Interpretation and Application of Reaction Class Transition State

Article pubs.acs.org/JPCA

Interpretation and Application of Reaction Class Transition State Theory for Accurate Calculation of Thermokinetic Parameters Using Isodesmic Reaction Method Bi-Yao Wang,‡ Ze-Rong Li,*,† Ning-Xin Tan,‡ Qian Yao,† and Xiang-Yuan Li‡ †

College of Chemistry and ‡College of Chemical Engineering, Sichuan University, Chengdu 610064, China S Supporting Information *

ABSTRACT: We present a further interpretation of reaction class transition state theory (RC-TST) proposed by Truong et al. for the accurate calculation of rate coefficients for reactions in a class. It is found that the RC-TST can be interpreted through the isodesmic reaction method, which is usually used to calculate reaction enthalpy or enthalpy of formation for a species, and the theory can also be used for the calculation of the reaction barriers and reaction enthalpies for reactions in a class. A correction scheme based on this theory is proposed for the calculation of the reaction barriers and reaction enthalpies for reactions in a class. To validate the scheme, 16 combinations of various ab initio levels with various basis sets are used as the approximate methods and CCSD(T)/CBS method is used as the benchmarking method in this study to calculate the reaction energies and energy barriers for a representative set of five reactions from the reaction class: RcCH(Rb)CRaCH2 + OH• → RcC•(Rb)CRaCH2 + H2O (Ra, Rb, and Rc in the reaction formula represent the alkyl or hydrogen). Then the results of the approximate methods are corrected by the theory. The maximum values of the average deviations of the energy barrier and the reaction enthalpy are 99.97 kJ/mol and 70.35 kJ/mol, respectively, before correction and are reduced to 4.02 kJ/mol and 8.19 kJ/ mol, respectively, after correction, indicating that after correction the results are not sensitive to the level of the ab initio method and the size of the basis set, as they are in the case before correction. Therefore, reaction energies and energy barriers for reactions in a class can be calculated accurately at a relatively low level of ab initio method using our scheme. It is also shown that the rate coefficients for the five representative reactions calculated at the BHandHLYP/6-31G(d,p) level of theory via our scheme are very close to the values calculated at CCSD(T)/CBS level. Finally, reaction barriers and reaction enthalpies and rate coefficients of all the target reactions calculated at the BHandHLYP/6-31G(d,p) level of theory via the same scheme are provided. the structure of the flame, or species profiles. The elementary reactions of the larger chemical species are usually generated according to reaction classes. Reactions in the same class have the same reactive moiety and thus are expected to have similarities in their potential energy surfaces along their reaction valleys.5 It is one of the most challenging tasks to provide accurate thermodynamic data and kinetic data for the detailed mechanisms. Many of the important chemical species involved in combustion are short-lived radicals and they are very reactive, so it is very difficult to obtain their thermodynamic properties and kinetic properties by experimental methods. Nowadays, theoretical methods are widely used to calculate the thermokinetic parameters for combustion of hydrocarbons. Among the methods used to calculate thermodynamic properties for modeling hydrocarbon combustion, group contribution methods6 may be the most widely used approach. In these methods, the molecular structure is represented by groups and the thermodynamic properties are obtained by the sum of the contribution of all groups within the molecular

1. INTRODUCTION Hydrocarbon fuels are some of the most common fuels in nature, with high energy density and low cost, and they are one of the main sources for obtaining energy in the present and in the future. In order to improve the combustion of these fuels while reducing the formation of pollutants, it is necessary to propose and validate kinetic reaction mechanisms for a detailed modeling of combustion. However, the combustion of hydrocarbon fuels is a complicated process: the detailed reaction mechanism of combustion usually includes hundreds of species and thousands of elementary reactions.1,2 Because of the complexity, detailed mechanisms for the combustion of large hydrocarbons are usually generated by automated computer software. The elementary reactions generated by the software, such as EXGAS,3,4 usually include two types of reactions: a reaction base, which includes all reactions involving radicals or molecules less than three carbon atoms (C0−C2 reaction base), less than five carbon atoms (C0−C4 reaction base), or less than seven carbon atoms (C0−C6 reaction base), and reactions involving larger species. The reaction bases are structured in a hierarchical manner with hydrogen−oxygen− carbon monoxide chemistry and optimized and validated by comparing simulated results with experimental results of combustion, such as the ignition delay time, the speed and © 2013 American Chemical Society

Received: January 27, 2013 Revised: March 13, 2013 Published: March 19, 2013 3279

dx.doi.org/10.1021/jp400924w | J. Phys. Chem. A 2013, 117, 3279−3291

The Journal of Physical Chemistry A

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molecules.13 Therefore, it is still a challenge in computational chemistry to accurately calculate energy barriers or reaction rates for large reaction systems. In this respect, Truong and coworkers38−44 recently have introduced reaction class transition state theory (RC-TST), which can accurately calculate the reaction rates for reactions in a class based on transition state theoty (TST) using low-level ab initio methods. TST theory was proposed by E. Eyring, M. G. Evans, and M. Polanyi in 1935.45 From the transition state theory framework the thermal rate coefficient can be expressed as

structure. These methods are fast and easy to implement. However, sometimes they are very approximate and the group contribution parameters for many radicals are lacking, so their applications are limited. Recently, many composite methods based on energy additivity schemes or extrapolation schemes for accurate prediction of thermochemical properties of molecules have been developed, and on average they can reach the so-called chemical accuracy of about 1 kcal/mol. Among these methods, the most successful and popular approaches may be the Gaussian-n (n = 1−4)7−14 family of model chemistries. The model chemistry of complete basis set (CBS) methods of Petersson and co-workers are successfully used for the calculations of the enthalpy of formation. Other composite methods, such as the W-n (Weizmann-1, Weizmann2, Weizmann-3) theory15−18 developed by Martin and de Oliveira et al., which aimed at “benchmark accuracy” of 1 kJ/ mol (0.24 kcal/mol), the High Accuracy Extrapolated ab initio Thermochemistry (HEAT) method by Stanton and coworkers,19−21 and the multicoefficient correlation method (MCCM) developed by Truhlar and co-workers,22−24 all exhibit chemical accuracy of 80 atoms).30 However, many functionals have been developed so far and the performance of DFT varies with the chemical system under study, as well with the functional itself. Many benchmarking studies of density functionals on different types of chemical systems are already available.31−37 It can be concluded from these benchmark comparisons that no functional outperforms all other functionals for all reaction systems; therefore, the choice of functional to tackle a chemical problem requires the comparison of the results of DFT functionals with those obtained from higher-level accurate computations. Moreover, in the calculation of electronic energies, most DFT methods usually have significant errors, and the errors increase significantly with the size of the

k(T ) = κ(T )σ

⎛ ΔV ⧧ ⎞ kBT Q ⧧(T ) ⎟ ⎜− exp h Φ R (T ) ⎝ kBT ⎠

(1)

where κ is the transmission coefficient accounting for the quantum mechanical tunneling effects, σ is the reaction symmetry number,29 kB and h are the Boltzmann and Planck constants, respectively, and T is the temperature. Q⧧ and ΦR are the total partition functions (per unit volume) of the transition state and the reactant, respectively, including translational, rotational, and vibrational contributions; ΔV⧧ is the classical barrier, the difference between the electronic energy of the transition state and that of the reactants. In eq 1, the partition function can be approximately written as a product of a translational, a rotational, and a vibrational partition function. The translational partition function and the rotational partition function can obtained directly from the geometry of the reactants or the transition state, which can be obtained by geometry optimization at a low-level ab initio method. The vibrational partition function requires the vibrational frequencies, which also can be obtained at the low-level ab initio method after proper scaling.48−50 In eq 1, the ΔV⧧ involves the difference of the electronic energies which can be calculated accurately only through high-level ab initio methods. Therefore, the most time-consuming step of the calculation of reaction rates is the calculation of the electronic energy. RC-TST theory proposed by Truong et al.38 is used to calculate the rate coefficients at modest levels of ab initio methods for reactions in a class. The basic idea of the RC-TST method38 has two reactions in a class: one reaction, called the target reaction, with rate coefficient kt that is unknown and needs to be calculated; another reaction, called the principal reaction, with rate coefficient kp that is known from an experiment or an accurate ab initio method. The ratio f(T) of the rate coefficients kp and kt for the two reactions can be factored into four factors according to eq 1: f (T ) = k t(T )/k p(T ) = fk fσ fQ fV

(2)

where f k, fσ, f Q, and f V are the ratio of the transmission coefficient, the symmetry number, the partition function, and the energy barrier, respectively. fκ (T ) = κ t(T )/κ p(T )

(3)

fσ = σt /σp

(4)

−1 ⎛ Q ⧧(T ) ⎞ ⎛ Q ⧧(T ) ⎞⎛ Q ⧧(T ) ⎞ p t ⎟ ⎜ ⎜ t ⎟ ⎟ fQ (T ) = ⎜⎜ R = ⎟ ⎜ Φ R (T ) ⎟ ⎜ Q ⧧(T ) ⎟ Φ T ( ) ⎝ t ⎠⎝ p ⎝ p ⎠ ⎠

⎛ ΦR (T ) ⎞−1 ⎜ t ⎟ ⎜ Φ R (T ) ⎟ ⎝ p ⎠ 3280

(5)



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fV (T ) = exp[(ΔV t⧧ − ΔV p⧧)/kBT ] = exp(−ΔΔV ⧧/kBT )

AP + BP + PT = A T + BT + PP

(6)

ΔΔH = Δr HP − Δr HT

⧧

ΔΔV is the difference of the energy barrier of the target reactions and the principal reaction. Truong and co-workers observed that the vibrational imaginary frequency of the transition state is a conserved quantity for different reactions in the same reaction class and the partition function factor f Q has a weak dependence on the temperature. More importantly, they found that ΔΔV⧧ can be accurately calculated from a relatively low level of theory; hence the potential energy factor f V has little dependence on the level of ab initio theory. The rate coefficient of the target reaction kt can be accurately predicted when the ratio f(T) is accurately predicted: k t(T ) = f (T ) k p(T )

where ΔΔH is the reaction enthalpy for reaction 12. Reaction 10 minus reaction 11 is AP + BP + TST = A T + BT + TSP ΔΔV ⧧ = ΔV P⧧ − ΔV T⧧

where ΔΔV is the reaction energy for reaction 13. It is apparent that reaction 12 is an isodesmic reaction because the numbers of bonds for each formal type are conserved in reactants and products. A typical bond type usually includes single bond, double bond, triple bonds, and aromatic bond. However, in a transition state, some bonds are usually extremely distorted and it is difficult to classify them into typical types of bond; hence we can not judge if reaction 13 is an isodesmic reaction according to the conservation of the numbers of bond for each type. A reaction system can be divided into two parts: the reaction center that involves the bond breaking and bond making, and the surrounding groups. The reactants and products in reaction 13 can also be divided into a reaction center that involves distorted bonds and surrounding groups that involve typical bond types. It is recognized that reactions in a given class have the same reaction center,38,39 and it can be easily shown that the numbers of bonds for each bond type in the surrounding groups in the two sides of reaction 13 are conserved. If the geometries of the reaction centers for reactions 10 and 11 are conserved, the calculated reaction energy for reaction 13 will not be much dependent on the level of ab initio calculation because the errors associated with incomplete basis sets and incomplete correction can be partially canceled, as in the case of an isodesmic reaction. Therefore, the traditional definition of isodesmic reaction can be expanded to the case for the reaction of the formation of transition state in this study. If reaction enthalpies ΔrHP′ and ΔrHT′ and energy barriers ΔV⧧P ′ and ΔV⧧T′ are the accurate values from experiment or from high-level ab initio calculations, and reaction enthalpies ΔrHP and ΔrHT and energy barriers ΔV⧧P and ΔV⧧T are the approximate values from low-level ab initio calculations, they should satisfy approximately the following equations from the theory of the isodesmic reaction:

It is well-known that the calculated reaction enthalpy for an isodesmic reaction is not much dependent on the level of ab initio calculation.49,50 An isodesmic reaction is defined as one reaction in which the numbers of bonds of each formal bond type are conserved in the reactants and products and hence a modest-level ab initio calculation can produce an accurate reaction enthalpy for an isodesmic reaction because the errors associated with incomplete basis sets and incomplete correlation energy can be partially canceled.51 Isodesmic reactions are usually designed to calculate the heats of formation for species at a relatively low level of theory. The isodesmic reaction approach has been used systematically for the calculation of transition states of reactions in a class by Knyazev.52,53 However, the approach should be further validated because distorted bonds are usually involved in the transition states and hence it is difficult to define the bond type. In this work, the definition of isodesmic reaction is expanded to reactions of the formation of transition states where both the number of bonds of each type outside the reaction center and the geometries of the reaction center are conserved. A further interpretation and validation of the RC-TST theory is also given by the comparison of the calculated energy barriers and rate constants using various ab initio methods.

2. VALIDATION OF THE ISODESMIC REACTION APPROACH FOR TRANSITION STATES Let us consider two elementary reactions in a class: the target reaction RT and the principal reaction RP. Their rate coefficients are kP and kT, respectively, for the two reactions. Δr HP

A T + BT = PT

Δr HT

(13)

⧧

(7)

AP + BP = PP

(12)

(8)

ΔΔV ⧧ = ΔV T⧧ ′ − ΔV P⧧ ′ = ΔV T⧧ − ΔV P⧧

(14)

ΔΔH = Δr HT′ − Δr HP′ = Δr HT − Δr HP

(15)

Hence we have (9)

ΔV T⧧ ′ = ΔV P⧧ ′ + (ΔV T⧧ − ΔV P⧧)

ΔrHP and ΔrHT are the reaction enthalpies for the principal reaction and the target reaction. The corresponding reactions for the formation of the transition states of the two reactions from the reactants can be written as AP + BP = TSP

ΔV P⧧

A T + BT = TST

ΔV P⧧

= ΔV T⧧ + (ΔV P⧧ ′ − ΔV P⧧)

(16)

Δr HT′ = Δr HP′ + (Δr HT − Δr HP) = Δr HT + (Δr HP′ − Δr HP)

(10)

(17)

In eqs 16 and 17 ΔV⧧P

ΔΔV P⧧ = ΔV P⧧ ′ − ΔV P⧧

(11)

ΔV⧧T

where and are the reaction energies for reactions 10 and 11, respectively, i.e., the energy barriers for reactions 8 and 9, respectively. Then, reaction 8 minus reaction 9 is

(18)

and ΔΔHP = Δr HP′ − Δr HP 3281

(19)



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are the corrections to the approximate reaction enthalpy and reaction energy from low-level ab initio calculations for any target reaction RT, and they are obtained from the principal reaction RP. Equations 16 and 17 show that the reaction enthalpy and the reaction energy for a target reaction can be obtained from their approximate values at low-level ab initio methods after correction and the corrected values would have the approximate accuracy of the accurate method used for the principal reaction. Equation 16 gives an interpretation of the little dependence of the potential energy factor f V (see eq 6) on the level of ab initio theory in RC-TST theory.

(CH3)2 CH(CH3)CCH 2 + OH• → (CH3)2 C•(CH3)CCH 2 + H 2O (CH3)2 CH(CH3)CCH 2 + OH• → (CH3)2 CH(CH 2•)CCH 2 + H 2O

→ CH3CH 2(CH3CH•)CCH 2 + H 2O

(R1)

(R2) •

(CH3)2 CCH 2 + OH → CH3(CH 2 )CCH 2 + H 2O (R3)

CH3CH 2CH 2CHCH 2 + OH

•

→ CH3CH 2CH•CHCH 2 + H 2O

(R4)

Etot = EHF + E cor

(CH3)2 CHCHCH 2 + OH• → (CH3)2 C•CHCH 2 + H 2O CH3CH 2(CH3)CCH 2 + OH• → CH3CH (CH3)CCH 2 + H 2O CH3CH 2(CH3)CCH 2 + OH

HF EXHF = E∞ + Be−aX

(R6)

(R7)

cor E∞ = EXcor + A /(X + 1/2)3

CH3CH 2CH 2CH 2CHCH 2 + OH• → CH3CH 2CH 2CH•CHCH 2 + H 2O

(R9)

tot HF cor E∞ = E∞ + E∞

CH3CH 2(CH3)CHCHCH 2 + OH• •

→ CH3CH 2(CH3)C CHCH 2 + H 2O

(R10)

(R11)

CH3CH 2CH 2(CH3)CCH 2 + OH• → CH3CH 2CH 2(CH 2•)CCH 2 + H 2O

(23)

Reactions R2−R15 are the target reactions. To test the reliability of the correction scheme, a test set containing five representative reactions, reactions R3, R4, R6, R8, and R9, is used to compare the reaction enthalpies and energy barriers calculated directly from ab initio methods of various levels with the results from the correction scheme. The reason for selecting the five reactions as the representative set is that both the distribution of the size of the reaction systems from C3 to C6 and the distribution of the chain branching of the reaction systems are considered. These ab initio methods which are 16

CH3CH 2CH 2(CH3)CCH 2 + OH• → CH3CH 2CH•(CH3)CCH 2 + H 2O

(22)

where X = 2, 3 for D, T extrapolation with the basis sets ccpVXZ, and A is a constant.63 Then the basis-set limit for the total single point energy is obtained by

(R8)

(CH3)2 CHCH 2CHCH 2 + OH• → (CH3)2 CHCH•CHCH 2 + H 2O

(21)

where X = 2, 3, 4 for D, T, Q extrapolation with the basis sets cc-pVXZ, and B and a are constants.60 The correlation energy is assumed to approach its CBS limit through extrapolation formulas of the form

•

→ CH3CH 2(CH 2•)CCH 2 + H 2O

(20)

The HF energy is assumed to approach its CBS limit by power laws:

(R5)

•

(R15)

The smallest reaction R1 is chosen as the principal reaction in this reaction class in this study. Becke’s half-and-half (BHandH)54 nonlocal exchange with Lee−Yang−Parr (LYP)55 nonlocal correlation functionals has been reported to be sufficiently accurate for predicting the transition state properties;39−43,56 thus the geometries for all species and transition states involved in the above reactions are optimized using this DFT functional with the 6-31G(d,p)57,58 basis set. In addition, vibration frequency calculations are also carried out to confirm the existence of transition states with one and only one imaginary frequency. Intrinsic reaction coordinate (IRC)59 calculations are carried out to confirm that the transition states are connecting the right minima between the reactants and the products. The accurate single point energies for reactants, products, and transition state for the principal reaction R1 are calculated by the CCSD(T) method extrapolated to the complete basis set (CBS) limit with the basis sets cc-pVXZ (X = D, T, Q), i.e., CCSD(T)/CBS method, which will be the benchmarking method for the accurate calculation of energy barriers and reaction enthalpies in this study. The detailed extrapolation schemes are given below.60−64 The total single point energy is a sum of the Hartree−Fock (HF) energy EHF and the correlation energy Ecor:

CH3CH 2CHCH 2 + OH• → CH3CH•CHCH 2 + H 2O •

(R14)

(CH3CH 2)2 CCH 2 + OH•

3. COMPUTATIONAL DETAILS The hydrogen abstraction reaction class between a hydroxyl radical (OH•) and an alkene (CCC) to form a water molecule (H2O) and an alkenyl radical (CCC•) is known to be an important reaction class in combustion processes of hydrocarbon fuels, especially in the high-temperature regime. Therefore, in this paper, the reason to choose this hydrogen abstraction reaction class to study is 2-fold: to test the applicability of the correction scheme and to provide accurate kinetic parameters for the reactions in this class which are important for the modeling of the combustion of hydrocarbons. All the reactions in this class are listed as follows: CH3CHCH 2 + OH• → CH 2•CHCH 2 + H 2O

(R13)

(R12) 3282



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generalized isodesmic reaction. For example, the formation reactions of the transition states for reactions R1 and R2 are

combinations of various ab initio levels with various basis sets, are labeled as L1, L2, ..., L16. (For details of these methods, see the footnotes of Table 2.) These approximate ab initio methods can be divided into five groups of ab initio levels: Hartree− Fock (L1), DFT (L2−L11), MP2 (L12 and L13), CCSD (L14), and CCSD(T) (L15 and L16). Reaction enthalpies and energy barriers for other target reactions that are not in the test set are first calculated by the low-level ab initio method of BHandHLYP/6-31G(d,p) and then are corrected by the correction scheme according to eqs 16 and 17. All the electronic structure calculations are done using the Gaussian 0365 program.

CH3CHCH 2 + OH• → CH 2···H···OH•CHCH 2 (R16)

CH3CH 2CHCH 2 + OH• → CH3CH···H···OH•CHCH 2

(R17)

Then their difference is CH3CHCH 2 + CH3CH···H···OH•CHCH 2 → CH 2···H···OH•CHCH 2 + CH3CH 2CHCH 2

4. RESULTS AND DISCUSSION 4.1. Conservation of the Reaction Centers. The labling of atoms and bonds involved in the reaction centers are given in Figure 1, where d1, d2, and d3 are the bond lengths and A1, A2,

(R18)

the typical bonds are C−H bond, C−C bond, and CC bond, and their numbers are 13, 2, and 2, respectively, in reactants and products. The reaction centers in the transition states are the parts shown by the virtual bonds, and they are conserved in the two sides of reaction R18. Therefore, all the geometric parameters are conserved in the two sides of the reaction and this reaction can be considered as a generalized isodesmic reaction. 4.2. Energy Barriers and Reaction Enthalpies for the Principal Reaction at Various ab Initio Levels. The smallest reaction R1 in the class is chosen as the principal reaction, and the accurate reaction barrier and the accurate reaction enthalpy are calculated by the CCSD(T)/CBS method in this study, because no experimental data have been reported for this reaction. Also, energy barriers and reaction enthalpies are also calculated at other levels of ab initio theory using various basis sets for the principal reaction R1 and the difference in energy barrier ΔΔV⧧P and the difference in reaction enthalpy ΔΔHP between the accurate CCSD(T)/CBS results and the approximate ab initio results are calculated. All the results are listed in Table 2. It is shown in Table 2 that large differences may exist in energy barriers and reaction enthalpies between the approximate methods and the accurate CCSD(T)/CBS method. The maximum absolute deviations of the reaction barrier and the enthalpy are 98.37 kJ/mol and 66.92 kJ/mol, respectively, which are calculated at the HF/6-31G(d,p) level and the

Figure 1. Labeling of atoms for the reaction centers.

and A3 are the bond angles. Ra, Rb, and Rc are hydrogen or alkyl substituents. The optimized geometric parameters of the reaction centers at transition state structures for the reaction class of RcCH(Rb)CRaCH2 + OH•→ RcC•(Rb)CRaCH2 + H2O are given in Table 1. From Table 1, it can be seen that the maximum absolute difference of the bond lengths and bond angles of the reaction centers at transition states are 0.091 Å and 7.5°, respectively. Hence the bonds in the reaction centers at the transition states are very well conserved among the reactions. Therefore, for any two transition state formation reactions in the class, both the numbers of the typical bonds and the reaction centers are conserved, hence their difference can be considered as a

Table 1. Optimized Geometric Parameters of Reaction Centers at Transition Statesa

a

TS

d1/Å

A1/deg

d2/Å

A2/deg

d3/Å

A3/deg

d4/Å

TS1 TS2 TS3 TS4 TS5 TS6 TS7 TS8 TS9 TS10 TS11 TS12 TS13 TS14 TS15 MADb

1.485 1.490 1.492 1.489 1.495 1.499 1.477 1.489 1.490 1.495 1.497 1.477 1.507 1.479 1.500 0.030

109.5 107.7 110.3 107.3 106.5 108.1 108.1 107.2 107.2 106.4 108.3 108.2 106.5 111.3 108.0 4.9

1.240 1.221 1.243 1.220 1.206 1.223 1.209 1.220 1.216 1.206 1.214 1.208 1.202 1.203 1.221 0.041

171.0 176.3 171.7 177.9 174.9 170.7 170.4 177.6 177.2 173.5 177.3 170.4 175.3 176.1 171.2 7.5

1.258 1.292 1.252 1.296 1.325 1.289 1.328 1.296 1.304 1.330 1.305 1.329 1.332 1.343 1.293 0.091

98.48 97.93 98.64 97.96 97.34 98.08 97.11 97.97 97.56 97.11 97.28 97.08 96.98 96.87 97.82 1.77

0.962 0.962 0.962 0.962 0.963 0.962 0.963 0.962 0.962 0.963 0.962 0.963 0.963 0.963 0.962 0.001

Bond lengths in angstroms and bond angles in degrees. bMAD: maximum absolute difference between different reactions. 3283



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For the Hartree−Fock (L1) method, the absolute deviations for the energy barrier and the reaction enthalpy for reactions R3, R4, R6, R8, and R9 calculated directly by the HF/631G(d,p) (L1) method are 99.97, 98.98, 99.65, 99.10, and 98.30 kJ/mol, and 46.94, 46.50, 48.97, 46.55, and 47.21 kJ/mol, respectively. After correction, the values are reduced to 1.60, 0.61, 1.28, 0.73, and 0.07 kJ/mol, and 3.99, 3.55, 6.02, 3.60, and 4.26 kJ/mol, respectively. The average absolute deviations for the energy barrier and the reaction enthalpy are 99.20 kJ/mol and 47.23 kJ/mol, respectively, before correction and are reduced to 0.86 kJ/mol and 4.28 kJ/mol, respectively, after correction, showing that the correction effect for the HF method is very significant. For DFT methods (L2−L11), large differences exist in the results between different DFT methods. The average values of the absolute deviation of the energy barrier by BHandHLYP/631G(d,p) (L2), BP86/6-31G(d,p) (L3), B3LYP/6-31G(d,p) (L 4 ), B3P86/6-31G(d,p) (L 5 ), O3LYP/6-31G(d,p) (L6),TPSS/6-31G(d,p) (L7), MPW1PW91/6-311+G(d,p) (L8), B971/TZVP (L9), B3PW91/cc-pVDZ (L10), B1B95/ccpVTZ (L11) are 21.12, 46.55, 16.02, 22.66, 14.33, 29.73, 9.11, 17.34, 20.85, and 10.21 kJ/mol, respectively, before correction and are 0.99, 0.24, 0.91, 0.79, 0.84, 0.41, 1.21, 1.14, 0.96, and 1.30 kJ/mol, respectively, after correction. The average values of the absolute deviation of reaction enthalpy by BHandHLYP/ 6-31G(d,p) (L2), BP86/6-31G(d,p) (L3), B3LYP/6-31G(d,p) (L 4 ), B3P86/6-31G(d,p) (L 5 ), O3LYP/6-31G(d,p) (L6),TPSS/6-31G(d,p) (L7), MPW1PW91/6-311+G(d,p) (L 8), B971/TZVP (L9 ), B3PW91/cc-pVDZ (L 10 ), and B1B95/cc-pVTZ (L11) are 24.45, 6.85, 69.24, 1.74, 2.36, 11.07, 4.15, 1.96, 5.19, and 4.56 kJ/mol, respectively, before correction and are reduced to 1.35, 4.86, 2.45, 3.56, 4.13, 3.53, 1.88, 1.66, 3.01, and 3.04 kJ/mol, respectively, after correction. Among the directly calculated energy barriers by DFT methods, the BP86/6-31G(d,p) (L3) method has the maximum average absolute deviation of 46.55 kJ/mol and has average absolute deviations of 46.23, 46.15, 47.08, 46.51, and 46.76 kJ/mol for reactions R3, R4, R6, R8, and R9,respectively. The MPW1PW91/6-311+G(d,p) (L8) method has the minimum average value of absolute deviation of 9.11 kJ/mol and has absolute deviations of 4.0, 9.07, 8.97, 8.99, and 8.80 kJ/mol for reactions R3, R4, R6, R8, and R9, respectively. After correction, the absolute deviations of the energy barrier for reactions R3, R4, R6, R8, and R9 obtained from the BP86/6-31G(d,p) method and the MPW1PW91/6-311+G(d,p) method are reduced to 0.46, 0.10, 0.39, 0.18, and 0.06 kJ/mol, and 0.59, 1.26, 1.35, 1.33, and 1.53 kJ/mol, respectively. The DFT method with the maximum average deviation for the reaction enthalpies is B3LYP/6-31G(d,p) (L4) method, which has the average absolute deviation of 69.24 kJ/mol and has the absolute deviations of 66.57, 70.35, 68.86, 70.27, and 70.13 kJ/mol for reactions R3, R4, R6, R8, and R9, respectively. The DFT method with the minimum average deviation for reaction enthalpy is the B3P86/6-31G(d,p) (L5) method, which has the average absolute deviation of 1.74 kJ/mol and has the absolute deviations of 7.04, 0.37, 0.60, 0.31, and 0.37 kJ/mol for reactions R3, R4, R6, R8, and R9, respectively. After correction, the average absolute deviations of the reaction enthalpy from the B3LYP/6-31G(d,p) method are reduced to 2.45 kJ/mol and the absolute deviations of the reaction enthalpy for reactions R3, R4, R6, R8, and R9 calculated from the B3LYP/631G(d,p) method are reduced to 0.34, 3.43, 1.94, 3.35, and 3.21 kJ/mol, respectively. After correction, the average absolute

Table 2. Reaction Enthalpies and Energy Barriers at Various ab Initio Levels for the Principal Reaction R1 (kJ/mol) ab initio levela

ΔV⧧P

ΔΔV⧧P b

ΔrHP

ΔΔHPb

CCSD(T)/CBS L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16

14.34 112.71 34.48 −32.36 −1.66 −9.31 −0.83 −15.12 4.01 −4.15 −7.47 2.83 6.16 6.46 34.78 26.26 16.12

− −98.37 −20.14 46.70 16.00 23.65 15.17 29.46 10.33 18.89 21.81 11.51 8.18 7.88 −20.44 −11.92 −1.78

−124.55 −81.59 −100.50 −127.85 −191.47 −121.12 −121.08 −111.35 −128.17 −125.06 −117.78 −127.09 −148.37 −150.62 −99.30 −99.90 −120.30

− −42.96 −24.05 3.30 66.92 −3.43 −3.47 −13.20 3.62 0.51 −6.77 2.54 23.82 26.07 −25.25 −24.65 −4.25

a L1, HF/6-31G(d,p); L2, BHandHLYP/6-31G(d,p); L3, BP86/631G(d,p); L4, B3LYP/6-31G(d,p); L5, B3P86/6-31G(d,p); L6, O3LYP/6-31G(d,p); L7, TPSS/6-31G(d,p); L8, MPW1PW91/6311+G (d,p); L9, B971/TZVP; L10, B3PW91/cc-pVDZ; L11, B1B95/ cc-pVTZ; L12, MP2/aug-cc-pVDZ; L13, MP2/aug-cc-pVTZ; L14, CCSD/cc-pVDZ; L15, CCSD(T)/cc-pVDZ; L16, CCSD(T)/aug-ccpVDZ. bDifference between the value at the CCSD(T)/CBS level and the corresponding approximate ab initio level.

B3LYP/6-31G(d,p) level, respectively. The minimum absolute errors of reaction barrier and the enthalpy are 1.78 kJ/mol and 0.51 kJ/mol, respectively, which are calculated at the CCSD(T)/aug-cc-pVDZ and the B971/TZVP levels, respectively. For various DFT methods with various basis sets, the maximum absolute errors of the reaction barrier and the enthalpy are 46.07 kJ/mol and 66.92 kJ/mol, respectively, which are calculated at the BP86/6-31G(d,p) level and the B3LYP/631G(d,p) level, respectively, and the minimum absolute errors of the reaction barrier and the reaction enthalpy are 11.51 kJ/ mol and 0.51 kJ/mol, respectively, which are calculated at the B1B95/cc-pVTZ level and the B971/TZVP level, respectively. However, for the B971/TZVP method, the calculated reaction barrier is up to 18.89 kJ/mol. These ΔΔV⧧P values and ΔΔHP values from the principal reaction will be used to correct the approximate reaction energies and reaction enthalpies at lower levels of ab initio methods for the target reactions. 4.3. Energy Barriers and Reaction Enthalpies for Representative Target Reactions. The reaction barriers and reaction enthalpies for representative target reactions in the test set are calculated by two schemes, respectively: the first scheme is to calculate the reaction barriers and reaction enthalpies directly from the various approximate ab initio methods listed in Table 2, and the second scheme is to correct the reaction barriers and reaction enthalpies from the first scheme by the correction scheme. The calculated reaction barriers and reaction enthalpies before and after correction are given in Tables 3 and 4, respectively. To see the effect of the correction scheme on the energy barrier and the reaction enthalpy, discussions are given according to the ab initio levels: Hartree−Fock (L1), DFT (L2−L11), MP2 (L12 and L13), CCSD (L14), and CCSD(T) (L15 and L16). 3284



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Table 3. Reaction Barriers of the Representative Reactions (kJ/mol) R3 Ia

computation level CCSD(T)/CBS L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 MADe

− 15.54 14.85 14.40 11.60 14.61 14.62 14.29 14.53 14.86 14.32 14.86 14.08 14.22 13.91 13.66 14.00 2.34

(1.60)c (0.91) (0.46) (2.34) (0.67) (0.68) (0.35) (0.59) (0.92) (0.38) (0.92) (0.14) (0.28) (0.03) (0.28) (0.06)

computation level CCSD(T)/CBS L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 MADd

I − 2.67 2.89 2.12 2.53 2.88 2.90 1.69 3.27 3.40 3.14 3.31 5.10 5.72 2.05 1.38 1.00 3.78

R4 Db

(0.73)c (0.95) (0.18) (0.59) (0.94) (0.96) (0.25) (1.33) (1.46) (1.20) (1.37) (3.16) (3.78) (0.11) (0.56) (0.94)

I

13.94 113.91 (99.97) 34.98 (21.04) −32.29 (46.23) −4.40 (18.34) −9.04 (22.98) −0.54 (14.48) −15.17 (29.11) 4.20 (9.74) −3.63 (17.57) −7.49 (21.43) 3.35 (10.59) 5.91 (8.03) 6.35 (7.59) 34.35 (20.41) 25.58 (11.64) 15.79 (1.85) 99.97 R8

− 2.76 3.03 2.25 2.62 2.97 3.00 1.83 3.41 3.40 3.35 3.29 5.38 5.88 2.40 1.78 1.52 3.73

R6 D

I − 5.58 5.51 3.91 4.89 4.71 5.03 4.12 5.65 4.95 4.35 5.31 7.67 8.32 4.05 3.39 4.01 4.02

2.15 101.13 (98.98) 23.17 (21.02) −44.44 (46.15) −13.37 (15.52) −20.68 (22.83) −12.16 (14.31) −27.63 (29.78) −6.92 (9.07) −15.09 (17.24) −18.45 (20.60) −8.22 (10.37) −2.80 (4.95) −1.99 (4.14) 22.85 (20.70) 13.70 (11.55) 3.30 (1.15) 98.98 R9

(0.61) (0.88) (0.10) (0.47) (0.82) (0.85) (0.32) (1.26) (1.25) (1.20) (1.14) (3.23) (3.73) (0.25) (0.37) (0.63)

D 4.30 103.95 (99.65) 25.65 (21.35) −42.78 (47.08) −11.11 (15.41) −18.94 (23.24) −10.13 (14.43) −25.34 (29.64) −4.67 (8.97) −13.53 (17.83) −17.45 (21.75) −6.20 (10.50) −0.51 (4.81) 0.45 (3.85) 24.50 (20.20) 15.31 (11.01) 5.79 (1.49) 99.65 AADe

(1.28) (1.21) (0.39) (0.59) (0.41) (0.73) (0.18) (1.35) (0.65) (0.05) (1.01) (3.37) (4.02) (0.25) (0.91) (0.29)

D

I

D

I

D

1.94 101.04 (99.10) 23.03 (21.09) −44.57 (46.51) −13.46 (15.40) −20.77 (22.71) −12.27 (14.21) −27.76 (29.70) −7.05 (8.99) −15.09 (17.03) −18.67 (20.61) −8.19 (10.13) −3.08 (5.02) −2.15 (4.09) 22.50 (20.56) 13.30 (11.36) 2.79 (0.85) 99.10

− 0.75 (0.07) 1.81 (0.99) 0.76 (0.06) 1.39 (0.57) 1.93 (1.11) 1.79 (0.97) −0.14 (0.96) 2.35 (1.53) 2.26 (1.44) 2.79 (1.97) 2.86 (2.04) 3.47 (2.65) 4.43 (3.61) 1.30 (0.48) 0.46 (0.36) −0.54 (1.36) 3.61

0.82 99.12 (98.30) 21.94 (21.12) −45.94 (46.76) −14.60 (15.42) −20.71 (21.53) −13.38 (14.20) −29.60 (30.42) −7.98 (8.80) −16.22 (17.04) −19.02 (19.84) −8.65 (9.47) −4.71 (5.53) −3.44 (4.26) 21.74 (20.92) 12.3 (11.48) 1.24 (0.42) 98.30

0.86 0.99 0.24 0.91 0.79 0.84 0.41 1.21 1.14 0.96 1.30 2.51 3.08 0.22 0.50 0.66 3.08

99.20 21.12 46.55 16.02 22.66 14.33 29.73 9.11 17.34 20.85 10.21 5.67 4.79 20.56 11.41 1.15 99.20

a

I: results are corrected by the correction scheme. bD: results are calculated directly from this ab initio method. cValues in parentheses are the absolute values of deviation between CCSD(T)/CBS method and Ln (n = 1, 2, ..., 16) method. dMAD is the maximum of the absolute values of the deviation among the 16 methods. eAAD is the average of the absolute deviation over the five representative reactions.

the absolute deviations for the energy barrier and the reaction enthalpy are reduced to 2.51 kJ/mol and 6.66 kJ/mol, respectively, by the MP2/aug-cc-pVDZ (L12) method and are reduced to 3.08 kJ/mol and 4.68 kJ/mol, respectively, by the MP2/aug-cc-pVTZ (L13) method. The absolute deviations for the energy barrier and the reaction enthalpy of reactions R3, R4, R6, R8, and R9 calculated by MP2/aug-cc-pVDZ (L12) are 8.03, 4.95, 4.81, 5.02, and 5.53 kJ/mol, and 18.98, 17.48, 15.64, 17.37, and 16.32 kJ/mol, respectively, before correction and the corresponding values are reduced to 0.14, 3.23, 3.37, 3.16, and 2.65 kJ/mol, and 4.84, 6.34, 8.19, 6.45, and 7.50 kJ/mol, respectively, after correction. The absolute deviations for the energy barrier and the reaction enthalpy of reactions R3, R4, R6, R8, and R9 calculated by MP2/aug-cc-pVTZ (L13) are 7.59, 4.14, 3.85, 4.09, and 4.26 kJ/mol, and 21.85, 21.71, 20.48, 21.70, and 21.24 kJ/mol, respectively, before correction, and the corresponding values are reduced to 0.28, 3.73, 4.02, 3.78,

deviation of the reaction enthalpy from B3P86/6-31G(d,p) is slightly increased to 3.56 kJ/mol. Overall, before correction, some of the DFT methods can give energy barriers or reaction enthalpies very close to the CCSD(T)/CBS and some of the DFT methods can give energy barriers or reaction enthalpies that have large deviations from the CCSD(T)/CBS results for reactions in this study, and after correction, all the DFT methods can give energy barriers and reaction enthalpies very close to the CCSD(T)/CBS results. For the MP2 methods (L12 and L13), the accuracy of calculations for the energy barriers is much better than the calculations for the reaction enthalpies. Before correction, the average values of the absolute deviations for the energy barrier and the reaction enthalpy are 5.67 kJ/mol and 17.16 kJ/mol, respectively, by the MP2/aug-cc-pVDZ (L12) method and are 4.79 kJ/mol and 21.40 kJ/mol, respectively, by the MP2/augcc-pVTZ (L13) method. After correction, the average values of 3285



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Table 4. Reaction Enthalpies for Reactions in the Test Set (kJ/mol) R3 computation level CCSD(T)/CBS L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 MADd computation level CCSD(T)/CBS L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 MADd

Ia − −116.87 −117.10 −117.41 −120.52 −117.25 −117.37 −117.36 −117.48 −117.60 −117.27 −117.70 −116.02 −116.63 −117.61 −117.60 −117.58 4.84

R4 Db

(3.99)c (3.76) (3.45) (0.34) (3.61) (3.49) (3.50) (3.38) (3.26) (3.59) (3.16) (4.84) (4.23) (3.25) (3.26) (3.28)

I − −134.44 −138.92 −143.56 −141.39 −141.78 −142.94 −141.98 −139.98 −139.68 −141.11 −140.79 −131.59 −133.66 −134.65 −134.85 −132.71 6.45

−120.86 −73.92 (46.94) −93.05 (27.81) −120.71 (0.15) −187.43 (66.57) −113.82 (7.04) −113.89 (6.97) −104.16 (16.70) −121.09 (0.23) −118.11 (2.75) −110.50 (10.36) −120.24 (0.62) −139.84 (18.98) −142.71 (21.85) −92.36 (28.50) −92.95 (27.91) −113.33 (7.53) 66.57 R8

− −134.00 −138.39 −143.08 −140.98 −141.35 −142.54 −141.51 −139.16 −139.20 −140.68 −140.48 −131.21 −133.19 −134.27 −134.44 −132.29 6.34

D −137.55 −91.05 (46.50) −114.34 (23.21) −146.39 (8.84) −207.90 (70.35) −137.92 (0.37) −139.06 (1.51) −128.31 (9.24) −142.78 (5.23) −139.71 (2.16) −133.91 (3.64) −143.02 (5.47) −155.03 (17.48) −159.26 (21.71) −109.02 (28.53) −109.78 (27.77) −128.04 (9.51) 70.35 R9

(3.55) (0.84) (5.53) (3.43) (3.80) (4.99) (3.96) (1.61) (1.65) (3.13) (2.93) (6.34) (4.36) (3.28) (3.11) (5.26)

D (3.60)c (0.88) (5.52) (3.35) (3.74) (4.90) (3.94) (1.94) (1.64) (3.07) (2.75) (6.45) (4.38) (3.39) (3.19) (5.33)

R6

I

I

−138.04 −91.49 (46.55) −114.87 (23.17) −146.87 (8.83) −208.31 (70.27) −138.35 (0.31) −139.47 (1.43) −128.77 (9.27) −143.59 (5.55) −140.19 (2.15) −134.33 (3.71) −143.33 (5.29) −155.41 (17.37) −159.74 (21.70) −109.40 (28.64) −110.19 (27.85) −128.46 (9.58) 70.27

− −131.77 −136.68 −141.53 −139.24 −139.83 −140.60 −140.03 −137.87 −137.30 −139.12 −139.41 −128.53 −131.20 −132.25 −132.30 −130.11 7.50

(4.26) (0.65) (5.50) (3.21) (3.80) (4.57) (4.00) (1.84) (1.27) (3.09) (3.38) (7.50) (4.83) (3.78) (3.73) (5.92)

I − −126.52 −131.92 −136.85 −134.48 −135.37 −135.22 −134.81 −133.19 −133.00 −134.73 −135.50 −124.35 −126.94 −128.23 −128.36 −126.45 8.19

D −132.54 −83.57 (48.97) −107.86 (24.68) −140.15 (7.61) −201.40 (68.86) −131.94 (0.60) −131.75 (0.79) −121.61 (10.93) −136.81 (4.27) −133.51 (0.97) −127.96 (4.58) −138.04 (5.50) −148.18 (15.64) −153.02 (20.48) −102.99 (29.55) −103.70 (28.84) −122.20 (10.34) 68.86 AADe

(6.02) (0.62) (4.31) (1.94) (2.83) (2.68) (2.27) (0.65) (0.46) (2.19) (2.96) (8.19) (5.60) (4.31) (4.18) (6.09)

D

I

D

−136.03 −88.82 (47.21) −112.63 (23.40) −144.84 (8.81) −206.16 (70.13) −136.40 (0.37) −137.13 (1.10) −126.83 (9.20) −141.49 (5.46) −137.81 (1.78) −132.35 (3.68) −141.94 (5.91) −152.35 (16.32) −157.27 (21.24) −107.00 (29.03) −107.65 (28.38) −125.86 (10.17) 70.13

4.28 1.35 4.86 2.45 3.56 4.13 3.53 1.88 1.66 3.01 3.04 6.66 4.68 3.60 3.49 5.18 6.66

47.23 24.45 6.85 69.24 1.74 2.36 11.07 4.15 1.96 5.19 4.56 17.16 21.40 28.85 28.15 9.43 69.24

a

I: results are corrected by the correction scheme. bD: results are calculated directly from this ab initio method. cValues in parentheses are the absolute values of deviation between CCSD(T)/CBS method and Ln (n = 1, 2, ..., 16) method. dMAD is the maximum of the absolute values of the deviation among the 16 methods. eAAD is the average of the absolute deviation over the five representative reactions.

and reaction enthalpies by the CCSD(T)/aug-cc-pVDZ (L16) method are more accurate than the results by the CCSD/ccpVDZ (L14) method and the CCSD(T)/cc-pVDZ (L15) method, showing that the accuracies of the results calculated directly by these methods are in accordance with their complexity. After correction, the average values of the absolute deviation for the energy barrier by L14, L15, and L16 methods are reduced to 0.22, 0.50, and 0.66 kJ/mol, respectively, and the average values of the absolute deviation for the reaction enthalpy by L14, L15, and L16 methods are reduced to 3.60, 3.49, and 1.18 kJ/mol, respectively, indicating that after correction the energy barriers and reaction enthalpies by the three approximate CC methods are very close to those by the benchmarking CCSD(T)/CBS method. Overall, the maximum values of MAD for the energy barrier and the reaction enthalpy obtained from the direct ab initio calculation methods are 99.97 kJ/mol and 70.35 kJ/mol,

and 3.61 kJ/mol, and 4.23, 4.36, 5.60, 4.38, and 4.83 kJ/mol, respectively, after correction. It can be seen that the effect of the correction on the energy barriers, especially on the reaction enthalpies, by MP2 methods is significant. The coupled-cluster (CC) methods used in this study include three approximate methods, CCSD/cc-pVDZ (L14), CCSD(T)/cc-pVDZ (L15), and CCSD(T)/aug-cc-pVDZ (L16), and the benchmarking CCSD(T)/CBS method. The average values of the absolute deviation for the energy barrier calculated directly by L14, L15, and L16 methods are 20.56, 11.41, and 1.15 kJ/mol, respectively, and the average values of the absolute deviation for the reaction enthalpy calculated directly by L14, L15, and L16 methods are 28.85, 28.15, and 9.43 kJ/mol, respectively. These results indicate that, for all the three approximate CC methods, the accuracies of the directly calculated energy barriers are much better than those of the directly calculated reaction enthalpies, and the energy barriers 3286



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equation, eq 24, for reactions in the training set and reactions in the test set are given in Tables 6 and 7, respectively.

respectively (see Tables 3 and 4, respectively). After correction, the maximum values of MAD for the energy barrier and the reaction enthalpy are reduced to 4.02 kJ/mol and 8.19 kJ/mol, respectively. In summary, the accuracy of the energy barriers and reaction enthalpies for the reaction in this study depends heavily on the level of ab initio method and the size of the basis set before correction and all the ab initio methods with various basis sets after correction by the correction scheme can give the energy barriers and reaction enthalpies for the reaction in this study close to the CCSD(T)/CBS results within the chemical accuracy of 2 kcal/mol. 4.4. Energy Barriers and Reaction Enthalpies for All Target Reactions. In this paper the reaction barriers and reaction enthalpies for all the target reactions are calculated approximately at the modest-level level ab initio method BHandHLYP/6-31G(d,p) and then are corrected by the correction scheme. The results are given in Table 5.

Table 6. Reaction Barriers Computed by the Correction Scheme and by the LER Equation (eq 24) for Reactions in the Training Set (in kJ/mol)

ΔV⧧′b/ (kJ/mol)

ΔV⧧a/ (kJ/mol)

ΔrH′b/ (kJ/mol)

ΔrHa/ (kJ/mol)

R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15

5.09 14.85 3.03 −2.02 5.51 1.39 2.89 1.81 −4.35 1.79 1.21 −3.55 −0.11 5.05

25.23 34.98 23.17 18.12 25.65 21.53 23.03 21.94 15.79 21.93 21.35 16.59 20.02 25.19

−140.38 −117.10 −138.39 −147.76 −131.92 −116.56 −138.92 −136.68 −147.55 −129.70 −115.91 −137.84 −120.95 −129.44

−116.32 −93.05 −114.34 −123.71 −107.86 −92.50 −114.87 −112.63 −123.50 −105.65 −91.85 −113.78 −96.89 −105.39

ΔV⧧IRMb

ΔV⧧LERc

ADd

R2 R5 R7 R10 R11 R12 R13 R14 R15 AADe

−117.67 −125.64 −93.04 −125.08 −106.94 −92.53 −115.96 −97.32 −107.09

5.09 −2.02 1.39 −4.35 1.79 1.21 −3.55 −0.11 5.05

−0.38 −1.21 2.18 −1.15 0.73 2.23 −0.21 1.73 0.72

5.47 0.81 0.79 3.20 1.06 1.02 3.34 1.84 4.33 2.43

Reaction energies calculated at BHandHLYP/6-31G(d,p). bReaction barriers calculated by the correction scheme. cReaction barriers calculated by LER equation, eq 24. dAD: absolute difference of energy barriers between the correction scheme and LER method. eAAD: average of AD.

Table 7. Reaction Barriers Computed by the Correction Scheme and the LER Method (eq 24) for Reactions in the Test Set (in kJ/mol) reaction

CCSD(T)/CBS

IRMa

LERb

ADLERc

ADIRMd

R3 R4 R6 R8 R9 AADe

13.94 2.15 4.30 1.94 0.82

14.85 3.03 5.51 2.89 1.81

2.04 −0.15 0.46 −0.20 0.05

11.90 2.30 3.84 2.14 0.87 4.21

0.91 0.88 1.21 0.95 0.99 0.99

a Calculated by the correction scheme. bLER method. cADLER: absolute difference between CCSD(T)/CBS and LER method. d ADIRM: absolute difference between CCSD(T)/CBS and correction scheme. eAAD: average of AD.

a Calculated directly with BHandHLYP/6-31G(d,p) level. bCalculated at BHandHLYP/6-31G(d,p) level and corrected by the correction scheme.

The correlation coefficient (R) and standard deviation (SD) of the fitted LER equation, eq 24, are about 0.40 and 3.32 kJ/ mol, respectively, and the average absolute difference of the energy barrier between the correction scheme and the LER method for reactions in the training set is 2.43 kJ/mol (see Table 6), indicating that the linear relationship between the energy barrier and the reaction energy is not very strong. As shown in Table 7, the average absolute difference of the energy barrier between the LER method and CCSD(T)/CBS for reactions in the test set is 4.21 kJ/mol, which is much larger than the average absolute difference of the energy barrier of 0.99 kJ/mol between the correction scheme and the CCSD(T)/CBS method. The maximum absolute difference of the energy barrier between the LER method and CCSD(T)/CBS for reactions in the test set is 11.90 kJ/mol for reaction R3, which is also much larger than the maximun absolute difference of the energy barrier of 1.21 kJ/mol between the correction scheme and the CCSD(T)/CBS method for reaction R6. These results indicates that the reaction barrier predicated by the correction scheme is much accurate than the LER method. 4.6. Rate Coefficients. In this paper, the high-pressurelimit rate coefficients for the studied reactions are calculated by conventional transition state theory using TheRate46 code and

4.5. Linear Energy Relationship. In the automatic generation of detailed mechanisms for the combustion of hydrocarbon fuels, the linear energy relationship (LER)66,67 method, which states that a linear relationship exists between the energy barrier and the reaction energy, is one of the approximated methods to roughly estimate the kinetic parameters for reactions in one class. In this study a LER for the energy barrier is also built. Nine reactions, R2, R5, R7, R10, R11, R12, R13, R14, and R15, are selected as the training set to build the LER, and the remaining reactions in the class, R3, R4, R6, R8, and R9, are used as the test set to test the applicability of the LER for the prediction of the energy barriers for the reactions in the class. The LER build for this reaction class is ΔV ⧧ = 11.854 + 0.104ΔE

ΔEa

a

Table 5. Reaction Barriers and Reaction Enthalpies reaction

reaction

(24)

⧧

where ΔE and ΔV are the reaction energy and energy barrier, respectively. In obtaining the above LER equation by leastsquares fitting from the training set, ΔE is the reaction energy calculated directly at the BHandHLYP/6-31G(d,p) level andΔV⧧ is the energy barrier calculated by the correction scheme. The reaction barriers calculated by the correction scheme and the reaction barriers calculated by the LER 3287



Article

Table 8. Comparison of Rate Coefficients by RC-TST and the CCSD(T)/CBS Method R3 kIRMa

T/K 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 maxc avgd

6.49 8.70 1.19 1.63 2.19 2.89 3.74 4.75 5.94 7.33 8.92 1.07 1.28 1.50 1.76 2.04

T/K

× × × × × × × × × × × × × × × ×

−13

8.07 1.04 1.40 1.87 2.47 3.22 4.13 5.20 6.46 7.92 9.59 1.15 1.36 1.60 1.86 2.15

10 10−13 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11

KIRM

500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 maxc avgd

R4

kCCSDb

6.43 6.10 6.42 7.10 8.06 9.25 1.07 1.23 1.42 1.63 1.87 2.12 2.41 2.72 3.05 3.42

× × × × × × × × × × × × × × × ×

a

× × × × × × × × × × × × × × × ×

KIRMa

f −13

10 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11

1.24 1.20 1.18 1.15 1.13 1.11 1.10 1.09 1.09 1.08 1.08 1.07 1.06 1.07 1.06 1.05 1.24 1.11 R8

6.52 6.20 6.54 7.25 8.24 9.48 1.10 1.24 1.46 1.68 1.92 2.19 2.48 2.80 3.15 3.53

× × × × × × × × × × × × × × × ×

−12

10 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11

10 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11

8.08 7.38 7.55 8.19 9.15 1.04 1.18 1.36 1.55 1.77 2.01 2.28 2.58 2.90 3.24 3.62

× × × × × × × × × × × × × × × ×

8.06 7.40 7.61 8.28 9.27 1.05 1.21 1.36 1.58 1.81 2.06 2.34 2.64 2.97 3.33 3.72

× × × × × × × × × × × × × × × ×

KIRMa

f

−12

10 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11

1.24 1.19 1.16 1.14 1.13 1.11 1.10 1.10 1.08 1.08 1.07 1.07 1.06 1.06 1.06 1.05 1.24 1.11

2.95 3.02 3.36 3.89 4.58 5.42 6.41 7.55 8.86 1.03 1.20 1.38 1.58 1.80 2.04 2.30

× × × × × × × × × × × × × × × ×

KCCSDb −12

10 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11

3.94 3.84 4.13 4.66 5.38 6.26 7.31 8.52 9.90 1.14 1.32 1.51 1.72 1.95 2.20 2.47

× × × × × × × × × × × × × × × ×

f

10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11

1.34 1.27 1.23 1.20 1.17 1.15 1.14 1.13 1.12 1.11 1.10 1.09 1.09 1.08 1.08 1.07 1.34 1.15

R9

KCCSDb −12

R6

KCCSDb

KIRMa

f

−12

10 10−12 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11

1.26 1.21 1.18 1.15 1.14 1.12 1.10 1.11 1.09 1.09 1.07 1.08 1.07 1.07 1.06 1.06 1.26 1.12

9.56 8.85 9.13 9.94 1.11 1.27 1.45 1.66 1.90 2.17 2.48 2.81 3.17 3.57 4.00 4.46

× × × × × × × × × × × × × × × ×

KCCSDb −12

10 10−12 10−12 10−12 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11

1.21 1.08 1.08 1.15 1.27 1.43 1.62 1.83 2.09 2.37 2.68 3.02 3.40 3.81 4.25 4.73

× × × × × × × × × × × × × × × ×

10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11 10−11

f 1.27 1.22 1.18 1.16 1.14 1.13 1.12 1.10 1.10 1.09 1.08 1.07 1.07 1.07 1.06 1.06 1.27 1.12

a

By the correction scheme. bBy the CCSD(T)/CBS method. cMax: maximum value of f over the temperature range. dAvg: average value of f over the temperature range.

the Eckart method68 is also employed to correct the quantum mechanical tunneling effect. According to this theory, the rate coefficient for the reaction kn

A+B→P

(25)

k = κ(kBT /h)(Q ⧧/Q AQ B) exp(−ΔV ⧧/RT )

(26)

rotational, and translational contributions are contained. Therefore, the ab initio level for the single point energy calculation only affects the reaction barrier ΔV⧧ in eq 26 if the effect of the transmission coefficient because of the different levels of the ab initio method is not taken into account. According to the correction scheme introduced earlier, if the reaction barriers by direct calculation method and by the correction scheme at the same ab initio level for a target reaction are ΔV⧧ and ΔV⧧′, respectively, and the corresponding rate coefficients are k and k′, respectively, then we have

is

where κ is the transmission coefficient accounting for the quantum mechanical tunneling effects; Q⧧, QA, and QB are the total partition functions (per unit volume) of the transition state, reactant A, and reactant B, respectively; ΔV⧧ is the difference value of electron energies between the transition state and the reactant, i.e., the energy barrier; T is the temperature; and kB and h are the Boltzmann and Planck constants, respectively. The partition functions in eq 26 do not contain the electronic contributions; only the vibrational,

k′/k = exp[(ΔV ⧧ − ΔV ⧧′)/RT ] = exp[( −ΔΔV ⧧)/RT ] (27)

Namely k′ = k exp[( −ΔΔV ⧧)/RT ]

(28)

where 3288


The Journal of Physical Chemistry A ΔΔV ⧧ = ΔV ⧧′ − ΔV ⧧

Article

Table 9. Kinetic Parameters (A, n, and E)a

(29)

is the correction of the energy barrier defined in eq 18. According to eq 28, an accurate rate coefficient for a target reaction can be obtained from a low-level ab initio calculation by multiplying a factor of exp[(−ΔΔV⧧)/RT]. The kinetic parameters for combustion modeling using software, such as Chemkin, are usually in the form of a modified Arrhenius equation: k = AT n exp( −E /RT )

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15

(30)

Hence, in this work, the kinetic parameters for the studied reactions are given in the form of A, n, and E, which are obtained by fitting to the calculated k at various temperatures through the least-squares method. 4.7. Comparison of the Rate Coefficients by RC-TST Theory and by the CCSD(T)/CBS Method. Because of the limitation of the experimental kinetic parameters for the reactions in this study, a comparison of the rate coefficients (kIRM) calculated by the new reaction method at the BHandHLYP/6-31G(d,p) level with the rate coefficients (kCCSD) calculated at the CCSD(T)/CBS level for the five target reactions R3, R4, R6, R8, and R9 are given to test the correction scheme. For convenience of comparison, a ratio factor is defined as

f = k max /k min

A

reaction 1.08 3.72 5.04 5.90 3.37 1.88 6.99 6.25 1.14 2.88 1.00 7.51 2.52 3.82 1.58

× × × × × × × × × × × × × × ×

n 10−24 10−23 10−25 10−23 10−22 10−23 10−23 10−23 10−22 10−22 10−22 10−23 10−22 10−23 10−23

4.00 3.51 4.03 3.43 3.21 3.54 3.37 3.42 3.38 3.14 3.37 3.37 3.16 3.30 3.51

E −1.24 −1.60 −1.21 −1.71 −1.91 −1.59 −1.80 −1.71 −1.75 −2.05 −1.77 −1.80 −2.00 −1.85 −1.60

× × × × × × × × × × × × × × ×

104 104 104 104 104 104 104 104 104 104 104 104 104 104 104

a

Kinetic parameters are reported in units consistent with J, s, mol, and cm.

barriers and reaction enthalpies for reactions in a class. The interpretation extends the isodesmic reaction method to the calculation of reaction barriers. We have calculated the reaction energies, energy barriers, and rate coefficient at various temperatures for reactions in a class: RcCH(Rb)CRaCH2 + OH• → RcC•(Rb)CRaCH2 + H2O. A comparison of these calculated values for five representative reactions in the class by direct ab initio methods with the results by the RC-TST method shows that the reaction energies, energy barriers, and rate coefficients are very sensitive to the level of the ab initio method and the choice of the basis set for the direct ab initio methods and they are not sensitive to the level of the ab initio method and the choice of the basis set for the new method; therefore, reaction energies, energy barriers, and rate coefficients for reactions in a class can be calculated accurately at a relatively low level of ab initio method using the new method. We have calculated the reaction energies, energy barriers, and rate coefficients for 15 reactions in this class using the new scheme at the BHandHLYP/6-31G(d,p) level. The results of this study indicate that RC-TST method is a very effective method for the accurate calculation of reaction energies,, energy barriers, and rate coefficients for reactions of large molecules.

(31)

where kmax and kmin are the maximum value and minimum value, respectively, between kIRM and kCCSD. The rate coefficients for these target reactions, their ratio factors, and the maximum values and the average values of the ratio factors over the temperature range 500−2000 K are listed in Table 8. It can be seen in Table 8 that the average values of f over the temperature range 500−2000 K are about 1.11, 1.11, 1.15, 1.12, and 1.12, respectively, for reactions R3, R4, R6, R8, and R9 and the maximum values of f over the temperature range 500−2000 K are 1.24, 1.24, 1.34, 1.26, and 1.27, respectively, for reactions R3, R4, R6, R8, and R9, indicating that the calculated rate coefficients of the five target reactions under the correction scheme at the BHandHLYP/6-31G(d,p) level are very close to results calculated directly at the CCSD(T)/CBS level. Therefore, accurate rate coefficients for the studied reactions can be obtained from the correction scheme at the BHandHLYP/631G(d,p) level. 4.8. Kinetic Parameters (A, n, E) for the Studied Reactions. For the principal reaction R1, the accurate rate coefficients at various temperatures are calculated from the single point energy at the CCSD(T)/CBS level. For the target reactions, the approximate rate coefficients k at various temperatures are calculated from the single point energy at the BHandHLYP/6-31G(d,p) level, and then the accurate rate coefficients k′ are obtained by correction of k through eq 28, where the correction of the energy barrier calculated as the difference between the CCSD(T)/CBS energy barrier and the BHandHLYP/6-31G(d,p) energy barrier for the principal reaction is −20.14 kJ/mol, and the approximate ΔV⧧ values are the values given in Table 5. The fitted Arrhenius parameters A, n, and E for all reactions in this study are given in Table 9.

■

ASSOCIATED CONTENT

S Supporting Information *

Detailed basis set extrapolation scheme, optimized Cartesian coordinates of reactants, transition states and products, and rate coefficients of all the target reactions calculated by BHandHLYP/6-31G(d,p) in Table S1. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

■

5. CONCLUSIONS We have given an interpretation of the RC-TST theory for the calculation of rate coefficients for reactions in a class and extended the theory to the accurate calculation of the reaction

ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (20973118 and 91016002). 3289



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III. Additional Improvements and Overview. J. Chem. Phys. 2008, 128, 114111. (22) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Multi-Coefficient Extrapolated Density Functional Theory for Thermochemistry and Thermochemical Kinetics. Phys. Chem. Chem. Phys. 2005, 7, 43. (23) Lynch, B. J.; Truhlar, D. G. Robust and Affordable Multicoefficient Methods for Thermochemistry and Thermochemical Kinetics: The MCCM/3 Suite and SAC/3. J. Phys. Chem. A 2003, 107, 3898. (24) Fast, P. L.; Truhlar, D. G. MC-QCISD: Multi-Coefficient Correlation Method Based on Quadratic Configuration Interaction with Single and Double Excitations. J. Phys. Chem. A 2000, 104, 6111. (25) Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron Affinities of the First-Row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96, 6976. (26) Harding, M. E.; Metzroth, T.; Gauss, J.; Auer, A. A. Parallel Calculation of CCSD and CCSD(T) Analytic First and Second Derivatives. J. Chem. Theory Comput. 2008, 4, 64. (27) Pollack, L.; Windus, T. L.; de Jong, W. A.; Dixon, D. A. Thermodynamic Properties of the C5, C6, and C8 n-Alkanes from ab Initio Electronic Structure Theory. J. Phys. Chem. A 2005, 109, 6934− 6938. (28) Vanden-Eijnden, E.; Tal, F. A. Transition State Theory: Variational Formulation, Dynamical Corrections, and Error Estimates. J. Chem. Phys. 2005, 123, 184103. (29) Duncan, W. T.; Bell, R. L.; Truong, T. N. TheRate: Program for ab Initio Direct Dynamics Calculations of Thermal and VibrationalState-Selected Rate Constants. J. Comput. Chem. 1998, 19, 1039− 1052. (30) Silvaa, P. J.; Ramos, M. Successes and Failures of DFT functionals in Acid/Base and Redox Reactions of Organic and Biochemical Interest. J. Comput. Theor. Chem. 2011, 966, 120−126. (31) Aguilera-Iparraguirre, J.; Curran, H. J.; Klopper, W.; Simmie, J. M. Accurate Benchmark Calculation of the Reaction Barrier Height for Hydrogen Abstraction by the Hydroperoxyl Radical from Methane. Implications for CnH2n+2 Where n = 2 → 4. J. Phys. Chem. A 2008, 112, 7047−7054. (32) Flener-Lovitt, C.; Woon, D. E.; Dunning, T. M.; Girolami, G. S. A DFT and ab Initio Benchmarking Study of Metal-Alkane Interactions and the Activation of Carbon-Hydrogen Bonds. J. Phys. Chem. A 2010, 114, 1843. (33) Csonka, G.; French, A. D.; Johnson, G. P.; Stortz, C. A. Evaluation of Density Functionals and Basis Sets for Carbohydrates. J. Chem. Theory Comput. 2009, 5, 679. (34) Wheeler, S. E.; Moran, A.; Pieniazek, S. N.; Houk, K. N. Accurate Reaction Enthalpies and Sources of Error in DFT Thermochemistry for Aldol, Mannich, and α-Aminoxylation Reactions. J. Phys. Chem. A 2009, 113, 10376. (35) Korth, M.; Grimme, S. “Mindless” DFT Benchmarking. J. Chem. Theory Comput. 2009, 5, 993. (36) Quintal, M. M.; Karton, A.; Iron, M. A.; Boese, A. D.; Martin, J. M. L. Benchmark Study of DFT Functionals for Late-Transition-Metal Reactions. J. Phys. Chem. A 2006, 110, 709−716. (37) Steinmann, S. N.; Corminboeuf, C. Comprehensive Benchmarking of a Density-Dependent Dispersion Correction. J. Chem. Theory Comput. 2011, 7, 3567−3577. (38) Truong, T. N. Reaction Class Transition State Theory: Hydrogen Abstraction Reactions by Hydrogen Atoms as Test Cases. J. Chem. Phys. 2000, 113, 4957−4964. (39) Huynh, L. K.; Ratkiewicz, A.; Truong, T. N. Kinetics of the Hydrogen Abstraction OH + Alkane → H2O + Alkyl Reaction Class: An Application of the Reaction Class Transition State Theory. J. Phys. Chem. A 2006, 110, 473−484. (40) Muszynska, M.; Ratkiewicz, A.; Huynh, L. K.; Truong, T. N. Kinetics of the Hydrogen Abstraction C2H3• + Alkane → C2H4 + Alkyl Radical Reaction Class. J. Phys. Chem. A 2009, 113, 8327−8336. (41) Bankiewicz, B.; Huynh, L. K.; Ratkiewicz, A. Kinetics of 1,4Hydrogen Migration in the Alkyl Radical Reaction Class. J. Phys. Chem. A 2009, 113, 1564−1573.

REFERENCES

(1) Simmie, J. M. Detailed Chemical Kinetic Models for the Combustion of Hydrocarbon Fuels. Prog. Energy Combust. Sci. 2003, 29, 599−634. (2) Pepiot-Desjardins, P.; Pitsch, H. An Efficient Error Propagation Based Reduction Method for Large Chemical Kinetic Mechanisms. Combust. Flame 2008, 154, 67−81. (3) Glaude, P. A.; Warth, V.; Fournet, R.; Battin-Leclerc, F.; Scacchi, G.; Come, G. M. Modeling of the Oxidation of n-Ocatane and nDecane Using an Automatic Generation of Mechanisms. Int. J. Chem. Kinet. 1998, 30, 949. (4) Fournet, R.; Battin-Leclerc, F.; Glaude, P. A.; Judenherc, B.; Warth, V.; Come, G. M.; Scacchi, G.; Ristori, A.; Pengloan, G.; Dagaut, P.; Cathonnet, M. The Gas-Phase Oxidation of n-Hexadecane. Int. J. Chem. Kinet. 2001, 33, 574. (5) Truong, T. N.; Duncan, W. T.; Tirtowidjojo, M. A Reaction Class Approach for Modeling Gas Phase Reaction Rates. Phys. Chem. Chem. Phys. 1999, 1, 1061−1065. (6) Cohen, N.; Benson, S. W. Estimation of Heats of Formation of Organic Compounds by Additivity Methods. Chem. Rev. 1993, 93, 2419. (7) Curtiss, L. A.; Raghavachari, K.; Pople, J. A. Gaussian-2 Theory Using Reduced Møller−Plesset Orders. J. Chem. Phys. 1993, 98, 1293. (8) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Pople, J. A. Assessment of Gaussian-2 and Density Functional Theories for the Computation of Enthalpies of Formation. J. Chem. Phys. 1997, 106, 1063. (9) Redfern, P. C.; Zapol, P.; Curtiss, L. A.; Raghavachari, K. Assessment of Gaussian-3 and Density Functional Theories for Enthalpies of Formation of C1−C16 Alkanes. J. Phys. Chem. A 2000, 104, 5850. (10) Curtiss, L. A.; Raghavachari, K.; Redfern, P. C.; Pople, J. A. Assessment of Gaussian-3 and Density Functional Theories for a Larger Experimental Test Set. J. Chem. Phys. 2000, 112, 7374. (11) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K.; Pople, J. A. Gaussian-3X (G3X) Theory: Use of Improved Geometries, Zero-Point Energies, and Hartree−Fock Basis Sets. J. Chem. Phys. 2001, 114, 108. (12) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K.; Pople, J. A. Gaussian-3X (G3X) Theory Using Coupled Cluster and Brueckner Energies. Chem. Phys. Lett. 2002, 359, 390. (13) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Assessment of Gaussian-3 and Density-Functional Theories on the G3/05 Test Set of Experimental Energies. J. Chem. Phys. 2005, 123, 124107. (14) Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-4 Theory. J. Chem. Phys. 2007, 126, 084108. (15) Martin, J. M. L.; de Oliveira, G. Towards Standard Methods for Benchmark Quality ab Initio ThermochemistryW1 and W2 Theory. J. Chem. Phys. 1999, 111, 1843. (16) Parthiban, S.; Martin, J. M. L. Assessment of W1 and W2 Theories for the Computation of Electron Affinities, Ionization Potentials, Heats of Formation, and Proton Affinities. J. Chem. Phys. 2001, 114, 6014. (17) Boese, A. D.; Oren, M.; Atasoylu, O.; Martin, J. M. L.; Kállay, M.; Gauss, J. W3 theory: Robust Computational Thermochemistry in the kJ/mol Accuracy Range. J. Chem. Phys. 2004, 120, 4129. (18) Karton, A.; Rabinovich, E.; Martin, J. M. L.; Ruscic, B. W4 Theory for Computational Thermochemistry: In Pursuit of Confident Sub-kJ/mol Predictions. J. Chem. Phys. 2006, 125, 144108. (19) Tajti, A.; Szalay, P. G.; Császár, A. G.; Kállay, M.; Gauss, J.; Valeev, E. F.; Flowers, B. A.; Vázquez, J.; Stanton, J. F. HEAT: High Accuracy Extrapolated ab Initio Thermochemistry. J. Chem. Phys. 2004, 121, 11599. (20) Bomble, Y. J.; Vázquez, J.; Kállay, M.; Michauk, C.; Szalay, P. G.; Császár, A. G.; Gauss, J.; Stanton, J. F. High-Accuracy Extrapolated ab Initio Thermochemistry. II. Minor Improvements to the Protocol and a Vital Simplification. J. Chem. Phys. 2006, 125, 064108. (21) Harding, M. E.; Vázquez, J.; Ruscic, B.; Wilson, A. K.; Gauss, J.; Stanton, J. F. High-Accuracy Extrapolated ab Initio Thermochemistry. 3290



Article

(63) Huh, S. B.; Lee, J. S. Basis Set and Correlation Dependent Extrapolation of Correlation Energy. J. Chem. Phys. 2003, 118, 3035− 3042. (64) Hwang, R.; Park, Y. C.; Lee, J. S. Binding Energies of HydrogenBonded Clusters from Extrapolation-Oriented Basis Sets. Theor. Chem. Acc. 2006, 115, 54−58. (65) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; et al.. Gaussian 03, revision B.01 [CP]; Gaussian, Inc.: Pittsburgh, PA, 2003. (66) Evans, M. G.; Polanyi, M. Further Considerations on the Thermodynamics of Chemical Equilibria and Reaction Rates. Trans. Faraday Soc. 1936, 32, 1333−1360. (67) Polanyi, J. C. Concepts in Reaction Dynamics. Acc. Chem. Res. 1972, 5, 161−168. (68) Eckart, C. The Penetration of a Potential Barrier by Electrons. Phys. Rev. 1930, 35, 1303−1309.

(42) Zhang, S. W.; Truong, T. N. Kinetics of Hydrogen Abstraction Reaction Class H + H−C(sp3): First-Principles Predictions Using the Reaction Class Transition State Theory. J. Phys. Chem. A 2003, 107, 1138−1147. (43) Kungwan, N.; Thanh, T. N. Kinetics of the Hydrogen Abstraction •CH3 + Alkane → CH4 + Alkyl Reaction Class: An Application of the Reaction Class Transition State Theory. J. Phys. Chem. A 2005, 109, 7742−7750. (44) Huynh, L. K.; Truong, T. N. Kinetics of the Hydrogen Abstraction CHO + Alkane → HCHO + Alkyl Reaction Class: An Application of the Reaction Class Transition State Theory. Theor. Chem. Acc. 2008, 120, 107−118. (45) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (46) Duncan, W. T.; Bell, R. L.; Truong, T. N. TheRate: Program for ab Initio Direct Dynamics Calculations of Thermal and VibrationalState-Selected Rate Constants. J. Comput. Chem. 1998, 19, 1039− 1052. (47) Baboul, A. G.; Curtiss, L. A.; Redfern, P. C.; Raghavachari, K. Gaussian-3 Theory Using Density Functional Geometries and ZeroPoint Energies. J. Chem. Phys. A 1999, 110, 7650−7657. (48) Khan, S. S.; Yu, X. R.; Wade, J. R.; Malmgren, R. D.; Broadbelt, L. J. Thermochemistry of Radicals and Molecules Relevant to Atmospheric Chemistry: Determination of Group Additivity Values Using G3//B3LYP Theory. J. Phys. Chem. A 2009, 113, 5176−5194. (49) Hehre, W. J.; Ditchfield, R.; Radom, L.; Pople, J. A. Molecular Orbital Theory of the Electronic Structure of Organic Compounds. V. Molecular Theory of Bond Separation. J. Am. Chem. Soc. 1970, 92, 4796−4801. (50) Pople, J. A.; Radom, L.; Hehre, W. J. Molecular Orbital Theory of the Electronic Structure of Organic Compounds. VII. Systematic Study of Energies, Conformations, and Bond Interactions. J. Am. Chem. Soc. 1971, 93, 289−300. (51) Wiberg, K. B.; Ochterski, J. W. Comparison of Different ab Initio Theoretical Models for Calculating Isodesmic Reaction Energies for Small Ring and Related Compounds. J. Comput. Chem. 1997, 18, 108−114. (52) Knyazev, V. D. Computational Study of the Reactions of H Atoms with Chlorinated Alkanes. Isodesmic Reactions for Transition States. J. Phys. Chem. A 2002, 106, 11603−11615. (53) Knyazev, V. D. Isodesmic Reactions for Transition States: Reactions of Cl Atoms with Methane and Halogenated Methanes. J. Phys. Chem. A 2003, 107, 11082−11091. (54) Becke, A. D. A New Mixing of Hartree-Fock and Local DensityFunctional Theories. J. Chem. Phys. 1993, 98, 1372. (55) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation Energy Formula into a Functional of the Electron Density. Phys. Rev. 1988, 37, 785. (56) Huynh, L. K.; Zhang, H. R.; Zhang, S.; Eddings, E.; Sarofim, A.; Law, M. E.; Westmoreland, P. R.; Truong, T. N. Kinetics of Enol Formation from Reaction of OH with Propene. J. Phys. Chem. A 2009, 113, 3177−3185. (57) Francl, M. M.; Pietro, W. J.; Hehre, W. J.; Binkley, J. S.; Gordon, M. S.; Defrees, D. J.; Pople, J. A. Self-Consistent Molecular Orbital Methods. XXIII. A Polarization-Type Basis Set for Second-Row Elements. J. Chem. Phys. 1982, 77, 3654−3665. (58) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (59) Gonzalez, C.; Schlegel, H. B. Reaction Path Following in MassWeighted Internal Coordinates. J. Phys. Chem. 1990, 94, 5523. (60) Truhlar, D. G. Basis-Set Extrapolation. Chem. Phys. Lett. 1998, 294, 45−48. (61) Halkier, A.; Helgaker, T.; Jorgensen, P. Basis-Set Convergence of the Energy in Molecular Hartree−Fock Calculations. J. Chem. Phys. Lett. 1999, 302, 437−446. (62) De Lara-Castells, M. P.; Krems, R. V.; Buchachenko, A. A. Complete Basis Set Extrapolation Limit for Electronic Structure Calculations: Energetic and Nonenergetic Properties of HeBr and HeBr2 van der Waals Dimers. J. Chem. Phys. 2001, 115, 10438−10449. 3291


Interpretation and Application of Reaction Class Transition State

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