 3000

9.05E-02 1.27E-02 1.47E-04 2.58E-07

kc 4.65E-11 6.50E-12 7.54E-14 1.32E-16

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FE 0

βc 1.00E-01 3.55E-02 1.89E-02 1.29E-02

= 1.15 kc 5.16E-11 1.83E-11 9.70E-12 6.64E-12

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1 Battin-Leclerc, F. Detailed chemical kinetic models for the low-temperature combustion of hydrocarbons with application to gasoline and diesel fuel surrogates. Prog. Energy Combust. Sci. 2008, 34, 440-498. 2 Bittner, J. D.; Howard, J. B. Composition profiles and reaction mechanisms in a near-sooting premixed benzene/oxygen/argon flame. Symp. Int. Combust. Proc. 1981, 18, 1105-1116. 3 Kislov, V. V.; Islamova, N. I.; Kolker, A. M.; Lin, S. H.; Mebel, A. M. Hydrogen abstraction acetylene addition and Diels-Alder mechanisms of PAH formation: A detailed study using first principles calculations. J. Chem. Theory Comput. 2005, 1, 908-924. 4 Brem, B. T.; Durdina, L.; Siegerist, F.; Beyerle, P.; Bruderer, K.; Rindlisbacher, T.; Rocci-Denis, S.; Andac, M. G.; Zelina, J.; Penanhoat, O.; et al. Effects of fuel aromatic content on nonvolatile particulate emissions of an in-production aircraft gas turbine. Environ. Sci. Technol. 2015, 49, 13149-13157. 5 Mebel, A. M.; Lin, M. C.; Yu, T.; Morokuma, K. Theoretical study of potential energy surface and thermal rate constants for the C6H5 + H2 and C6H6 + H reactions. J. Chem. Phys. A. 1997, 101, 3189-3196. 6 Giri, B. R.; Bentz, T.; Hippler, H.; Olzmann, M. Shock-tube study of the reactions of hydrogen atoms with benzene and phenyl radicals. Z. Phys. Chem. 2009, 223, 539-549. 7 Dean, A. M. Predictions of pressure and temperature effects upon radical addition and recombination reactions. J. Phys. Chem. 1985, 89, 4600-4608. 8 Smith, S.C.; McEwan, M.J.; Gilbert, R.G. The relationship between recombination, chemical activation and unimolecular dissociation rate coefficients. J. Chem. Phys. 1989, 90, 4265-4273. 9 Tsang, W.; Bedanov, V.; Zachariah, M.R. Master equation analysis of thermal activation reactions: Energy-transfer constraints on falloff behavior in the decomposition of reactive intermediates with low thresholds. J. Phys. Chem. A 1996, 100, 4011-4018. 10 Chen, C.-J.; Bozzelli, J.W. Analysis of tertiary butyl radical + O2, isobutene + HO2, isobutene + OH, and isobutene-OH dducts + O2: A Detailed tertiary butyl oxidation mechanism. J. Phys. Chem. A 1999, 103, 9731-9769. 11 Shestov, A. A.; Kostina, S. A.; Knyazev, V. D. Thermal decomposition of dichloroketene and its reaction with H atoms. Proc. Combustion Inst. 2005, 30, 975-983. 12 Jasper, A. W.; Miller J. A.; Klippenstein, S. J. Collision efficiency of water in the

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Nonmonotonic Temperature Dependence of the Pressure-Dependent

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