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Multistructural Variational Transition State Theory: Kinetics of the Hydrogen Abstraction from Carbon‑2 of 2‑Methyl-1-propanol by Hydroperoxyl Radical Including All Structures and Torsional Anharmonicity Xuefei Xu, Tao Yu, Ewa Papajak, and Donald G. Truhlar* Department of Chemistry and Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States S Supporting Information *

ABSTRACT: We calculated the forward and reverse rate constants of the hydrogen abstraction reaction from carbon-2 of 2-methyl-1-propanol by hydroperoxyl radical over the temperature range 250−2400 K by using multistructural canonical variational transition state theory (MS-CVT) including both multiple-structure and torsional potential anharmonicity effects by the multistructural torsional anharmonicity (MS-T) method. In these calculations, multidimensional tunneling (MT) probabilities used to compute the tunneling transmission coefficients were evaluated by the small-curvature tunneling (SCT) approximation. Comparison with the rate constants obtained by the single-structural harmonic oscillator (SS-HO) approximation shows that multistructural anharmonicity increases the forward rate constants for all temperatures, but the reverse rate constants are reduced for temperatures lower than 430 K and increased for higher temperatures. The neglect of multistructural torsional anharmonicity would lead to errors of factors of 1.5, 8.8, and 13 at 300, 1000, and 2400 K, respectively, for the forward reaction, and would lead to errors of factors of 0.76, 3.0, and 6.0, respectively, at these temperatures for the reverse reaction.

1. INTRODUCTION Global warming and declining fossil fuel resources have led to intensive studies of alternative fuels. The biofuels produced from biomass, such as methanol and ethanol, as alternatives have been widely used in the engines. However, recently some problems have been reported1,2 in the use of bioethanol; for example, it may worsen local air quality. 2-Methyl-1-propanol (isobutanol) is being considered as a replacement for ethanol as a fuel-blending component for use in internal combustion engines. Compared to ethanol and methanol, isobutanol is much closer to gasoline in some of its properties, such as, energy density, octane value, and Reid vapor pressure (a measurement of volatility), and it can be blended in any ratio with gasoline as a result of low water affinity, which makes it possible to use isobutanol directly in current engines without modification.3 Isobutanol is a green renewable fuel that can be produced from agricultural products, and it burns clean like a hydrocarbon.3 Because it is potentially an important fuel in the future, interest in the combustion chemistry of isobutanol is rising. Unfortunately, there are very few combustion investigations including isobutanol,4−8 and little kinetic data, such as reaction rate constants for combustion reactions involving isobutanol have been reported. In the present study, we investigate the hydrogen abstraction reaction from carbon-2 of 2-methyl-1-propanol (isobutanol) by © 2012 American Chemical Society

hydroperoxyl radical (•O2H), which plays an important role in combustion chemistry, to produce 3-hydroxy-2-methyl-2propyl radical and hydrogen peroxide (H2O2) by the theoretical method: (CH3)2 CHCH 2OH + •O2 H •

→ (CH3)2 CCH 2OH + H 2O2

(R1)

The •O2H/H2O2 chemistry to a certain extent determines autoignition behavior in the temperature region 600−1300 K because H2O2 is the primary source of •OH radical.9 Multistructural anharmonicity due to molecular torsions have been found10−15 to have a large effect on partition functions, thermal reaction rate constants, and the equilibrium constants of reactions. In the present case, the reactant 2-methyl-1propanol and product 3-hydroxy-2-methyl-2-propyl radical both have four torsions. In each case, two of them are the torsions of methyl groups, which do not generate additional distinguishable structures and do not contribute the multistructural anharmonicity. The other two torsions, which are H−O−C−C and O−C−C−C torsions, lead to nine minima on Received: July 29, 2012 Revised: September 28, 2012 Published: September 28, 2012 10480

dx.doi.org/10.1021/jp307504p | J. Phys. Chem. A 2012, 116, 10480−10487

The Journal of Physical Chemistry A

Article

the potential energy surface for 2-methyl-1-propanol12 and seven minima for 3-hydroxy-2-methyl-2-propyl13 radical. Even H2O2 has two conformers due to the torsion of the O−O bond.16 The transition state of this H-abstraction reaction has five torsional motions (the two torsions of the methyl groups are not counted in these five torsions) and, consequently, a larger number of conformers, as will be discussed in the next section. Thus, the multistructural torsional anharmonicity cannot be ignored in the calculations of reaction rate constants of the reaction R1. Hence, in the present work, we use multistructural canonical variational transition state theory (MS-CVT)14,15 including the effect of multistructural torsional (MS-T) anharmonicity (in particular, both multiple-structure anharmonicity and torsional potential anharmonicity) to calculate the forward and reverse rate constants of the reaction R1 over the temperature range 250−2400 K. In the calculations, the multidimensional tunneling (MT) probabilities used to compute the transmission coefficients were evaluated by the small-curvature tunneling (SCT)17 approximation. In general, one should also consider large-curvature tunneling paths,17b,18 but SCT tunneling paths were been found to make the dominant tunneling contributions in a similar reaction, namely, hydrogen abstraction from methanol by hydroperoxyl radical,15 and so we only consider SCT tunneling here. This decision is reasonable in light of the significantly asymmetric nature of the transition state. The final rate constants obtained using the present method are denoted as MS-CVT/SCT rate constants.

calculations of partition functions and zero-point energies, as is done in the present article (as explained in section 3).

3. COMPUTATIONAL DETAILS The exhaustive search for all conformational structures of the transition state of reaction R1 is the first step. The M08-HX19/ MG3S 20 method, which is highly recommended21 for calculating transition state structure, is used in this step. The MG3S basis set is the same as the 6-311+G(2df,2p) basis set for systems containing only H, C, and O. The single-point energies of the located conformers are recalculated by the more accurate CCSD(T)-F12a22/jul-cc-pVTZ23 method. The FTS MS‑T factor for the transition state has been calculated using all generated conformers by the MSTor program.24 The reactant 2-methyl-1propanol and the product 3-hydroxy-2-methyl-2-propyl radical have been investigated at the same level in previous papers,12,13 and so here we just use those reported structures and MS-T partition functions directly. The equilibrium structures of the reactant •O2H and product H2O2 are optimized by the M08HX/MG3S method. Using the global minima geometries of the reactants,12 products,13 and transition state (this work) obtained by M08-HX/MG3S, the reaction energy and the forward and reverse barrier heights of reaction R1 are calculated using the various Minnesota density functionals (M05,25 M052X,26 M06,27 M06-2X,27 M08-SO,19 and M08-HX) combined with three basis sets (MG3S, ma-TZVP,28 and maug-ccpVTZ29) and also the coupled cluster (CCSD(T)-F12a) method with the calendar basis sets30 (jun-cc-pVDZ, jun-ccpVTZ, and jul-cc-pVTZ) and the dual-level jun-jun31 method. The CCSD(T)-F12a/jul-cc-pVTZ results are taken as the benchmark to validate the other methods by calculating the mean unsigned error (MUE) of the three values (energy of reaction and forward and reverse barrier heights). The combination of density functional and basis set with the best performance is used in the later calculations of the potential energy along the minimum energy path, VMEP, and the vibrationally adiabatic ground-state potential energy curve, VGa , via the multiconfiguration Shepard interpolation method (MCSI)32 by using the MCSI program.33 It will be shown in Section 4.2 that the combination with the smallest MUE is M08-HX/MG3S. Therefore, a frequency scale factor of 0.97334 is used in all calculations of partition functions and the VGa curve to reduce the average error in zero-point energies calculated by the local harmonic approximation. In this way, nontorsional anharmonicity is introduced to improve the accuracy of the low-temperature results. The final VMEP and VGa curves are obtained by the MCSI method using 13 nonstationary Shepard points close to the MEP plus the transition state structure and well structures in the reactant and product valleys. This method is denoted as MCSI-13. The 13 Shepard points are chosen by following a similar strategy to that presented in a previous paper.32b Seven of the 13 Shepard points are located on the reactants side of the transition state with energies 0.70, 2.02, 2.43, 3.10, 3.93, 7.44, and 11.74 kcal/mol below the transition state, and the other six are located on the products side with energies 1.04, 2.76, 3.67, 5.07, 6.83, and 7.56 kcal/mol below the transition state. The reaction coordinate s is scaled to a reduced mass of 1 amu. The MEP is followed using the Page−McIver method35 with a step size of 0.001 Å along the reaction coordinate (s = −1.5  1.5 Å). The parameters for the molecular mechanics force field used in the MCSI calculations are those of the MM3 force field,36 with modifications needed for this work, and the

2. MS-CVT/SCT THEORY The MS-CVT/SCT rate constant is calculated by k MS‐CVT/SCT = F MS‐T(T )κ SCT(T )kCVT(T )

(1)

where kCVT is the single-structure harmonic CVT rate constant, κSCT is the transmission coefficient obtained by the SCT tunneling approximation, and FMS‑T is the reaction MS-T factor, which is the ratio of the multistructural torsional anharmonicity factor of the transition state to that of reactants. The FMS‑T factor includes both the multiple-structure anharmonicity and torsional anharmonicity and is defined by F MS‐T =

TS FMS ‐T(T ) 2 R, i ∏i = 1 FMS ‐T(T )

(2)

where FαMS‑T is the multistructural torsional anharmonicity factor of species α, where α is the transition state (TS) or one of the two reactants (R), i. Each factor in eq 2 can be decomposed into a multiple-structure local-harmonic (MS-LH) component FαMS‑LH and a torsional (T) component FαT. α α α FMS ‐T = FMS‐LH(T )FT (T ) ⎛ Q MS‐LH (T ) ⎞⎛ Q MS‐T (T ) ⎞ con‐rovib, α ⎟⎜ con‐rovib, α ⎟ = ⎜⎜ SS‐HO ⎟⎜ Q MS‐LH (T ) ⎟ Q ( T ) ⎠⎝ con‐rovib, α ⎝ rovib,GM, α ⎠

(3)

In eq 3, “Q” is the partition function for a given species α, “rovib” denotes rotational−vibrational, “con-rovib” denotes conformational−rotational−vibrational, GM denotes the global minimum geometry of the species, and SS-HO stands for the single-structure harmonic oscillator approximation. Note that the MS-LH component would more appropriately be called a multiple-structure locally quasi-harmonic (MS-LQ) component when a scaling factor is used for vibrational frequencies in 10481



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Table 1. Names of Structures, Torsion Angles (in degrees), and Their Relative Conformational Energies (in kcal/mol) for Transition State of the H-Abstraction from Carbon-2 of 2-Methyl-1-propanol by •O2H Radicala relative conformational energy angles of torsions Structureb −

+

− + +

G GC Ca G−G+C+G+a+ G−G+G−C−a− T+G+G−G−g+ G−G−C+G−g+ T−G−C+G−g+ G−G−G+G+g− G−G+C+G+a− G+G−G+G+g− G−G+G−g−a+ T−G+T+C+a+ T+T−G+C+a+ G−T+T−C−a− T+G−T−C−a+ G−T−G+C+a+ g−T+G−C−a− T+T+T−C−a+ G−G−T−C−a− G−T+G−C+a+ G−T−T+C+a+ G−G−T−C+a+ G−T−G+C+a− T−T+G−C−a+ G−G−G−C+a− T+G−A+C−a+ T+G−T+G−a+

M08-HX/MG3S

CCSD(T)-F12a/jul-cc-pVTZ

ϕ(4,1,6,7)

ϕ(6,1,4,5)

ϕ(1,4,5,16)

ϕ(4,5,16,17)

ϕ(5,16,17,18)

BOc

0 Kd

BOc

−72.61 −63.01 −56.30 176.95 −65.09 −162.17 −73.55 −41.49 73.06 −60.90 −176.75 177.26 −69.92 177.89 −73.08 −77.45 178.38 −71.09 −70.78 −70.76 −66.25 −69.44 −178.40 −69.86 164.09 168.34

61.15 56.46 65.85 57.89 −54.61 −56.36 −59.46 53.82 −45.47 52.87 66.67 −175.49 178.48 −63.72 −175.91 174.85 180.07 −66.82 175.93 −179.34 −62.10 −177.29 177.31 −72.48 −67.55 −68.92

−9.25 17.41 −50.35 −34.97 27.57 12.64 31.28 10.02 46.10 −54.82 161.72 48.66 −176.96 −167.58 52.04 −54.38 −177.71 −163.90 −70.19 178.86 −179.75 59.51 −58.64 −74.73 114.34 159.39

18.73 48.17 −13.61 −50.72 −52.96 −35.21 49.04 51.36 59.36 −80.63 21.08 17.58 −4.88 −6.91 12.15 −13.58 −0.83 −18.83 7.16 3.06 5.77 4.46 −6.41 13.59 −24.21 −54.26

97.29 96.53 −94.17 75.07 80.23 79.02 −77.98 −99.69 −82.75 96.11 96.79 96.76 −97.64 99.22 97.69 −98.29 97.60 −98.78 100.55 99.66 99.55 −100.32 100.98 −99.16 97.16 97.96

0.00 0.11 0.16 0.49 0.58 0.76 0.86 0.94 1.12 1.22 2.88 3.17 3.28 3.40 3.42 3.44 3.48 3.51 3.59 3.61 3.65 3.68 3.75 3.85 4.30 4.31

0.00 0.17 0.16 0.51 0.63 0.73 0.94 1.15 1.38 1.16 2.53 2.85 3.08 2.98 3.20 3.19 3.12 3.21 3.37 3.38 3.30 3.41 3.32 3.54 3.61 3.66

0.00 0.25 0.12 0.68 0.58 0.74 0.95 1.13 1.44 1.05 2.77 2.77 2.81 3.23 2.83 2.88 3.06 3.34 3.04 3.06 3.44 3.12 3.33 3.68 3.82 3.77

a

The geometries are obtained by the M08-HX/MG3S method. The notation is explained in detail in refs 12 and 41; in short, G or g, C, A, or a, and T denote respectively gauche, cis, anti, and trans. bOnly one of each pair of mirror image structures is given in this table, and the corresponding mirror structure has the opposite sign for each torsion angle and the same energy. cBO denotes Born−Oppenheimer, i.e., exclusive of zero point energy. d0 K denotes zero Kelvin, i.e., inclusive of zero point energy.

Figure 1. Global minimum structure G−G+C−C+a+/G+G−C+C−a− of the transition state of the hydrogen abstraction from carbon-2 of 2-methyl-1propanol by the hydroperoxyl radical. The five torsions of Table 1 are shown in green. 10482



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Table 2. Multistructural Torsional Anharmonicity Factors for Species Involved in the H-Abstraction from Carbon-2 of 2Methyl-1-propanol by •O2H Radicala,b 2-methyl-1-propanol (R1)

TS

3-hydroxy-2-methyl-2-propyl (P1)

H2O2 (P2)

T/K

R1 FMS ‐T

R1 FMS ‐ LH

FTR1

TS FMS −T

TS FMS − LH

FTTS

P1 FMS ‐T

P1 FMS ‐ LH

FTP1

P2 FMS ‐T

P2 FMS ‐ LH

FTP2

250 298.15 300 400 600 800 1000 1500 2000 2400

7.52 8.07 8.09 8.99 10.06 10.36 10.16 8.73 7.13 6.03

5.63 5.84 5.85 6.14 6.48 6.66 6.77 6.92 7.00 7.04

1.34 1.38 1.38 1.46 1.55 1.56 1.50 1.26 1.02 0.86

9.36 12.03 12.14 20.14 44.11 69.75 89.09 103.86 92.22 78.18

6.73 7.99 8.04 11.47 21.33 33.08 44.84 70.13 89.06 100.70

1.39 1.51 1.51 1.76 2.07 2.11 1.99 1.48 1.04 0.78

6.10 7.49 7.54 9.98 12.59 13.05 12.48 9.90 7.58 6.16

3.98 4.66 4.69 5.97 7.91 9.25 10.20 11.70 12.55 13.00

1.53 1.61 1.61 1.67 1.59 1.41 1.22 0.85 0.60 0.47

2.10 2.13 2.13 2.19 2.28 2.33 2.35 2.30 2.20 2.12

2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00

1.05 1.06 1.06 1.09 1.14 1.17 1.17 1.15 1.10 1.06

a The results in this table are obtained using CCSD(T)-F12a/jul-cc-pVTZ energies with the M08-HX/MG3S geometries. bThere is only one structure for the •O2H radical, so the three factors of that radical are unity.

sition state and the five torsions are shown in Figure 1. The naming convention for labeling the structures is explained in refs 12 and 41. As shown in Table 1, the largest zero-pointexclusive energy difference between the 26 pairs structures is 4.31 kcal/mol at the M08-HX/MG3S level, and 3.82 kcal/mol at the CCSD(T)-F12a/jul-cc-pVTZ level. The 10 pairs of structures with the lowest energy are all within 1.22 kcal/mol at the M08-HX/MG3S level (1.44 kcal/mol at the CCSD(T)F12a/jul-cc-pVTZ level), and they make the largest contributions to the partition function. The M08-HX/MG3S energies are very similar to those of CCSD(T)-F12a/jul-cc-pVTZ single-point energies for these 20 structures, so that the multistructure torsional factors of the transition state calculated with the two methods are very similar. Therefore, only multistructure torsional factors FTS MS−T and their components TS (FTS MS‑LH and FT ) calculated with CCSD(T)-F12a/jul-cc-pVTZ energies are given in Table 2. For comparison, Table 2 also gives the multistructure torsional factors of the reactant that has a torsion and the products. Table 2 shows that the transition state has considerably larger multiple-structure anharmonicity (FαMS‑LH) at each temperature (particularly for temperatures higher than 600 K) than either the reactant 2-methyl-1propanol or the product 3-hydroxy-2-methyl-2-propyl radical (due to more conformers), but it has a similar torsional anharmonicity (FαT). The FTS MS‑T factor of the transition state including MS-T anharmonicity increases from 9.36 at 250 K to a maximum value (103.86) at 1500 K. The reaction MS-T factors of eq 2 are tabulated in Table 3 for both the forward and reverse reactions. The multistructural torsional anharmonicity increases the reaction rate of the forward reaction at all the temperatures (250−2400 K) we investigated. The factors in Table 3 indicate that the multistructural torsional anharmonicity enhancement of the reaction rate gradually increases when the temperature is raised, and it becomes more important at high temperatures for both the forward and the reverse reactions. At the low temperatures (250−430 K), the FMS‑T factors of the reverse reaction are smaller than unity, which means that the multistructural torsional anharmonicity reduces the reaction rate of the reverse reaction in this temperature region. At high temperatures, the multistructural torsional anharmonicity is much more important for the forward reaction than the reverse reaction. 4.2. Validations of Methods and Barrier Height. Using the lowest-energy structures of the reactants, products, and the transitions state optimized by M08-HX/MG3S, the zero-point-

modified parameters are provided in the Supporting Information. Direct dynamics calculations were performed for the reaction path in the small region of s = −0.3−0.3 Å to verify the VGa curves obtained by the MCSI-13 method in this critical region. Our rate calculations are carried out for the low-pressure limit. We do not need to consider the prereactive and postreactive intermediate noncovalent complexes (van der Waals complexes) because the only aspect of the calculations that they affect are the tunneling transmission coefficients.11b In the low-pressure limit of the forward reaction, there are insufficient termolecular collisions to populate those states of the prereactive intermediate with energies lower than the lowest-energy state of two reactants, and there are insufficient collisions to form products from any directly produced product−valley complexes with energies lower than the ground-state energy of products. Thus the existence of noncovalent intermediates along the reaction path has no effect on the final MS-CVT/SCT low-pressure rate constants. Based on the generated VMEP and VGa curves, kCVT and κSCT of eq 1 are calculated for the temperature range 250−2400 K by using the MC-TINKERATE program.37 Then the final MSCVT/SCT rate constants are obtained using eq 1. All Minnesota density functional calculations are performed using a locally modified version, MN-GFM,38 of Gaussian09.39 The grid for density functional integrations has 99 radial shells around each atom and 974 angular points in each shell. The CCSD(T)-F12a calculations are carried out with the Molpro program.40

4. RESULTS AND DISCUSSION 4.1. Conformers and MS-T Anharmonicity of Transition State and Reactions. The multiple conformers and MS-T partition functions of the reactant 2-methyl-1-propanol and product 3-hydroxy-2-methyl-2-propyl radical have been reported in previous papers.12,13 Here we focus on the structure searching and MS-T factors for the transition state and their use for rate constant calculations. We found 26 pairs of optically active structures (52 structures) of the transition state by the M08-HX/MG3S method. The names of structures, the five torsions angles that can generate distinguishable structures, and the relative conformational energies are shown in Table 1. The global minimum structure G−G+C−C+a+/G+G−C+C−a− of the tran10483



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Table 3. Multistructural Torsional Anharmonicity Factor Ratios for the H-Abstraction from Carbon-2 of 2-Methyl-1Propanol by •O2H Radicala forward reaction T/K 250 298.15 300 400 600 800 1000 1500 2000 2400

F

MS‑T

1.24 1.49 1.50 2.24 4.38 6.73 8.77 11.90 12.94 12.97

F

MS‑LH

1.20 1.37 1.37 1.87 3.29 4.97 6.62 10.13 12.72 14.30

Table 4. Energies of Reaction and Forward and Reverse Barrier Heights (Zero-Point-Exclusive, in kcal/mol) for the H-Abstraction from Carbon-2 of 2-Methyl-1-propanol by •O2H Radicala

reverse reaction F

T

1.04 1.09 1.09 1.20 1.33 1.35 1.32 1.18 1.02 0.91

F

MS‑T

0.73 0.75 0.76 0.92 1.53 2.29 3.04 4.56 5.52 5.98

FMS‑LH

FT

0.85 0.86 0.86 0.96 1.35 1.79 2.20 3.00 3.55 3.87

0.86 0.88 0.88 0.96 1.14 1.28 1.38 1.52 1.56 1.54

CCSD(T)-F12a/jul-ccpVTZ CCSD(T)-F12a/jun-ccpVTZ CCSD(T)-F12a/jun-ccpVDZ CCSD(T)-F12a/jun-jun M08-HX/MG3S M06/ma-TZVP M08-HX/maug-ccpVTZ M06−2X/ma-TZVP M08-SO/MG3S M06−2X/maug-ccpVTZ M05-2X/maug-ccpVTZ M06−2X/MG3S M08-HX/ma-TZVP M06/maug-cc-pVTZ M05/MG3S M05/ma-TZVP M05-2X/MG3S M05/maug-cc-pVTZ M08-SO/maug-ccpVTZ M08-SO/ma-TZVP M05-2X/ma-TZVP M06/MG3S

a

The results in this table are obtained by CCSD(T)-F12a/jul-ccpVTZ energies with the M08-HX/MG3S geometries.

exclusive energy of reaction and the forward and reverse barrier heights of reaction R1 as calculated by 18 combinations of six Minnesota functionals and three basis sets and by the CCSD(T)-F12a method with four basis sets are shown in Table 4. The CCSD(T)-F12a/jul-cc-pVTZ results are taken as the benchmark, and the corresponding MUEs of the three values of the other 21 electronic model chemistries are calculated and given in Table 4. The M08-HX/MG3S has the smallest MUE (0.15 kcal/mol) of all the density methods, and is chosen to be used for frequencies and Hessians in the later dynamic calculations. The CCSD(T)-F12a/jul-cc-pVTZ (M08-HX/MG3S) calculated energies of reaction and forward and reverse barrier heightswithout zero point energiesare 10.83 (11.05), 15.13 (15.27), and 4.31 (4.22) kcal/mol, respectively. Adding zero point contributions calculated by the M08-HX/MG3S method with the vibrational frequency scale factor mentioned above gives 9.93, 12.87, and 2.95 kcal/mol for the 0 K energy of reaction and forward and reverse barrier heights with the CCSD(T)-F12a/jul-cc-pVTZ//M08-HX/MG3S method and gives 10.15, 13.01, and 2.86 kcal/mol for the 0 K energy of reaction and forward and reverse barrier heights with the M08HX/MG3S method that is used for the dynamics calculations. 4.3. Dynamic Calculations for the Rate Constants. The calculated potential energy along the minimum energy path, VMEP, and the ground-state vibrationally adiabatic potential curve, VGa , are shown in Figure 2. The MCSI-13 method obtains a smooth VMEP curve that is in excellent agreement with the direct dynamics calculation; in fact the two curves cannot be distinguished in Figure 2. The MCSI-13 VGa curve has some slight ups and downs along the path in the vicinity of the transition state, but it also agrees well with the one obtained by the direct dynamics calculation as shown in Figure 2. Using these potential curves, the single-structure harmonic (in fact, it is quasiharmonic because nontorsional anharmonicity has been introduced by using the scaling factor in the zero-point-energy calculations) CVT rate constants kCVT and corresponding transmission coefficient κSCT for both forward and reverse reactions are evaluated and tabulated in Table 5, as well as their products, kCVT/SCT. The tunneling increases the reaction rates, especially for the low temperatures, for example, κSCT = 10.65 at 250 K and κSCT = 5.59 at 300 K. Above 800 K, the tunneling becomes less important (κSCT ≤ 1.32) for this Habstraction reaction.

energy of reaction

forward barrier

reverse barrier

MUEb

10.83

15.13

4.31

0.00

10.74

15.14

4.40

0.06

11.13

14.95

3.82

0.32

10.28 11.05 10.35 10.78

14.60 15.27 14.61 15.82

4.31 4.22 4.25 5.04

0.36 0.15 0.35 0.49

11.17 10.16 10.21

14.34 15.77 13.79

3.17 5.61 3.58

0.76 0.87 0.90

11.19

14.03

2.84

0.98

10.68 11.94 9.23 9.22 10.39 11.64 10.00 10.54

13.64 16.65 13.95 15.07 16.37 14.01 16.32 16.86

2.96 4.70 4.73 5.84 5.99 2.37 6.32 6.33

1.00 1.01 1.06 1.07 1.12 1.29 1.34 1.35

11.45 12.43 8.61

17.22 14.64 12.92

5.77 2.20 4.30

1.39 1.40 1.48

a

All results in this table are calculated with the geometries obtained by the M08-HX/MG3S method, employing the G−G+C−C+a+ structure for the transition state and the lowest-energy structures (refs 12 and 13) for reactants and products. bThe mean unsigned errors in the three energetic quantities are calculated with respect to the CCSD(T)F12a/jul-cc-pVTZ results.

The final kMS‑CVT/SCT rate constants including both multistructural torsional anharmonicity and tunneling are calculated using eq 1 and shown in Table 5, and their common logarithms are plotted as functions of temperature in Figure 3. For comparison, the common logarithms of kCVT/SCT are also plotted in Figure 3. The figure clearly shows the large deviations at high temperatures of the single-structure harmonic rate constants (kCVT/SCT) for both forward and reverse reactions as compared to the more accurate kMS‑CVT/SCT rate constants. Based on the multistructural torsional factor ratios calculated in section 4.1, the neglect of multistructural anharmonicity in rate constants calculations would lead to errors of factors of 1.50, 8.77, and 13.0 at 300, 1000, and 2400 K, respectively, for the forward reaction, and would lead to errors of factors of 0.76, 3.04, and 5.98 at 300, 1000, and 2400 K, respectively, for the reverse reaction. The present forward MS-CVT/SCT rate constants are fitted with the four-parameter expression in ref 42, and can be represented as ⎛ T ⎞n −E(T + T0)/ R(T 2 + T02) MS‐CVT/SCT ⎟ × e k forward = A⎜ ⎝ 300 ⎠ 10484

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Figure 3. The temperature dependence of CVT/SCT and MS-CVT/ SCT rate constants for both the forward and reverse reactions for the hydrogen abstraction from carbon-2 of 2-methyl-1-propanol by hydroperoxyl radical.

Table 6. The Fitting Parameters Obtained from the FourParameter Expressions Eqs 4 and 5 for the Forward and Reverse Reactionsa Fitting parameters

Figure 2. Calculated potential energy along the minimum energy path (VMEP) and ground-state vibrationally adiabatic potential curve (VGa ) versus the reaction coordinate s, where the reaction coordinate s is scaled to a reduced mass of 1 amu.

directions forward reverse

3

A (cm molecule

−1

−1

s )

2.55 × 10−15 6.58 × 10−16

T0 (K)

n

E(kcal/mol)

156.89 425.50

3.956 3.019

9.383 4.081

a

Equation 4 is used for the forward reaction, and eq 5 is used for the reverse reaction.

where R in eqs 4 and 5 is the gas constant. Equation 4 ensures that the rate constant for the endothermic direction goes to zero at 0 K. The reverse reaction is exothermic, and its rate constants are fitted using the four-parameter expression of ref 11b. MS‐CVT/SCT k reverse

⎛ T + T0 ⎞n −E(T + T0)/ R(T 2 + T02) ⎟ × e = A⎜ ⎝ 300 ⎠

plots,43 and these slopes may be calculated analytically from the above fits, which yield Ea(forward) = E

(5)

T 4 + 2T0T 3 − T02T 2 (T 2 + T02)2 4

Ea(reverse) = E

The fitting parameters (A, T0, n, and E) are given in Table 6. (We remind the reader that these are least-squares fits, and they do not fit the data exactly.) The corresponding Arrhenius activation energies are given as the local slopes of Arrhenius

T + 2T0T 2

(T +

+ nRT (6)

3

− T02T 2 T02)2

+ nR

T2 T + T0

(7)

The resulting activation energies are given in table 7, which shows a very large temperature dependence for both the

Table 5. Calculated Reaction Rates (in cm3 molecule−1 s−1) and κSCT Factors (Tunneling) for the H-Abstraction from Carbon-2 of 2-Methyl-1-propanol by •O2H Radicala MS-T resultsb

SS-HO results by M08-HX/MG3S forward reaction T/K 250 298.15 300 400 600 800 1000 1500 2000 2400

CVT 2.23 1.88 2.17 7.99 4.47 1.45 1.42 4.33 3.07 8.92

× × × × × × × × × ×

10−26 10−24 10−24 10−22 10−19 10−17 10−16 10−15 10−14 10−14

reverse reaction

CVT/SCT 2.37 1.07 1.21 2.22 7.21 1.91 1.69 4.68 2.77 8.12

× × × × × × × × × ×

10−25 10−23 10−23 10−21 10−19 10−17 10−16 10−15 10−14 10−14

CVT 3.27 8.90 9.20 3.85 2.22 6.99 1.65 7.61 2.14 3.98

× × × × × × × × × ×

CVT/SCT

10−18 10−18 10−18 10−17 10−16 10−16 10−15 10−15 10−14 10−14

3.47 5.07 5.14 1.07 3.58 9.19 1.97 8.22 1.93 3.62

× × × × × × × × × ×

10−17 10−17 10−17 10−16 10−16 10−16 10−15 10−15 10−14 10−14

forward reaction

reverse reaction

MS-CVT/SCT

MS-CVT/SCT

2.95 1.59 1.82 4.97 3.16 1.29 1.48 5.57 3.58 1.05

× × × × × × × × × ×

10−25 10−23 10−23 10−21 10−18 10−16 10−15 10−14 10−13 10−12

2.54 3.83 3.89 9.88 5.50 2.10 6.00 3.75 1.06 2.17

× × × × × × × × × ×

10−17 10−17 10−17 10−17 10−16 10−15 10−15 10−14 10−13 10−13

κSCT 10.65 5.70 5.59 2.79 1.62 1.32 1.20 1.08 1.05 1.03

a

The results in this table are obtained with the M08-HX/MG3S geometries. bThe MS-T factors are obtained with CCSD(T)-F12a/jul-cc-pVTZ energies. 10485



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Table 7. The Activation Energy Ea (in kcal/mol) of the Forward and Reverse Reaction for the H-Abstraction from Carbon-2 of 2-Methyl-1-propanol by •O2H Radical T/K

forward reaction

reverse reaction

250 298.15 300 400 600 800 1000 1500 2000 2400

11.0 12.6 12.6 14.6 16.7 18.1 19.4 22.8 26.4 29.4

0.96 1.54 1.57 2.96 5.57 7.55 9.09 12.2 15.0 17.3

3.04, and 5.98 at 300, 1000, and 2400 K for the reverse reaction. The forward reaction of the hydrogen abstraction from carbon-2 of 2-methyl-1 propanol by hydroperoxyl radical is found to be much faster than the hydrogen abstraction reaction of methanol by hydroperoxyl radical over the temperature range of 250−2400 K, and has a similar rate to that of the hydrogen abstraction from carbon-3 of 1-butanol by hydroperoxyl radical over the temperature range of 600−2400 K.

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ASSOCIATED CONTENT

S Supporting Information *

The Cartesian coordinates and absolute energies in hartrees of the transition structures and the modified MM3 parameters used in MCSI calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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forward and reverse reactions. This provides yet one more example of how dangerous it is to assume that the Arrhenius activation energy is approximately equal to either the classical barrier height or the vibrationally adiabatic barrier height, both of which are independent of temperature. As far as we know, no rate constants have been measured for reaction R1. We can, however, compare the calculated rate constants for reaction R1 with those of the H-abstraction reaction (R2) from methanol by •O2H,15 and those of Habstraction reaction (R3) from carbon-3 of 1-butanol by •O2H.31 In the whole temperatures region from 250 to 2400 K, the forward rate constants of R1 are much larger than those of R2: at 300 K, 1.82 × 10−23 (R1) versus 3.21 × 10−28 (R2); at 1000 K, 1.48 × 10−15 (R1) versus 6.41 × 10−17 (R2); and at 2400 K, 1.05 × 10−12 (R1) versus 4.29 × 10−13 (R2). At low temperatures (250−400 K), the forward rate constants of R1 are 1 order of magnitude larger than those of R3. Over the temperature range from 600 to 2400 K, the forward reaction rate constants of the two reactions are of the same order of magnitude, and when the temperature is higher than 1000 K, reaction R1 is calculated to be slightly slower than reaction R3; for example, at 600 K, 3.16 × 10−18 (R1) versus 1.41 × 10−18 (R3); at 1000 K, 1.48 × 10−15 (R1) versus 1.68 × 10−15 (R3); and at 2400 K, 1.05 × 10−12 (R1) versus 1.30 × 10−12 (R3). These comparisons imply that the H-abstraction from methyl groups (carbon-3) of 2-methyl-1-propanol are probably smaller than those for H-abstraction from carbon-2 (the present investigation) or from carbon-1. The H-abstraction from carbon-1 of isobutanol is an interesting case for future study.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to Jingjing Zheng and John Alecu for help with the direct dynamics calculations and for helpful discussions. This work was supported in part by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Grant No. DE-FG02-86ER13579 and as part of the Combustion Energy Frontier Research Center under Award Number DE-SC0001198.

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REFERENCES

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5. SUMMARY The thermal forward and reverse reaction rate constants for hydrogen abstraction from carbon-2 of 2-methyl-1-propanol by hydroperoxyl radical over the temperature range of 250−2400 K have been estimated using MS-CVT/SCT theory including both multiple-structure and torsional potential anharmonicity effects by the multistructural torsion (MS-T) method and including quantal effects on the reaction coordinate by the multidimensional small-curvature tunneling (SCT) method. Comparison with the rate constants obtained by the singlestructural harmonic oscillator (SS-HO) approximation shows that multistructural anharmonicity increases the forward rate constants for all temperatures and increases the reverse rate constants for temperatures higher than 430 K. The neglect of multistructural anharmonicity would lead to errors of factors of 1.50, 8.77, and 13.0 at 300, 1000, and 2400 K, respectively, for the forward reaction, and would lead to errors of factors of 0.76, 10486



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