Potential Energy Surface for Large Barrierless Reaction Systems

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Potential Energy Surface for Large Barrierless Reaction Systems: Application to the Kinetic Calculations of the Dissociation of Alkanes and the Reverse Recombination Reactions Qian Yao, Xiao-Mei Cao, Wen-Gang Zong, Xiao-Hui Sun, Ze-Rong Li, and Xiang-Yuan Li J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b00877 • Publication Date (Web): 14 May 2018 Downloaded from http://pubs.acs.org on May 15, 2018

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The Journal of Physical Chemistry

Potential Energy Surface for Large Barrierless Reaction Systems: Application to the Kinetic Calculations of the Dissociation of Alkanes and the Reverse Recombination Reactions Qian Yao,† Xiao-Mei Cao,‡ Wen-Gang Zong,§ Xiao-Hui Sun,† Ze-Rong Li,*,† and Xiang-Yuan Li§ †

College of Chemistry, ‡College of Aeronautics & Astronautics and §College of Chemical Engineering, Sichuan University, Chengdu 610064, China

Abstract: The isodesmic reaction method is applied to calculate the potential energy surface (PES) along the reaction coordinates and the rate constants of the barrierless reactions for unimolecular dissociation reactions of alkanes to form two alkyl radicals and their reverse recombination reactions. The reaction class is divided into 10 subclasses depending upon the type of carbon atoms in the reaction centers. A correction scheme based on isodesmic reaction theory is proposed to correct the PESs at UB3LYP/6-31+G(d,p) level. To validate the accuracy of this scheme, a comparison of the PESs at B3LYP level and the corrected PESs with the PESs at CASPT2/aug-cc-pVTZ level is performed for 13 representative reactions and it is found that the deviations of the PESs at B3LYP level are up to 35.18 kcal/mol and are reduced to within 2 kcal/mol after correction, indicating that the PESs for barrierless reactions in a subclass can be calculated meaningfully accurately at a low level of ab initio method using our correction scheme. High-pressure limit rate constants and pressure dependent rate constants of these reactions are calculated based on their corrected PESs and the results show the pressure dependence of the rate constants cannot be ignored, especially at high temperatures. Furthermore, the impact of molecular size on the pressure-dependent rate constants of decomposition reactions of alkanes and their reverse reactions has been studied. The present work provides an effective method to generate meaningfully accurate PESs for large molecular system. 1. INTRODUCTION Hydrocarbon fuels are the most common fuels in nature, with high energy density and low cost, and they are one of the main sources for obtaining energy now and in the future. In order to improve the combustion efficiency of these fuels whilst reducing the formation of pollutants, it is necessary to construct a reliable detailed kinetic reaction mechanism for the modeling of the combustion of hydrocarbon fuels. However the combustion of hydrocarbon fuels is a complicated process and the detailed combustion mechanism usually includes hundreds of species and thousands of elementary reactions,1,2 therefore they are usually automatically generated by Automated Reaction Mechanism Generators(ARMG)3-6 according to reaction classes. In the pioneering work of Curran et al.7 and Westbrook et al.,8 the oxidation mechanism of alkanes contains 16 reaction classes for low-temperature region and 9 reaction classes that are common for low-temperature and high-temperature regions: 1. Unimolecular fuel decomposition (e.g. RR′ → R• + R′•; RH → R• + H•) 2. H-atom abstraction from the fuel (e.g. RH + X• →R• + XH) 3. Alkyl radical decomposition 1 ACS Paragon Plus Environment

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4. Alkyl radical isomerization 5. H-atom abstraction from alkenes 6. Addition of radical species to olefin 7. Alkenyl radical decomposition 8. Olefin decomposition 9. Alkyl radical+O2 to produce olefin+HO2 directly. The barrierless unimolecular dissociation reactions of alkanes to form two alkyl radicals and their reverse recombination reactions are two of the most important reaction classes in the combustion mechanisms of hydrocarbon fuels and have been investigated in a large variety of experimental and theoretical studies9-50 and the overview of the theoretical and experimental kinetic studies is presented in Table 1. However, there are few of studies on dissociations of large alkanes involving more than 8 carbon atoms. These reactions are usually pressure dependent reactions, but there are only a few of theoretical studies on the pressure dependence of rate constants for these reactions. Moreover, the pressure dependence of the rate constants is related to the molecular size,51 so the studies of the influence of the molecular size on the pressure dependent rate constants are significant to derive rate rule for the automatic generation of the combustion mechanisms for hydrocarbons. Table 1. Overview of the Recent Theoretical and Experimental Kinetic Studies of Dissociation Reactions of Alkane and Their Reverse Recombination Reactions Reaction system

Referencea

Method

Pressure

Davidson 1995 [29]

Shock Tube

0.5-1.5 atm

Du 1996 [20]

Incident Shock-Wave Experiments

1.1-2.3 atm

Cody 2002 [30]

Discharge Flow-Mass Spectrometer

0.6-2.0 torr

Cody 2003 [31]

Discharge Flow-Mass Spectrometer

0.6-1.5 torr

Laser Photolysis

0.6-10 torr

CH3+CH3→C2H6

Wang 2003 [32]

MRCI//CASSCF

Pressure Dependent

Sangwan 2015 [33]

Laser Photolysis

1-100 bar

Hessler 1996 [34]

Monte Carlo

Pressure Dependent

Pesa 1998 [21]

CFTST

High Pressure Limit

Klippenstein 1999 [35]

MRCI//CASSCF

Pressure Dependent


CASPT2//B3LYP

High Pressure Limit

Golubeva 2007 [36]

EOM-SF-CCSD

-

Zheng 2008 [37]

M06-L

High Pressure Limit

Li 2008 [38]

CASPT2//CASSCF

High Pressure Limit

Ge 2010 [39]

CR-CC(2,3)//MP2

High Pressure Limit

Blitz 2015 [40]

Linearized Second-Order Master Equation

High Pressure Limit

Organometallic Vapor-Phase Epitaxy

All Pressure Regimes

RRKM

Pressure Dependent

Oehlschlaeger 2005 [10]

Shock Tube

0.13-8.4 atm

Kiefer 2005 [11]

Shock Tube

70-5700 torr

C2H6→CH3+CH3 Cardelino 2003 [41]

2 ACS Paragon Plus Environment

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Yang 2009 [42]

Shock Tube

20-280 torr

Lorant 2001 [22]

B3LYP

High Pressure Limit

Li 2008 [38]

CASPT2//CASSCF

High Pressure Limit

Ge 2010 [39]

CR-CC(2,3)//MP2

High Pressure Limit

Knyazev 2001 [25]

Laser Photolysis

High Pressure Limit


CASPT2//B3LYP

High Pressure Limit

Mousavipour 2003 [15]

UMP2/6-311+G(2d,2p)

High Pressure Limit

Zhu 2004 [16]

G2M(CC2)

Pressure Dependent


Shock Tube

0.13-8.4 atm

Shock Tube

6-31 torr

CASPT2

Pressure Dependent

Mousavipour 2003 [15]

UMP2/6-311+G(2d,2p)

High Pressure Limit

Zhu 2004 [16]

G2M(CC2)

Pressure Dependent

Atkinson 1997 [43]

Laser Photolysis

637−1059 pa

Shafir 2003 [44]

Laser Photolysis

High Pressure Limit


CASPT2//B3LYP

High Pressure Limit


CASPT2//B3LYP

High Pressure Limit


Shock Tube

0.2-8.8 atm


Shock Tube

0.2-8.8 atm


Shock Tube

0.2-8.8 atm

Shock Tube

0.3-1 atm

CCSD(T)

High Pressure Limit


CASPT2//B3LYP

High Pressure Limit


CASPT2//B3LYP

High Pressure Limit

Shock Tube

0.3-1 atm

CCSD(T)

High Pressure Limit

CASPT2//B3LYP

High Pressure Limit

CH3+C2H5→C3H8

C3H8→CH3+C2H5

Sivaramakrishnan 2011 [17]

C2H5+C2H5→n-C4H10

CH3+i-C3H7→i-C4H10

n-C4H10→C2H5+C2H5

n-C4H10→CH3+n-C3H7

i-C4H10→CH3+i-C3H7

Sivaramakrishnan 2012 [46] CH3+t-C4H9→t-C5H12

C2H5+i-C3H7→i-C5H12

t-C5H12→CH3+t-C4H9 Sivaramakrishnan 2012 [46] C2H5+t-C4H9→C6H14 Klippenstein 2006 [18] i-C3H7+i-C3H7→C6H14



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CASPT2//B3LYP

High Pressure Limit


CASPT2//B3LYP

High Pressure Limit

Ding 2011 [47]

CCSD(T)//B3LYP

Pressure Dependent

Ding 2011 [47]

CCSD(T)//B3LYP

Pressure Dependent

Ding 2011 [47]

CCSD(T)//B3LYP

Pressure Dependent


CASPT2//B3LYP

High Pressure Limit

Ning 2015 [48]

CASPT2//CASSCF

Pressure Dependent

Ning 2015 [48]

CASPT2//CASSCF

Pressure Dependent

Ning 2015 [48]

CASPT2//CASSCF

Pressure Dependent

Ning 2015 [48]

CASPT2//CASSCF

Pressure Dependent

Zhao 2017 [49]

Approximate Calculation

Pressure Dependent

Zhao 2017 [49]


Pressure Dependent

Zhao 2017 [49]


Pressure Dependent

Zhao 2017 [49]


Pressure Dependent

Zhao 2017 [49]


Pressure Dependent

Zhao 2017 [50]


Pressure Dependent

Zhao 2017 [50]


Pressure Dependent

Zhao 2017 [50]


Pressure Dependent

Zhao 2017 [50]


Pressure Dependent

i-C3H7+t-C4H9→C7H16

n-C7H16→CH3+n-C6H13

n-C7H16→C2H5+n-C5H11

n-C7H16→n-C3H7+n-C4H9

C8H18→t-C4H9+t-C4H9

iso-C8H18→CH3+1-C7H15

iso-C8H18→CH3+2-C7H15

iso-C8H18→i-C3H7+C5H11

iso-C8H18→i-C4H9+t-C4H9 n-C10H22 →ĊH3 + n-Ċ9H19

n-C10H22 →Ċ2H5 + n-Ċ8H17

n-C10H22 →n-Ċ3H7 + n-Ċ7H15

n-C10H22 →n-Ċ4H9 + n-Ċ6H13

n-C10H22 →n-Ċ5H11 + n-Ċ5H11

n-C12H26 →ĊH3 + n-Ċ11H23

n-C12H26 →Ċ2H5 + n-Ċ10H21

n-C12H26 →n-Ċ3H7 + n-Ċ9H19

n-C12H26 →n-Ċ4H9 + n-Ċ8H17

n-C12H26 →n-Ċ5H11 + n-Ċ7H15


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n-C12H26 →n-Ċ6H13 + n-Ċ6H13 Zhao 2017 [50] a


Pressure Dependent

Only the first author is listed.

Currently, the estimation of kinetic parameters in combustion models generated by ARMGs is mainly obtained through structure/reactivity correlations or analogies with similar reactions. These approaches are empirical and probably lead to significant uncertainties in the rate constants.52 With the increasing demand for accuracy in kinetic parameters, the reaction-class transition state theory (RC-TST) by Truong and co-workers is widely employed.53-62 Knyazev has developed a method named RESLIR for reaction systems with tight transition states which is close to the RC-TST.63-65 So far, the RC-TST method has been applied to various reaction classes with tight transition states containing H-atom abstraction from the fuel, alkyl radical decomposition, alkyl radical isomerization and H-atom abstraction from alkenes. Our group presented an interpretation of the RC-TST based on isodesmic reactions and proposed a simple correction scheme for accurate determinations of the energy barriers and rate constants from the principal reactions with tight transition states.61 An isodesmic reaction is traditionally defined as a chemical reaction that the numbers of bonds of each formal bond type are conserved in the reaction and a low-level ab initio calculation can provide an accurate reaction enthalpy for the isodesmic reaction as the errors associated with incomplete correlation energy and incomplete basis sets can be partially canceled66 and our previous work61 extended the definition of the isodesmic reaction to reactions involving transition states with deviations of bond distances and angles from normal values. It should be noticed that the RC-TST method is just used to calculate rate constants for reaction class systems with tight transition states and we extended the isodesmic reaction method to meaningfully accurately calculate reaction barriers, reaction enthalpies and rate constants for such reaction class systems in our prior work.61 In the combustion mechanism of hydrocarbons, many reaction classes are barrierless reactions and the rate constants for the barrierless reactions are usually calculated with the variational-transition-state theory (VTST).67-71 The Variable Reaction Coordinate Transition State Theory (VRC-TST) version of the RRKM theory involves selecting a range of dividing surfaces at different positions along the reaction coordinate, evaluating sum of states appropriate to the given energy and angular momentum at each position, and giving an array of rate constants as a function of the reaction cordination and temperature, and finally the reaction rate constant at a specific temperature is determined as the minimum value along the reaction coordination. One of the challenges involved in the accurate calculation of rate constants for barrierless reactions is the construction of accurate PESs along the reaction coordinates.67-68,72 Accurate PESs for these reactions are usually required for the region that the forming bond distance is from 1.5 Å to 6.0 Å and for all orientations of the two fragments in the dissociation reactions. A difficulty to calculate PESs in this region, especially in the region that the bond distance is larger than 2 Å, is due to the multireference character73 of the electronic wave function for systems with partially broken bonds. Thus the suitable electronic structure methods are required for multireference systems. Multireference wave function methods are used to calculate the PESs for the barrierless reactions. Unlike singlereference electronic structure methods, these methods are not black-box methods and the best choice of the reference state depends on systems. Generally, multireference wave function methods that include dynamical correlation energy are very computationally expensive, and therefore, they are only suitable for small molecular system and calculation of accurate PES with large molecular system is still a challenge in computational chemistry. In this article, isodesmic reaction method is generalized to construct accurate PESs along the reaction coordinates for barrierless reactions involving large molecules, which are required for the calculation of the reaction rate constants using variational



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transition state theory and this method is applied to the meaningfully accurate calculation of the highprssure limit rate constants and pressure-dependent rate constants for the dissociation reactions of alkanes and their reverse reactions, which are very important reaction classes in the combustion modeling of alkanes. 2. METHODS 2.1. PESs Calculation for Barrierless Reactions Using Isodesmic Reaction Approach. Reactions in the same class have the same reactive moiety, thus are expected to have similarities in their potential energy surfaces along their reaction valleys.74 In the RC-TST theory of Truong, one reaction in the studied reaction class with the rate constant that is known from an accurate quantum chemical calculation or from an experiment is chosen as the principal reaction (also called the reference reaction) and other reactions are called target reactions, with rate constants are unknown and need to be calculated. In our previous work,61 we have shown that, for the principal reaction RP and a target reaction RT in the reaction class, AP = BP + CP (1) AT = BT + CT (2) where AP is the reactant and BP and CP are the products for the principal reaction and AT is the reactant and BT and CT are the products for the target reaction, the transition state formation reaction for reaction (1) minus the transition state formation reaction for reaction (2) can be considered as an isodesmic reaction if the gemoetris of the transition states for reaction (1) and reaction (2) are conserved. In VTST theory, PES along the reaction coordinate is required for barrierless reactions and in this study, we extend the application of the isodesmic reaction theory in the the transition state formation reactions to the any points along the reaction coordinates. Truong et al.62 proposed the RC-TST theory for reactions with barriers in a reaction class, however, the term “reaction class” has not been rigorously defined in their work. In our previous work,61 we have shown that two reactions can be considered as in a reaction class when the geometries of the transition states for them are conserved and hence, the difference for their transition state formation reaction is an isodesmic reaction. The conservation of the geometries of the transition states includes the conservation of the geometries of the reaction centers and the number of the bonds of each formal bond type in the surrondings of the reaction centers. In this study, for the unimolecular dissociation of alkanes to form two alkyl radicals, the reaction coordinate s can be defined as the relative distance between the two atoms (C–C) in the broken bond with the reaction coordinate being 0 for the reactant and s being the deviation of the C–C distance from its value at reactant geometry. The energy difference between the reactant and a point in the PES with reaction coordinate s can be considered as the reaction energy in the following reaction: A(0) = A( s) (3) For the principal reaction and target reaction with same reaction coordinate s, we have AP (0) = AP ( s ) ∆VP (s ) (4) AT (0) = AT ( s ) ∆VT (s ) (5) where ∆VP (s ) and ∆VT (s ) are the potential differences between reaction coordinate s and reaction coordinate 0 for the principal reaction and target reaction, respectively. Then, reaction (4) minus reaction (5) is AT (0) + AP ( s ) = AT ( s ) + AP (0) ∆∆V = ∆VT ( s ) − ∆VP ( s ) (6) where ∆∆ V is the potential difference for reaction (6). When the reaction coordinate is not on its equilibrium point along the PES, it is difficult to classify the 6 ACS Paragon Plus Environment

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bonds of the reacting system into typical types of bond; hence it is hard to judge if reaction (6) is an isodesmic reaction according to the conservation of the numbers of bonds for each bond type. According to our previous work,61 the reactants and products in reaction (6) can be divided into a reaction center involving distorted bonds and surrounding groups involving typical bond types. It is usual that the numbers of bonds for each bond type in the surrounding groups in the two sides of reaction (6) are conserved and so if the geometries of the reaction centers for reaction (6) are conserved, reaction (6) can be considered as an isodesmic reaction and the calculated reaction energy will not be much dependent on the level of ab initio theory because the errors associated with incomplete correlation energy and incomplete basis sets can be partially cancelled in the case of an isodesmic reaction. If potential difference ∆VP (s )′and ∆VT (s )′ are the accurate values from experiment or from highlevel ab initio method and potential difference ∆VP (s) and ∆VT (s ) are the approximate values from lowlevel ab initio method. They should satisfy approximately the following equations according to the isodesmic reaction method: ∆∆V = ∆VT ( s )′− ∆VP ( s )′= ∆VT ( s ) − ∆VP ( s ) (7) Hence we have ∆VT ( s )′= ∆VP ( s )′+ ( ∆VT ( s ) − ∆VP ( s ))

(8)

= ∆VT ( s ) + ( ∆VP ( s )′− ∆VP ( s ))

In equation (8)

V(s)

∆∆VP ( s ) = ∆VP ( s )′− ∆VP ( s ) (9) ∆∆VP (s ) is the correction to the approximate potential energy from low-level ab initio method for any target reaction RT and it is obtained from the principal reaction RP. Equation (9) shows that the potential energy for a target reaction can be obtained from correcting their approximate values at low-level ab initio methods and the corrected value would have the approximate accuracy of the accurate method used for the principal reaction. Therefore, PESs for barrierless reactions in a class can be calculated meaningfully accurately at a low level of ab initio method using our correction scheme and for illustration the correction scheme is given in Figure 1. Finally, the traditional isodesmic reaction method is expanded to the case for the barrierless reactions in this study.

∆∆VP

target reaction low-level ab initio method calculated by the correction scheme

principal reaction

∆∆VP

PES

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


low-level ab initio method high-level ab initio method

Reaction coordinates

s

Figure 1. Correction scheme



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2.2. Computational details. In this study, UB3LYP/6-31+G(d,p) method with the Gaussian 09 program75 is used as the approximate method for the single point energies along the PES which is obtained by a relaxed scan at the same level. “Scan” means that the reaction coordinates are changed gradually and “relaxed” means that the reaction coordinates are fixed, while other coordinates that are orthogonal to the reaction coordinates are relaxed during the energy minimization. The complete active space with second-order perturbation theory (CASPT2(2e,2o)/aug-cc-pVTZ) with the Molpro2012 softwares76-77 is used as the high-level ab initio method for the single point energy along the PES obtained by a relaxed scan at the UB3LYP/631+G(d,p) level. The PESs are obtained by a set of fixing C-C distance at CASPT2(2e,2o)/aug-cc-pVTZ // B3LYP/6-31+G(d,p) level without zero-point energy (ZPE). Only the ZPEs for the stationary points are considered. The ZPEs are obtained at B3LYP/6-31+G(d,p) level and are scaled by a factor of 0.98.78 All reactions studied in this work are listed as follows: C2H6 → ĊH3 + ĊH3 (R1) C3H8 → ĊH3 + Ċ2H5 (R2) n-C4H10 → ĊH3 + n-Ċ3H7 (R3) n-C4H10 →Ċ2H5 + Ċ2H5 (R4) i-C4H10 → ĊH3 + i-Ċ3H7 (R5) n-C5H12 →ĊH3 + n-Ċ4H9 (R6) n-C5H12 →Ċ2H5 + n-Ċ3H7 (R7) i-C5H12 →Ċ2H5 + i-Ċ3H7 (R8) i-C5H12 →ĊH3· + CH3CH2(CH3)ĊH (R9) t-C5H12 →ĊH3 + (CH3)3Ċ (R10) n-C6H14 →ĊH3 + n-Ċ5H11 (R11) n-C6H14 →n-Ċ3H7 + n-Ċ3H7 (R12) CH3(CH3CH2)2CH →Ċ2H5 + CH3CH2(CH3)ĊH (R13) CH3CH(CH3)CH(CH3)CH3 →i-Ċ3H7 + i-Ċ3H7 (R14) (CH3)3CCH2CH3 →ĊH3 + CH3CH2(CH3)2Ċ (R15) (CH3)3CCH2CH3 →Ċ2H5 + (CH3)3Ċ (R16) (CH3)3CCH2CH2CH3 →n-Ċ3H7 + (CH3)3Ċ (R17) CH3CH(CH3)CH(CH3)CH2CH3 →i-Ċ3H7 + CH3CH2(CH3)ĊH (R18) CH3CH(CH3)C(CH3)2CH3 →i-Ċ3H7 + (CH3)3Ċ (R19) CH3CH(CH3)C(CH3)2CH2CH3 →i-Ċ3H7 + CH3CH2(CH3)2Ċ (R20) (CH3)3CC(CH3)3 →(CH3)3Ċ + (CH3)3Ċ (R21) (CH3)3CC(CH3)2CH2CH3 →(CH3)3Ċ + CH3CH2(CH3)2Ċ (R22) n-C7H16 →ĊH3 + n-Ċ6H13 (R23) n-C7H16 →Ċ2H5 + n-Ċ5H11 (R24) n-C7H16 →n-Ċ3H7 + n-Ċ4H9 (R25) (CH3)3CCH2(CH3)CHCH3 →ĊH3 + (CH3)3CCH2(CH3)ĊH (R26) (CH3)3CCH2(CH3)CHCH3 →ĊH3 + (CH3)2ĊCH2(CH3)CHCH3 (R27) (CH3)3CCH2(CH3)CHCH3 →i-Ċ3H7 + (CH3)3CĊH2 (R28) (CH3)3CCH2(CH3)CHCH3 →(CH3)3Ċ + ĊH2(CH3)CHCH3 (R29) n-C10H22 →ĊH3 + n-Ċ9H19 (R30) n-C10H22 →Ċ2H5 + n-Ċ8H17 (R31) n-C10H22 →n-Ċ3H7 + n-Ċ7H15 (R32) n-C10H22 →n-Ċ4H9 + n-Ċ6H13 (R33) n-C10H22 →n-Ċ5H11 + n-Ċ5H11 (R34)


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n-C12H26 →ĊH3 + n-Ċ11H23 (R35) n-C12H26 →Ċ2H5 + n-Ċ10H21 (R36) n-C12H26 →n-Ċ3H7 + n-Ċ9H19 (R37) n-C12H26 →n-Ċ4H9 + n-Ċ8H17 (R38) n-C12H26 →n-Ċ5H11 + n-Ċ7H15 (R39) n-C12H26 →n-Ċ6H13 + n-Ċ6H13 (R40) In this study, the decomposition reactions of alkane and their reverse reactions without an intrinsic barrier are treated with VRC-TST method in Variflex code.79 Variational effects are incorporated by the canonical variational transition-state theory (CVT), in which the flux is minimized for a canonical ensemble. The CVT rate constant, k CVT (T ) , at temperature T, can be obtained as the minimum of the generalized transition-state theory rate constant, k GT (T , s ) , as a function of s, that is,

k CVT (T ) = min k GT (T , s ) s

(11)

In the VRC-TST, the fixed points are located at the atoms involved in the breaking bond and a bond length reaction coordinate is obtained. The basis of this implementation is the assumed separation of modes into the “conserved” modes (which have little variation in character during the reaction process and are typical vibrations) and the “transitional” modes (which do have a considerable change in character and are typical rotations and hindered rotations).80 The function that is commonly used to fit the potential energies along the reaction coordinate for a barrierless association/decomposition process is a Varshni potential function: R V ( R ) = De {1 − e exp[ − β ( R 2 − Re2 )]}2 − De (12) R where De is the dissociation energy for the breaking bond without the zero-point energy, R is the reaction coordinate (the distance between the two bonding atoms; C–C bond in this work), Re is the equilibrium bond length, De, β and Re are fitting parameters. The potential for the modes orthogonal to the reaction coordinate (which is also named bending potential) is given by V0 (1 − cos(a − a0 ) ∗ cos(b − b0 ) ^ n ) (13) where V0 is the Varshni potential for the bonding interaction, a and b denote bending angles between the two fragments, n is specified in the input file of Variflex. Pressure dependent rate constants are calculated using the RRKM/ME theory for temperatures between 500 and 2000 K and pressures between 0.01 and 100 atm.81-82 The master-equation method is used to calculate pressure-dependent rate constants. The master equation can be written in the matrix form: dρ = Bρ + R (14) dt All the eigenvalues of the matrix B are negative and the overall pressure-dependent thermal rate constant kuni(T, P) can be derived as the negative of the largest eigenvalue of the matrix B.83 This eigenvalue is the chemically significant eigenvalue, which corresponds to the slow mode describing the “chemical reaction”, and for multi-reaction channels case, there is one chemically significant eigenvalue for each reaction channel. All calculations are carried out with a series of energy grain sizes and variations in the maximum system energy to ensure that the results have converged. Argon is taken as the bath gas, with Lennard-Jones (L-J) collision diameter parameter σ=3.330 Å and well-depth ɛ=94.8 cm-1. The L-J parameters for other species involving in this study are derived from the work of Sun et al.84 Energy transfer coefficients are treated with the exponential down model with



Page 10 of 24

the average downward transfer parameter < ∆ E > down = 120 (T / 300 ) 0.9 K-1 cm-1. This function form is reasonable since the similar form has been used in the work of Kiefer.11 However, it should be noted that low pressure rate constants are very sensitive to parameter of the exponential down model,85-87 which depends significantly on the molecular size. As there are few studies about the relationship of parameter of exponential down model with the molecular size, so the same value for all the studied systems are used approximately because the lack of different parameters for systems with different sizes. 3. RESULTS AND DISCUSSION 3.1. Conservation of the Reaction Centers. In the reactions of the dissociation of alkanes and their reverse recombination reactions, the reaction center includes the broken C–C bond and its bonded atoms. The labeling of atoms and bonds involved in the reaction centers are given in Figure 2, where d1, d2, d2, d3, d4, d5 and d6 are the bond lengths and A1, A2, A3, A4, A5 and A6 are the bond angles. Ra, Rb, Rc, Rd, Re and Rf are hydrogen or alkyl substituents. Considering the atom types for the two C atoms of the broken C–C bond in the reaction centers, the studied reactions are divided into 10 subclasses as follows: pp, ps, pt, pq, ss, st, sq, tt, tq and qq, where p, s, t, q represent primary carbon atom connecting with three hydrogen atoms, secondary carbon atom connecting with two hydrogen atoms, tertiary carbon atom connecting with one hydrogen atom and quaternary carbon atom connecting with no hydrogen atom, respectively. The R1, R2, R4, R5, R8, R10, R14, R16, R19 and R21 are chosen as the principal reactions, which are the smallest reactions, for the pp-, ps-, pt-, pq-, ss-, st-, sq-, tt-, tq- and qq-subclasses, respectively. In the list of the studied reactions, R3, R6, R11, R23, R30 and R35 belong to ps-subclass. R9 and R26 belong to pt-subclass. R15 and R27 belong to pq-subclass. R7, R24, R25, R31, R32, R33, R34, R36, R37, R38, R39 and R40 belong to ss-subclass. R13 and R28 belong to st-subclass. R17 and R29 belong to sq-subclass. R18, R20 and R22 belong to tt-, tq- and qq-subclass, respectively. It should be noticed that pp-subclass contains only one reaction R1.

Figure 2. Labeling of atoms for the reaction centers at reaction coordinate s. The bond lengths d1, d2, d2, d3, d4, d5 and d6 and bond angles A1, A2, A3, A4, A5 and A6 for all studied reactions are given in Table S1. Maximum absolute deviation (MAD) and average absolute deviation (AAD) of the bond lengths and bond angles in the reaction centers for each subclass at a set of fixed C– C distance values from 2.0 to 5.0 Å, are listed in Table 2. From Table 2, it can be seen that the maximum absolute differences of the bond lengths d1, d2, d2, d3, d4, d5 and d6 and bond angles A1, A2, A3, A4, A5 and A6 of the reaction centers are 0.022 Å, 0.002 Å, 0.003 Å, 0.0011 Å, 0.017 Å, 0.014 Å, 7.5°, 3.4°, 5.4°, 7.2°, 6.1° and 4.9°, respectively. Hence the geometric parameters in the reaction centers are very well conserved along the reaction coordinates. Therefore, for any two reactions with the same distance of carbon–carbon bond in the subclass, both the numbers of the typical bonds and the reaction centers 10 ACS Paragon Plus Environment

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are conserved. Hence their difference at the same distance of carbon–carbon bond can be considered as a generalized isodesmic reaction. For example, the reactions R2 and R3 at the reaction coordinate s are CH3CH2CH3 → CH3•••CH2CH3(s) (R2´) CH3CH2CH2CH3 → CH3•••CH2CH2CH3(s) (R3´) Then their difference is CH3CH2CH3 + CH3•••CH2CH2CH3(s) → CH3•••CH2CH3(s) + CH3CH2CH2CH3 (∆R) The typical bonds are C−H bond and C−C bond and their numbers are 18 and 4, respectively, in reactants and products. The reaction centers at reaction coordinate s are the parts shown by the virtual bonds and they are conserved in the two sides of reaction ∆R. Therefore, all the geometric parameters are conserved in the two sides of the reaction at reaction coordinate s and this reaction can be considered as a generalized isodesmic reaction. Table 2. Maximum Absolute Difference (MAD) and Average Absolute Difference (AAD) of the Bond Lengths and Bond Angles in the Reaction Centers for Each Subclass Subclasses

d1/(Å)

d2/(Å)

d3/(Å)

d4/(Å)

d5/(Å)

d6/(Å)

MAD

AAD

MAD

AAD

MAD

AAD

MAD

AAD

MAD

AAD

MAD

AAD

ps

0.000

0.000

0.000

0.000

0.000

0.000

0.001

0.001

0.001

0.001

0.003

0.001

pt

0.000

0.000

0.000

0.000

0.000

0.000

0.003

0.002

0.017

0.006

0.002

0.001

pq

0.000

0.000

0.000

0.000

0.000

0.000

0.002

0.001

0.002

0.001

0.014

0.008

ss

0.002

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.001

0.002

0.001

st

0.022

0.008

0.002

0.001

0.003

0.001

0.004

0.002

0.003

0.001

0.007

0.003

sq

0.018

0.005

0.001

0.001

0.001

0.001

0.001

0.001

0.002

0.001

0.001

0.001

tt

0.000

0.000

0.001

0.000

0.002

0.001

0.011

0.006

0.002

0.001

0.002

0.001

tq

0.000

0.000

0.001

0.000

0.001

0.000

0.001

0.001

0.012

0.009

0.001

0.001

qq

0.001

0.001

0.001

0.001

0.001

0.001

0.003

0.001

0.013

0.008

0.002

0.001

Subclasses

A1/(°)

A2/(°)

A3/(°)

A4/(°)

A5/(°)

A6/(°)

MAD

AAD

MAD

AAD

MAD

AAD

MAD

AAD

MAD

AAD

MAD

AAD

ps

0.7

0.2

0.6

0.2

0.6

0.2

0.7

0.3

0.8

0.3

1.5

0.4

pt

1.1

0.4

1.0

0.2

1.3

0.3

4.1

0.9

2.6

1.8

4.3

0.2

pq

1.8

0.4

1.8

0.3

2.9

0.5

3.9

1.9

3.4

1.4

1.0

0.6

ss

1.2

0.3

2.1

0.3

1.6

0.3

2.2

0.3

2.0

0.3

2.6

0.5

st

7.5

3.2

3.0

1.1

3.5

1.4

7.2

2.3

6.1

1.9

1.8

0.6

sq

5.3

2.2

3.0

1.4

3.5

1.1

5.7

1.2

5.0

1.9

2.7

1.0

tt

3.8

1.0

1.7

0.9

5.4

2.5

2.0

1.1

5.6

2.5

4.9

1.5

tq

2.1

0.7

1.0

0.4

1.0

0.4

1.6

1.1

1.6

1.2

1.7

1.1

qq

3.1

1.4

3.4

1.1

2.0

0.7

0.9

0.6

3.9

1.4

4.0

1.6

3.2. PESs for Principal Reactions. A comparison of PES for R1 calculated in this study at CASPT2/aug-cc-pVTZ//B3LYP/6-31+G(d,p) level with that calculated by Klippenstein et al.18 using CAS+1+2/aug-cc-pVTZ//B3LYP/6-31G* and CASPT2/aug-cc-pVTZ//B3LYP/6-31G* is given in Figure 3 and it can be seen that their values are close to our calculated PES, indicating that the CASPT2/aug-cc-pVTZ is accurate for describing the PES for the dissociation of alkanes. 11 ACS Paragon Plus Environment


0 -2 V(r)/kcal mol-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-4 -6 this work CAS+1+2+QC/aug-cc-pVTZ18 CASPT2/aug-cc-pVTZ18

-8

3.0

3.5

4.0 r(C-C)/angstrom

4.5

5.0

Figure 3. Plot of the PESs for ethane dissociation. The correction terms in equation (8) are the differences between the values at the CASPT2/aug-ccpVTZ and the UB3LYP/6-31+G(d,p) for the principal reactions and the values are given in Table 3. It is shown in Table 3 that large differences exist in potential energies between the approximate UB3LYP/631+G(d,p) methods and the accurate CASPT2/aug-cc-pVTZ. These values will be used to correct potential energies calculated by the approximate UB3LYP/6-31+G(d,p) method for target reactions according to equation (8). Table 3. Values of the Correction Terms (∆∆Va) for the Ten Subclasses ∆∆V(kcal/mol) pp(R1) ps(R2) pt(R5) pq(R10) ss(R4) st(R8) sq(R16) tt(R14) tq(R19) qq(R21) 2.0 39.27 35.61 31.91 28.47 33.02 29.66 26.18 26.32 23.17 18.47 2.2 38.50 35.08 31.71 28.56 32.72 29.68 26.53 26.68 23.96 19.84 2.4 36.74 33.62 30.58 27.77 31.48 28.72 25.98 26.23 23.82 20.05 2.6 34.01 31.21 28.55 26.12 29.29 26.84 24.51 24.76 22.78 19.54 2.8 30.56 28.09 25.80 23.76 26.37 24.21 22.28 22.51 20.98 18.23 3.0 26.70 24.57 22.64 20.93 23.04 21.23 19.58 19.79 18.64 16.31 3.2 22.73 20.92 19.30 17.89 19.59 17.98 16.64 16.87 16.01 13.99 3.4 18.87 17.37 16.03 14.88 16.22 14.93 13.77 13.94 13.32 11.67 3.6 15.29 14.07 12.97 12.04 13.10 12.09 11.18 11.21 10.81 9.53 3.8 12.07 11.10 10.24 9.46 10.31 9.51 8.81 8.77 8.50 7.55 4.0 9.23 8.49 7.82 7.21 7.87 7.28 6.69 6.70 6.51 5.81 4.2 6.76 6.23 5.73 5.17 5.78 5.32 4.89 4.86 4.80 4.26 4.4 4.64 4.30 3.96 3.59 3.98 3.66 3.31 3.32 3.29 2.87 4.6 2.85 2.63 2.41 2.21 2.44 2.23 2.00 2.00 2.03 1.73 4.8 1.32 1.21 1.11 1.01 1.13 1.03 0.89 0.92 0.99 0.87 5.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 a Difference between the value at the CASPT2/aug-cc-pVTZ and the UB3LYP/6-31+G(d,p). r(C−C)/Å


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3.3. PESs for a Set of Representative Target Reactions. In order to validate the accuracy of the correction scheme for the ten subclasses on the potential differences, a set of 13 representative target reactions R3, R6, R7, R9, R11, R12, R13, R15, R17, R18, R20, R22 and R24 are selected. Three kinds of PES for the 13 reactions are obtained: the PES at CASPT2/aug-cc-pVTZ level, the PES at UB3LYP/6-31+G(d,p) level and the PES obtained by correcting the PES at UB3LYP/6-31+G(d,p) level using the isodesmic reaction method and they are labled as PES(CAS), PES(DFT), PES(IRM), respectively and are illustrated in Figure S1. A comparison of the potential energies at some points along the reaction coordinates for these PESs are given in Table S2. It can be seen from Table S2 that the deviations of the potential energies along PES(DFT) from the potential energies along PES(CAS) are up to 35.18 kcal/mol. However, after coorection, the deviations of the potential energies along PES(IRM) for all target reactions are within 1.14 kcal/mol, which are within the so-called chemical accuracy of 1 ~ 2 kcal/mol, indicating that the potential energies by a lowlevel ab initio method can be corrected by our correction scheme to obtain meaningfully accurate values. From above discussion it can be concluded that the correction scheme is meaningfully accurate for the calculation of the potential enegies for unimolecular dissociation reaction class of alkanes. 3.4. High-Pressure Limit Rate Constants for Studied Reactions. The high-pressure limit rate constants of the decomposition reactions of alkanes and their reverse reactions involved in this study using the VTST theory on PES(CAS), PES(IRM) at different temperatures are calculated and are labeled as k(CAS) and k(IRM), respectively. For the 20 principal reactions, the accurate rate constants are calculated on the PES(CAS) and for the 60 target reactions, the rate constants are calculated on the PES(IRM). These rate constants are fitted to the modified Arrhenius equations and the obtained A, n and E parameters for the decomposition reactions of alkanes and their reverse reactions are given in Table 4 and Table 5, respectively. A comparison of the rate constants for principal reactions R1 and R2 on PES(CAS) with the high pressure experimental and theoretical data in literatures10-17 is made to test the accuracy of CASPT2/augcc-pVTZ method and the values are given in Figure 4. A comparison of the rate constants for the reverse reactions of the principal reactions (labled as re-R1 and re-R2, where re-Rn represents the reverse reaction of Rn) with the high pressure experimental and theoretical data in literatures14-16, 18-21,23 is given in Figure 5. It can be seen from Figure 4 and Figure 5 that the rate constants calculated by CASPT2/augcc-pVTZ method are close to the values from literatures, indicating that the CASPT2/aug-cc-pVTZ method is accurate for calculation of the potential energies and rate constants of the unimolecular dissociation reactions of alkanes and their reverse reactions. To test the accuracy of our correction scheme for the calculation of the high-pressure limit rate constants, rate constants on PES(CAS) for the 13 representative reactions and their reverse reactions are also calculated. For convenience of comparison, a ratio factor is defined as: (13) f = k max / k min where kmax and kmin are the maximum value and minimum value, respectively, between k(CAS) and k(IRM). The rate constants and the ratio factors for these representative target reactions and for their reverse reactions over the temperature range 500−2000 K are listed in Table S3 and Table S4, respectively. It can be seen from Table S3 that the maximum values of f for the target reactions R3, R6, R7, R9, R11, R12, R13, R15, R17, R18, R20, R22 and R24 over the temperature range 500−2000 K are 1.67, 1.43, 1.99, 1.58, 1.44, 4.13, 1.94, 1.74, 3.40, 2.31, 2.11, 1.82 and 2.21 and from Table S4 that the maximum values of f for reactions re-R3, re-R6, re-R7, re-R9, re-R11, re-R12, re-R13, re-R15, re-R17, re-R18, re-R20, re-R22 and re-R24 over the temperature range 500−2000 K are 1.00, 1.01, 1.00, 13 ACS Paragon Plus Environment


Page 14 of 24

1.00, 1.00, 1.01, 1.00, 1.00, 1.00, 1.00, 1.01, 1.01 and 1.00, respectively, indicating that the calculated high-pressure limit rate constants on PES(IRM) are very close to the results calculated on PES(CAS). Therefore, meaningfully accurate high-pressure limit rate constants for the studied reactions can be obtained from the correction scheme at the UB3LYP/6-31+G(d,p) level. Table 4. Kinetic Parameters (A, n and E) for Decomposition Reactions of Alkanes Reaction

Log A (s-1)

n

Ea (cal mol-1)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24 R25 R26 R27 R28 R29 R30 R31 R32 R33 R34 R35 R36 R37

26.75 28.23 27.67 28.71 27.40 27.29 29.30 29.25 27.04 27.79 26.53 29.29 28.71 29.77 27.90 29.59 29.72 28.61 29.82 29.40 30.31 29.13 24.32 26.77 27.23 25.67 27.90 28.68 28.60 29.75 30.42 31.10 30.71 30.78 38.06 39.83 40.77

-2.76 -2.92 -2.73 -2.87 -2.80 -2.64 -2.87 -2.95 -2.74 -2.91 -2.42 -2.68 -2.83 -2.99 -2.84 -3.00 -2.90 -2.82 -3.02 -2.78 -2.93 -2.46 -1.83 -2.10 -2.21 -2.33 -2.84 -2.66 -2.69 -3.12 -3.21 -3.20 -3.21 -3.17 -5.79 -5.88 -5.88

99843.8 98522.4 98193.5 96053.4 97500.8 97629.5 95800.6 94935.4 96751.5 95700.0 97230.6 95181.9 93929.8 92673.0 94241.5 93595.0 92780.1 90836.5 90113.8 87523.0 87029.6 83352.9 96395.1 94124.2 93970.4 92437.6 94241.5 89276.0 89595.4 99024.5 96507.1 96243.5 95806.3 95767.1 102836.1 100291.2 100141.9


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R38 R39 R40

39.94 40.05 39.97

-5.88 -5.83 -5.86

99711.4 99728.3 99579.3

Table 5. Kinetic Parameters (A, n and E) for the Reverse Reactions Reaction Log A (cm3molecule-1s-1) n Ea (cal mol-1) re-R1 re-R2 re-R3 re-R4 re-R5 re-R6 re-R7 re-R8 re-R9 re-R10 re-R11 re-R12 re-R13 re-R14 re-R15 re-R16 re-R17 re-R18 re-R19 re-R20 re-R21 re-R22 re-R23 re-R24 re-R25 re-R26 re-R27 re-R28 re-R29 re-R30 re-R31 re-R32 re-R33 re-R34 re-R35 re-R36 re-R37 re-R38

-7.76 -7.55 -7.45 -7.36 -7.62 -7.85 -7.28 -7.48 -7.79 -7.82 -8.44 -8.05 -7.84 -7.66 -7.93 -7.80 -8.13 -8.23 -8.10 -8.83 -8.87 -10.49 -10.24 -9.73 -9.62 -9.46 -9.41 -8.44 -8.99 -5.38 -5.93 -5.96 -6.03 -6.24 3.44 3.23 3.21 3.18

-0.84 -0.91 -0.90 -0.98 -0.89 -0.82 -0.94 -0.94 -0.85 -0.86 -0.60 -0.73 -0.84 -0.89 -0.80 -0.88 -0.76 -0.74 -0.80 -0.57 -0.63 -0.15 -0.02 -0.19 -0.26 -0.43 -0.43 -0.66 -0.56 -1.33 -1.30 -1.27 -1.26 -1.24 -3.97 -3.96 -3.96 -3.95

768.9 847.5 828.4 935.3 807.8 714.0 866.4 867.8 758.2 788.6 342.9 497.1 732.5 781.7 706.3 800.7 587.7 549.5 657.8 277.0 386.4 -440.5 -487.2 -244.1 -157.9 50.8 44.1 294.5 267.8 2037.8 2134.6 2115.9 2040.2 2037.4 6024.3 5907.6 5915.5 5970.6



re-R39 re-R40

8

-3.91 -3.92

2 0

Oehlschlaeger et al.10 Dean12 17 Sivaramakrishnan et al. Zhu et al.16 Mousavipour et al.15 this work

6 4

-2 -4

2 0 -2 -4

-6 -8

5947.2 5911.6

8

Log(k/s-1)

4 Log(k/s-1)

2.93 3.11

Oehlschlaeger et al.10 Tsang14 Kiefer et al.11 Dean12 Stewart et al.13 this work

6

-6 R1

-8 0.6

0.8 1.0 1000K/T

R2

1.2

0.6

0.8

1.0 1000K/T

1.2

Figure 4. Comparison of theoretical predictions and experimental measurements of the high pressure limit rate constants for R1 and R2.

re-R2 -10.0 log(k/cm3molecule-1s-1)

log(k/cm3molecule-1s-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 24

-10.5

-11.0

Klippenstein et al.18 Teng et al.23 19 Waage et al. Pesa et al.21 Du et al.20 this work

re-R1 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

-10.5

Klippenstein et al.18 Tsang14 Zhu et al.16 Mousavipour et al.15 this work

-11.0

0.6

1000K/T

0.8

1.0

1.2 1.4 1000K/T

1.6

1.8

2.0

Figure 5. Comparison of theoretical predictions and experimental measurements of the high pressure limit rate constants for re-R1 and re-R2.

3.3.5. Pressure Dependent Rate Constants for Studied Reactions. Pressure-dependent rate constants for the 10 principle reactions and 30 target reactions and their reverse reactions are calculated and listed in Table S5 and Table S6 in the form of modified Arrhenius paramteres(A, n, E). A comparison of the pressure dependent rate constants for R1 calculated in this study with the experimental results by Davidson et al.27 is given in Table 6. It can be seen that the difference between our calculated rate constants and the experimental results by Davidson et al. is within a factor of 6, indicating that the our calculated values are in good agreement with measurements. A representative set of reactions including R12, R17, R18 and their reverse reactions are chosen to test our correction scheme for the calculation of the pressure dependent rate constants. The pressure dependent rate constants k(IRM), k(CAS) and the ratio factors f over the temperature range 500−2000 K at pressure 0.01, 1 and 100 atm are listed in Table S7, Table S8 and Table S9, respectively, for R12, 16 ACS Paragon Plus Environment

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R17, R18 and their reverse reactions. It can be seen that the values of f for R12, R17 and R18 and their reverse reactions are within 4.00, indicating that the correction scheme is suitable for the prediction of the pressure dependent rate constants. Ratios k(T,P)/k∞(T) for decomposition reactions of alkanes and their reverse reactions at the temperature 500, 1000, 1500, 2000 K and pressure 0.01, 0.1, 1, 10 and 100 atm, while k(T,P) is the temperature and pressure-dependent rate constant and k∞(T) is high-pressure limit rate constant, are given in Table S10. In general, if k(T,P)/ k∞(T) < 1/2, pressure dependence cannot be ignored.49 It can be seen from Table S10, the influence of pressure on the rate constants increases as temperature increases and it becomes significant at the temperature larger than 1500K. While at temperature larger than 1500K, rate constants are less than one-half of the high-pressure limit values in most cases. The ratios k(T,P)/k∞(T) for decomposition reactions and their reverse reactions are nearly identical, indicating that the pressure dependence of the rate constants for decomposition reactions and their reverse reactions are the same. From the Table S10 it can be seen that ratios k(T,P)/k∞(T) for the reactions with the same carbon atoms which belong to different subclasses are different, indicating that the pressure dependence of the rate constants is also influenced by the type of the reactions. It is also observed that the pressure dependence of the rate constants is different for different molecular size with the same type of the reactions. Table 6. Comparison of the Experiment Rate Constants by Davidson et al. with Our Calculated Values by PES(CAS) for R1. T(K) P(atm) kexp(s-1)

k(CAS)(s-1)

1348 1485 1626 1793 1912 1448 1654 1821 1828 1999 1471 1488 1531 1586 1601 1617 1759 1797 1518 1602 1675 1945 1397

8.31E+00 1.40E+02 1.33E+03 1.06E+04 3.21E+04 9.24E+01 2.95E+03 2.12E+04 2.30E+04 1.08E+05 1.34E+02 1.93E+02 4.21E+02 1.02E+03 1.25E+03 1.65E+03 1.07E+04 1.54E+04 3.52E+02 1.34E+03 3.84E+03 7.06E+04 4.91E+01

0.757 0.725 0.676 0.681 0.635 1.29 1.31 1.2 1.21 1.22 1.1 1.2 1.23 1.2 1.14 1.21 1.21 1.09 1.35 1.25 1.26 1.24 3.97

3.30E+01 4.20E+02 2.40E+03 1.00E+04 3.00E+04 3.00E+02 4.50E+03 2.10E+04 3.60E+04 5.00E+04 4.90E+02 7.70E+02 1.00E+03 2.40E+03 2.60E+03 3.90E+03 2.00E+04 3.00E+04 6.70E+02 2.90E+03 5.60E+03 3.70E+04 3.00E+02

kmax/kmin 4.0 3.0 1.8 1.1 1.1 3.3 1.5 1.0 1.2 2.2 3.7 4.0 2.4 2.4 2.1 2.4 1.9 2.0 1.9 2.2 1.5 2.0 6.1



1598 1666 1529 1680 1747 1946

3.81 3.64 4.14 3.79 3.81 3.89

4.90E+03 1.20E+04 1.50E+03 1.10E+04 2.40E+04 1.40E+05

2.22E+03 6.08E+03 7.13E+02 7.56E+03 1.84E+04 1.57E+05

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2.2 2.0 2.1 1.5 1.3 1.1

4. SUMMARY AND CONCLUSIONS We applied the isodesmic reaction method to the meaningfully accurate calculation of the PESs along the reaction coordinates and the rate constants of the barrierless reactions for unimolecular dissociation reactions and their reverse reactions. The reaction class is divided into 10 subclasses depending upon the type of carbon atoms in the reaction centers. A correction scheme based on isodesmic reaction theory is proposed for the correction of the potential energies at B3LYP level for barrierless reactions in a class. To validate the accuracy of this scheme, a comparison of the PESs at B3LYP level and the corrected PESs with the PES at CASPT2 level is performed for 13 representative reactions and it is found that the deviations of the PES at B3LYP level are up to 35.18 kcal/mol and are reduced to within 2 kcal/mol after correction, indicating that the potential energies for barrierless reactions in a subclass can be calculated meaningfully accurately at a low level of ab initio method using the correction scheme. Highpressure limit rate constants are calculated based on their corrected PESs and the values for 13 representative target reactions are within a factor of 4.2 of the rate constants based on PESs at CASPT2 level, indicating that the calculated high-pressure limit rate constants on the corrected PESs are close to the results calculated on PESs at CASPT2 level. The pressure dependent rate constants of these reactions are also calculated and the results show the pressure dependence of the rate constants cannot be ignored, especially at high temperatures. Furthermore, the impact of molecular size on the pressure-dependent rate constants of decomposition reactions of alkanes and their reverse reactions has been discussed and it is found that the pressure dependence of the rate constants is influenced by the type of the reactions and the molecular size. This approach can be readily generalized to the dissociations of other systems, e.g. C–O and C–H. ASSOCIATED CONTENT Supporting Information Bond lengths and bond angles of reaction centers, PES curves for studied reactions, high-pressure limit rate constants and pressure dependent rate constants and the optimized geometries of reactants and products are provided in the Supporting Information. AUTHOR INFORMATION Corresponding Author * E-mail: [email protected](Z.R.Li) Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS We express our gratitude to the reviewers for many helpful suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 91441114).


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V(s)

TOC Graphic

∆∆VP

target reaction

PES

low-level ab initio method calculated by the correction scheme

principal reaction

∆∆VP

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low-level ab initio method high-level ab initio method

Reaction coordinates

s

This study focuses on the potential energy surface for large barrierless reaction systems: application to the kinetic calculations of the dissociation of alkanes and the reverse recombination reactions.


Potential Energy Surface for Large Barrierless Reaction Systems

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