Performance of First-Principles-Based Reaction Class Transition State

Subscriber access provided by UNIV OSNABRUECK

Article

Performance of First-Principles based Reaction Class Transition State Theory Thanh N. Truong, Lam K Huynh, and Artur Ratkiewicz J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.5b09564 • Publication Date (Web): 11 Jan 2016 Downloaded from http://pubs.acs.org on January 15, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 35

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

The Journal of Physical Chemistry

Performance of First-Principles Based Reaction Class Transition State Theory Artur Ratkiewicz,a Lam K. Huynh,b,c and Thanh N. Truongd* a

Chemistry Institute, University of Bialystok, Ciolkowskiego 1K 15-245 Bialystok, Poland Institute for Computational Science and Technology at Ho Chi Minh City, Vietnam c International University, VNU-HCMC, Thu Duc District, HCMC, Vietnam d Henry Eyring Center for Theoretical Chemistry, Department of Chemistry, University of Utah, 315 S. 1400 E. Rm. 2020, Salt Lake City, Utah 84112 b

*

Corresponding author email: [email protected] phone: 801-581-4301

1 ACS Paragon Plus Environment


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

Abstract Performance of the Reaction Class Transition State Theory (RC-TST) for prediction of rates constants of elementary reactions is examined using data from its previously applications to a number of different reaction classes. The RC-TST theory is taking advantage of the common structure denominator of all reactions in a given family combining with structure activity relationships to provide a rigorous theoretical framework to obtain rate expression of any reaction within reaction class in a simple and cost-effective manner. This opens the possibility for integrating this methodology with Automated Mechanism Generator for ‘on-thefly’ generating accurate kinetic models of complex reacting systems.

Keywords: Reaction Class Transition State Theory (RC-TST), Automated Mechanism Generation, Linear Energy Relationship, H abstraction, H migration, C–C bond β scission


Page 2 of 35

Page 3 of 35

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


1. Introduction Despite the push for alternative and renewable energies due to the recent energy crisis, the demand for hydrocarbon fuels – particularly for transportation – has not dropped since 1970.1 In addition, global warming, in which combustion processes are a large contributor, is becoming a serious threat to Earth sustainability. The quest for better fuel efficiency and lower pollutant emissions for combustion systems is more important than ever, thus the ability to perform detailed simulations of combustion processes would have significant impacts on many industries as well as on environment. Within combustion research, reactor modeling is the key step bridging the molecular science and Computational Fluid Dynamics (CFD) simulations of macroscopic systems, both of which have been recognized as critical enabling technologies in the “2020 Technological Vision for the Chemical Industry”2. An essential component of reactor modeling is the mechanism or kinetic model. The kinetic mechanism/model consists of all or at least important reactions involving chemical species existing in the system with associated thermodynamic and kinetic parameters. The more accurate model can significantly aid the design of better combustion processes. Existing kinetic models for traditional and alternative fuels (namely, hydrocarbons and oxygenated compounds) can in most cases satisfactorily capture the main combustion parameters (e.g., auto-ignition delay times, laminar flame speed, heat release, fuel consumption) with temperature and pressure, and the formation of the major products such as oxides of carbon and ethylene. However, they have been scarcely tested to simulate the formation of minor combustion products, such as high molecular weight alkenes, dienes, aromatic compounds, aldehydes, ketones, alcohols, acids and soot, which can have a deleterious impact on both the environment and human health. Even for simple compounds such as the lower alkanes, it is surprising to see that existing oxidation mechanisms differ substantially in both the number and type of elementary chemical reactions that they include. Moreover, there are also striking differences in the rate constants/coefficients for identical reactions that they utilize. These differences between models are exacerbated when trying to predict the formation of the above-mentioned minor products. This is partly due to an incomplete understanding of possible reaction channels and a lack of accuracy in the rate coefficients. Particularly, a common relevant feature from these kinetic models is the use of the generic rate expression of all reactions in a given class as mentioned above. Such a common engineering approach can yield large errors in thermal rate constants. For an example, variation in the barrier heights for reactions in the OH + alkanes reaction class is in the order of 8 kcal/mol.3, which would yield 3 ACS Paragon Plus Environment


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

the variation in the rate constants on the order 3x103 even at 500 K. This leads to a number of complications. For instance, it is difficult to gauge the accuracy of the kinetic models when applied to different initial conditions. In addition, as the number of kinetic models steadily increases, the need for unifying these models also increases. However, there has been no systematic path that can help to unify these mechanisms since the same reaction may have very different rate expressions in different mechanisms which were tailored to different experimental conditions. There are two key challenges in developing accurate detailed kinetic models. One is its completeness, which depends on how the associated reactions are generated, and the other is the accuracy of their rate constants.

1.1. Mechanism Generation The problem arises from the fact that modeling the oxidation of large molecular-weight fuels like those present in transportation fuels is difficult, since the process goes through a large number (hundreds to thousands) of chemical species and consequently a large number of reactions. For example, there are 3662 reactions involving 470 species considered in simulations of n-hexane combustion by Battin-Leclerc4, 5 and coworkers and 479,206 reactions and 19,052 species in simulations of tetradecane combustion performed by Broadbelt and coworkers6. When modeling such a systems, the use of computer generated systems is no longer just an alternative to manual mechanism construction, but is a necessity. This task is accomplished by specialized, based on the symbolic algebra operations7 in software tools known as Automated Reaction Mechanism Generators (ARMG)6, 8-10. Its main advantage is that the user can characterize an entire complex mechanism in terms of initial species (seed molecules) and a small set of generic molecular transformation rules, called reaction patterns. Expanding these to the full set of reactions is done automatically. Automated model generation is appealing as it can overcome the common flaws in the conventional approach of manually generated reactions. Several ARMG software packages were developed and successfully applied to different chemical systems.8-18

1.2. Rate Constant Generation Currently the estimation of kinetic parameters in models generated by the ARMG’s is mainly obtained through structure/reactivity correlations and analogies with similar reactions leading to a lack of accuracy and certainty in the used data. A better approach is to employ the Evans-Polanyi linear free energy relationship between the activation energies and bond disso4 ACS Paragon Plus Environment

Page 4 of 35

Page 5 of 35

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


ciation energies or heats of reaction of similar reactions to estimate the unknown activation energy. Both approaches are empirical and suffer from significant uncertainties in the estimated constants19. To address the increasing demand for accuracy in kinetic parameters the automated first-principles based such as the Group Additivity (GA) scheme and the ReactionClass Transition State Theory (RC-TST) can be employed. Group Additivity (GA) scheme, initially developed Benson et al.20, was further adopted to the fast rate constants generation purpose by Green and coworkers21-23 (Group Additivity for Transition States – TS-GA). This approach relies on forming new thermochemical groups, called super-groups, corresponding to the reactive moiety in the transition structures of a given reaction class. Such super-groups together with existing thermochemical groups for stable moieties enable the modeler to derive the free energy of activation of any reaction in a given reaction class, and, consequently, its rate constants. The TS-GA method has the advantage that the super-group value can easily be refined to make better generic rate prediction when experimental data become available on even a single reaction belonging to that family. Similar schemes were also applied by Ghent10, 24-27, Kang’s28, 29 and other30-35 groups, which used ab initio group contribution method for activation energies for radical additions. Alternatively, a set of rate rules based on the high level calculations of the model systems, which can be applied to a range of analogous reactions, was proposed by the Dean and coworkers36-43. Systematic, based on the high level CBS-Q, G2, and G4 composite computational methods, study of H migrations in hydrocarbons, including also cyclic species, were reported by Davis et al.44-46 The Reaction Class Transition State Theory (RC-TST) was developed by Truong and co-workers14, 47-53. This theory utilizes the reaction class concept to group similar reactions into classes. The Transition State Theory (TST) is then used as the framework to obtain rate expression. All reactions that have the same common structural denominator, the reactive moiety, form a class. There is no unique definition of a reaction class since it depends on how one defines the common structural denominator and the remaining chemical space. For example, alkanes + H• → alkyl + H2 is a reaction class which is a subset of the H-C(sp3) + H• → •C(sp3) + H2 reaction class. The first class restricts the substituents R1, R2, and R3 to be alkyl groups while no such restriction occurs on the latter. The next question raised is how reactions are classified into classes. Let us consider the following two reactions that are classified in the same class, e.g., hydrogen abstraction by the same radical R, as pictured in Figure 1.


The Journal H of Physical Chemistry

H

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

C

H

H

+ R

C

H

H

H

R1

R1

H

R

Page 6 of 35

products

Transition State

C

R2

H

+

R

C

R2

R3

H

R

products

R3

Figure 1. Examples of reactions belonging to the same reaction class of hydrogen abstraction by R radical at C(sp3) site. It is easily noticed that these two reactions in the same class have the same reactive moiety, which consists of atoms whose nature of chemical bonding was changed during the course of the reaction. Here the reactive moieties are in the shaded boxes and the rest are considered as substituents. Other reactions that have the same reactive moiety and different substituents, for example the reaction of C2H6 + R → CH3CH2 + RH, also belong to this reaction class. Recently, Wang et al. presented a novel interpretation of the RC-TST based on isodesmic reactions, and proposed a simple correction scheme for accurate determinations of the energy barriers from the principle reactions28, 29, 33, 39. This review provides a systematic examination on the performance of RC-TST after it was applied to a number of different reaction classes of different types.

2. Reaction Class Transition State Theory (RC-TST) Within the RC-TST framework, the high pressure limit of the rate constant of an arbitrary reaction (denoted as ka) in a given reaction class is proportional to the rate constant of a reference reaction of that class, kr: k a (T ) = f (T ) × k r (T )

(2)

Usually, the simplest reaction within a given class (called also principal reaction) is used as a reference reaction. However, this is not always the case – for some classes better results are obtained with different choice of the reference reaction50, 51. The key idea of the RC-TST method is to factor f(T) into different components under the TST framework: f (T ) = f σ × f κ (T ) × f Q (T ) × f V (T ) × f HR (T )

(3)

where fσ , fκ , f Q , fV and f HR are the symmetry number, tunneling, partition function, potential energy and hindered rotations factors, respectively. These factors are simply the ratios of the corresponding components in the TST expression for the two reactions:

fσ =

σa σr

(4) 6 ACS Paragon Plus Environment

Page 7 of 35

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


f κ (T ) =

κ a (T ) κ r (T )

(5)

 Q‡ (T )   Q‡ (T )   aR   a‡   Φ (T )   Q (T )  = r  fQ (T ) =  a  Q‡ (T )   ΦR (T )   rR   aR   Φ (T )   Φ (T )   r   r 

(6)

 (∆V ‡ − ∆V ‡ )   ∆∆V ‡  a r fV (T ) = exp  −  = exp  −  k BT    k BT  f HR (T ) =

c HR , a (T )

(7)

(8)

c HR , r (T )

where κ(T) is the transmission coefficient accounting for the quantum mechanical tunneling effects; σ is the reaction symmetry number; Q‡ and Φ R are the total partition functions in which the vibrational partition functions are computed using the RRHO (Rigid Rotor Harmonic Oscilator) approximation (per unit volume) of the transition state and reactants, respectively; ∆V ‡ is the classical reaction barrier height; cHR symbolizes the correction to the total partition function due to the hindered rotation treatment, T is the temperature in Kelvin. Among these, only the symmetry factor is temperature independent and can be easily calculated from the molecular topology of the reactant. Obtaining exact value of four other factors requires structures, energies, and vibrational frequencies of both reactants and transition states for the reaction investigated. This exact methodology is called full (explicit) RC-TST. In fact, it is a reformulation of the ratio of TST rate constants of two reactions in a given class. The key significance of its formulation is in providing a rigorous framework for deriving a more cost effective approach referred to as RC-TST/SAR which utilizes Structure-Activity Relationships to approximate the above reaction class factors rather than explicitly calculated. In the RC-TST/SAR method, transition state structure search for the potential energy factor is not required though some efforts were made in obtaining on-the-fly TS search54 for possible applications in the ARMG schemes. Instead, SAR is employed where the classical reaction barrier height ∆V ‡ for the arbitrary reaction is obtained using the LER (Linear Energy Relationship) approximation, which is similar to the well-known Evans-Polanyi linear free energy relationship,55, 56 between classical barrier heights and reaction energies of reactions in a given reaction class. Other factors are calculated from simplified analytical expressions (RC-TST correlations), derived from using a representative (training) set belonging to 7 ACS Paragon Plus Environment


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

the reaction class. If the training set is chosen careful to cover different types of substituents then these simplified expressions can be used for all reactions in the class having these substituent types.

Since RC-TST formulation is based on the relative rate in which particular

reaction factors capture only changes of a given property (namely barrier height, tunneling, hindered rotations etc.) from those of the reference reaction to any other process within a given family, thus it can take advantage of mutual error cancellations as shown further in this study. Note the symmetry factor fσ,, the only one exactly known, is calculated in the same way (i.e. by dividing symmetry number of a given reaction by the symmetry number of the reference reaction) in all RC-TST variants. It is important to note difference between LER and SAR acronyms used in this study. Whereas the former one means the approximation of the unknown barrier by linear relationship and is used only to estimate the fV (potential energy) factor, the SAR means approximations of all factors (including fV) of the unknown factors with approximate correlations, LER is a special case of SAR. Explicit or full RC-TST is equivalent to full TST calculations of the rate factors with DFT potential energy surface information. In RC-TST/SAR approximation, if LER is used then only reaction energy (not enthalpy) needs to be calculated. Other approximations are discussed in more detailed below. Within the combustion of hydrocarbon fuels, Curran et al.57-60 employed a group of reaction classes for oxidation of n-heptane and isooctane and obtained encouraging agreement with experimental data. For high temperature combustion of hydrocarbons, it is well accepted that the following reaction 9 types are important: 1. Unimolecular hydrocarbon decomposition (RR’ → R• + R’•; RH → R• + H•) 2. Hydrogen atom abstraction from the fuel (RH + X• → R• + XH where X=H•, O•, •OH, •CH3, O2, •C2H5, •C2H3) 3. Alkyl radical decomposition via β-scission 4. Alkyl radical + O2 → olefin + HOO• 5. Alkyl radical isomerization via H-shift 6. Olefin abstraction reactions by •OH, H•, O•, •CH3 7. Olefin addition reactions by •OH, H•, O•, •CH3 8. Alkenyl radical decomposition 9. Olefin decomposition


Page 8 of 35

Page 9 of 35

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


So far, the RC-TST theory was applied to nineteen reaction classes, which are detailed in Table 1 and tested against both experimental and theoretical data available in the NIST database61. Using the methodology of Curran et al.58, those classes can be though as belong to the reaction type pattern defined in their study. For example, patterns corresponding to the metatheses involving alkane, as belonging to Curran reaction type 2 (H abstraction from the fuel). In Table 1, the second column represents the index of Curran pattern60, for which the reaction class on column 1 belongs.

2.1. Reference Reaction As mentioned above, the RC-TST rate for any reaction in the given class can be thought as an extrapolation from the rate of the reference reaction. For that reason, the quality of the thermal rate constants used for the reference reaction is important for the overall accuracy of the RCTST method for a particular reaction class. In our first few applications of RC-TST47, we suggested the use of the smallest reaction, i.e. the principal reaction of the class to be the reference reaction since its rate constants can be calculated accurately from first principles or are often known experimentally. However, we found later that the use of the smallest reaction as the reference reaction is not always the best choice. For example, for the OH + alkane reaction class, we found that the hydrogen abstraction by hydroxyl radical at ethane, OH + C2H6, is the better reference reaction than the principal OH + CH43. In particular, based on our analyses the use of the OH + C2H6 reaction gives better RC-TST factors than the OH + CH4 reaction, especially for vibrational partition function factors. In fact, the OH + C2H6 process is the smallest reactions that have all elements of this class, namely the reactive moiety and the C–C bond of the alkyl group. Similar results were also found for some others metatheses reaction classes62-64 (see Table 1). It is interesting to observe that, for families corresponding to radical processes (i.e. intramolecular H migrations and C-C bond β scissions), the simplest reaction within a given class (principal reaction) is the best reference reaction however (see Table 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

Page 10 of 35

Table 1. Detailed list of reaction classes for which RC-TST formulas are available. In the table, PAH means Polycyclic Aromatic Hydrocarbon and PAR = Polycyclic Aromatic Radical. Reaction class

Curran pattern

Reference Reaction

Electronic theory level

Rate calculation method

1

R-CH3 + H = R-CH2• + H247

2

CH4 + H• = CH3• + H2

PMP4/cc-pVTZ//BH&HLYP/cc-pVDZ

CVT/SCT/HR47

2

R-CH3 + OH = R-CH2• + H2O3

2

C2H6 + OH =C2H5• + H2O

—

Experimental65

PMP4/cc-pVTZ//BH&HLYP/cc-pVDZ

CVT/SCT/HR66

CCSD(T)/cc-pVDZ//BH&HLYP/cc-pVDZ

CVT/SCT/HR67

3 4

5

6

7

8 9 10

R-CH3 + CH3 = R-CH2• + CH4

66

R-CH3 + C2H5• = R-CH2• + C2H667 R-CH3 + O = R-CH2• + OH63 R-CH3 + CHO = R-CH2• + 62

HCHO

R-CH3 + C2H3 = R-CH2• + C2H464 R-CH3 + Cl = R-CH2• + HCl70 PAH + H = PAR +

H272

R-CH=CH2 + H = R-CH=CH•

2 2

2

2

2

C2H6 + CH3• = C2H5• + CH4 C2H6 + C2H5• = C2H5• + C2H6 C2H6 + •O• = C2H5• + •OH C2H6 + •CHO = C2H5• + HCHO

BH&HLYP/cc-pVDZ

CVT/SCT/HR with Large Curvature Tunneling correction63, 68

CCSD(T)/cc-pVTZ//BH&HLYP/cc-pVDZ

CVT/SCT/HR62

G269

CVT/SCT/HR69

C2H6 + C2H3• = C2H5• + C2H4 C2H6 + Cl=

CCSD(T)/cc-pVnZ extrapolated to CBS

C2H5• + HCl

limit (D,T)

—

C6H6 + H• = C6H5• + H2

BH&HLYP/6-31G(d,p)//B3LYP/6-31G(d,p)

TST/Eckart72

6

C2H4 + H• = C2H3• + H2

CCSD(T)/cc-pVTZ//BH&HLYP/cc-pVDZ73

CVT/SCT/HR73

2

10

ACS Paragon Plus Environment

µVT/SCT/HR71

Page 11 of 35

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


+ H273

11

12*

R-CH=CH2 +OH = R-CH=CH• 74

+ H2O

CH2=CH-R + OH = CH2=CH-R• + H2O

33

6

6

13

R-OH + H = R•-OH + H251

2

14

R-OH + H = R-O• + H250

2

15* 16 17 18 19

H(C=O)O-R + H = H(C=O)OR• + H2

29

1,4 intramolecular H migrations in alkyl radicals48, 53 1,5 intramolecular H migrations 53, 76

in alkyl radicals

1,6 intramolecular H migrations in alkyl radicals53, 77 Beta scission of the C-C bond in alkyl radicals52, 78

2

C2H4 + OH• = C2H3• + H2O CH3=CH-CH3 + OH = CH3=CH-CH2• + H2O CH3CH2OH + H = CH3CH•OH + H2 CH3CH2OH + H = CH3CH2O• + H2 H(C=O)O-CH3 + H = H(C=O)O-CH2• + H2

CCSD(T)/cc-pVTZ//BH&HLYP/cc-pVTZ

CVT/SCT/HR74

CCSD(T)/CBS

TST/Eckart33

G2M(rcc,MP2)//B3LYP/6-311+G(d,p)75

CVT/SCT75

G2M(rcc,MP2)//B3LYP/6-311+G(d,p)75

CVT/SCT75

CCSD(T)/CBS29

TST29

5

C4H9• = C4H9•


CVT/SCT/HR53

5

C4H11• = C4H11•


CVT/SCT/HR53

5

C6H13• = C6H13•


CVT/SCT/HR53

3

C3H7• = C2H4 + CH3•


CVT/SCT/HR52

*

Results reported by B. Y. Wang et al.33 and Q.Y. Wang et. al.29 with modified RC-TST schemes

11

ACS Paragon Plus Environment


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

In the Table 1, first fifteen classes (1-15) are molecular metatheses reactions (hydrogen abstraction from alkanes (1-8), polycyclic aromatic hydrocarbons (PAH’s) (9), olefins (1012), alcohols (13-14) and esters (15). Classes 16-19 are unimolecular radical processes – intermolecular H migrations (isomerization) and degradation via C-C bond β-scission. In kinetics, it is crucial to have accurate energetic information (barrier height and reaction energy) for chemical reactions in order to get accurate absolute rate constants. For example, under the TST framework, an error of 2.0 kcal/mol in the reaction barrier height gives an uncertainty factor of 29 to the thermal rate constant at room temperature. Thus, there is a need of calculating the barrier height within the uncertainty of 2 kcal/mol. In terms of firstprinciple calculations, this can be achieved by using high levels of correlated molecular orbital theory (MO) and sufficiently large basis sets containing high angular momentum functions. Within the RC-TST framework, only the relative barrier heights are needed, however. Our previous studies (references listed in Table 1) have shown that, in the most of the cases, these relative barrier heights can be accurately predicted by the BH&HLYP method. Moreover, we have shown that the BH&HLYP functional is far superior to B3LYP with the same basis set to predict the potential barrier height62, and also still performs better in some cases in comparison67 with the newer DFT Mxx family of functionals79, designed especially for chemical kinetics. The rate of the reference reaction could be obtained either from state of the art theoretical calculation or from the high quality experimental data. As one may see from the Table 1 most of the rates for the reference reactions, used in our studies, were calculated theoretically on the CVT/SCT/HR level, where CVT stands for canonical variational transition state, SCT for centrifugal-dominant small-curvature multidimensional tunneling, and HR for hindered rotor. These results were obtained employing the TheRate80 and POLYRATE81 kinetic packages. In these calculations, overall rotations were treated classically and vibrations were treated quantum-mechanically within the harmonic approximation, except for the modes corresponding to the internal rotations of –CH3 groups around the C–C bond at the reactants, TS, and products, which were treated as hindered rotations using the approach suggested by Ayala and Schlegel82. For the β-scission reaction class also rotations of the ethyl groups were taken into account (see Section 2.2.3). This formalism optimizes the accuracy of treating a single rotor to minimize the compound errors in the case of multiple internal rotors.


Page 12 of 35

Page 13 of 35

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


2.2 Reaction Class Parameters In this section, we review on how the RC-TST factors can be derived using the representative reaction set. The goal here is to establish some accuracy matrix for different factors based results obtained from the reaction classes that had already been studied.

2.2.1. Potential energy factor The potential energy factor can be calculated using eq. (7), where ∆Va≠ and ∆Vr≠ are the barrier heights of the arbitrary and reference reactions, respectively. The potential-energy factor can be calculated using the reaction barrier heights of the arbitrary reaction and the reference reaction. The RC-TST/SAR method uses the linear energy relationship (LER) similar to the well-known Evans Polanyi linear free-energy relationship between classical barrier heights and reaction energies of reactions to estimate reaction barriers. Furthermore, this variant of the RC-TST method uses averaged values of particular factors (except for symmetry factor) rather than exact values defined by Eqs 5−8. As a consequence, RC-TST/SAR rates constants are estimated using only the reaction energy and reactants topology information; no transition state and frequency calculation are needed. This feature makes the RC-TST/SAR method applicable to the different automated mechanisms generation (ARMG) schemes. Note that the reaction energies needed to derive LER can be obtained from different methods, which are not necessarily the same as for the barrier heights. In fact, the AM1 semiempirical method was also employed to illustrate this point. It is worth noting that: •

AM1 method is used only to calculate the reaction energy ∆E, not the classical barrier height ∆V#. Both kinds of LER equations approximate the BH&HLYP calculated barrier heights.

•

AM1 reaction energy is only used to extract accurate barrier height from the LER’s, it does not directly involved in any rate calculations.

Alternatively, it is possible to approximate all reactions at the same type of carbon atom site as having the same barrier height, namely the average value such as H abstraction at all tertiary carbon sites would have the same barrier height. This approximation is referred to as the Barrier Height Grouping (BHG) approximation. Whereas the RC-TST/SAR method needs the reaction energy info, the BHG approach does not require any additional calculation, thus making its implementation fast and easy. The performance of LER and BHG methods for different reaction types are detailed in the Table 2. For a particular reaction type, the mean absolute deviations (MAD’s) of the fitted barrier heights calculated with the LER or BHG approximation from those explicitly calculated are shown in Figure 2. Both in Table and Fig13 ACS Paragon Plus Environment


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 14 of 35

ure 2, MAD is calculated as an absolute difference between the exact (i.e. BH&HLYP/ccpVDZ) barrier of particular reaction and that estimated from the LER or BHG approximation.

Table 2. Mean absolute deviations (MAD’s) of barrier heights obtained with the LER or BHG approximation to the BH&HLYP/cc-pVDZ barriers for different reaction classes (in kcal/mol). Values in parentheses are for cyclic species. Reaction class 3

R-CH3 + OH = R-CH2• + H2O R-CH3 + CH3 = R-CH2• + CH466

LER (BH&HLYP/ccpVDZ) 0.12 0.21

LER (AM1) 0.22 0.13

0.20 0.09 (0.27) 0.19 0.11 0.21 0.33 0.28

0.27 0.24 0.13 0.12 0.45 0.43 —

0.23 0.15 0.09 0.24 0.26 0.79 1.45

0.24

—

1.13

0.07 0.17 (0.44)

0.12 0.26

0.18 0.28

R-CH3 + C2H5• = R-CH2• + C2H667 C2H464

R-CH3 + C2H3 = R-CH2• + R-CH3 + CHO = R-CH2• + HCHO62 R-CH=CH2 + H = R-CH=CH• + H273 1-4 H migration 1-5 H migration 1-6 H migration Beta scission of the C-C bond in alkyl radicals52, 78 R-OH + H = R-O• + H250 R-OH + H = R•-OH + H251 .


BHG — —

Page 15 of 35

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


Figure 2. Mean absolute deviations (MAD’s) of barrier heights obtained with the LER or BHG approximation to the BH&HLYP/cc-pVDZ barriers for different reaction classes (in kcal/mol). As from Table 2 and Figure 2, the performance of the LER approximation depends on the types of reaction class. However, MAD errors are less than 0.5 kcal/mol in all cases. Generally, LER’s based on BH&HLYP energies are more accurate than those based on AM1 reaction energies though the differences are only noticeable in the H-migration reaction type. BHG approximation as expected yields larger error compared to the LER approach. The differences are small for the H-abstraction reaction type but are larger for the H-migration and βscission reaction types, which are radical reactions. The largest MAD error of 1.45 kcal/mol is found for the 1,6 H-migration class. The LER equations may be dependent on the topology of the reactants/product. For example, there are two kinds of LER equations for the R-CH3 + H = R-CH2• + H247 family, namely for alkanes and for alkenes, with a double bond in R. Another kind of topological issue appears for oxygenated compounds, namely classes 2 and 5 from Table 1. For these families, LER for specific reaction depends on the existence of so-called beta carbons electronic effect. This effect is referred to as β-radical stabilization; that is, the existence of alkyl groups at the β-positions from the radical position of the product alkyl would lead to larger stabilization of the transition state and thus lower the barrier for hydrogen abstraction3, 63. However, it should be noted, that topology dependent LER is an exception rather than a rule. Most of the reaction classes investigated so far do not have noticeable topology dependence in LER’s. Extendibility of the LER approach As described above, simple LER’s between barrier heights and reaction energies work well for simple H-abstractions, the performance decreases for reaction types that are more complicated. For instance, when applied to cyclic species (for example, classes 4, 7 and 13 from Table 1), only DFT-based LER’s were obtained. AM1-based LER’s and BHG are not applicable, however. In addition, as shown in ref.51, there are noticeable differences in reaction barriers for reactions at the α carbon compared to those at other carbon sites such as β, γ, and δ. For these reasons, different LER’s were derived, for α and β, γ, δ H abstraction sites, respectively. Furthermore, other form of LER’s than simple dependence on the reaction energy can also employed.

For instance, for O-addition reaction to aromatic hydrocarbons (benzene,

naphthalene, phenanthrene and pyrene) a linear relationship between absolute hardness and 15 ACS Paragon Plus Environment


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

barrier heights has been found83. Interestingly, it was shown, that the barrier heights do not show a strong linear correlation with its reaction energies for this reaction family. GomesBalderas et al. have also shown that for certain types of radical additions to double bonds, the barrier does not correlate well with the heat of reaction, but with other properties of the reactants84, which was partially validated by experiments85.

2.2.2. Partition Function Factor The partition factor is the product of the translational, rotational, internal rotation, vibrational, and electronic component. To have a good understanding of the properties of this factor, we illustrated eq. 6 in Figure 3 for an arbitrary reaction that has three substituents (R1, R2, and R3) with respect to the principal reaction. Because the total partition function is a product of translational, rotational, electronic, and vibrational partition functions and due to the specific forms of the translational and rotational partition functions, the temperature dependence of translational and rotational components of partition functions is canceled in fQ. Furthermore, because most gas-phase reactions take place in the electronic ground state, the electronic partition functions do not contribute to fQ. Thus, the temperature dependence of fQ comes solely from the vibrational component. Furthermore, it was shown (see, for example, references48,

49, 67, 73, 76, 77

), that the contributions to the vibrational partition functions from the

principal components of the reactive moiety and of the substituents are canceled. Therefore, the main factors that govern the temperature dependence of the fQ factor are the differences in the vibrational frequencies due to the coupling of substituents with the reactive moiety. Thus, even in flexible molecules where it is difficult to calculate accurately the partition functions of the low-frequency vibrational modes such as hindered rotations, many of the vibrational partition functions effectively cancel out in the calculation of fQ.

Figure 3. Schematic illustration of the factor of partition function fQ. Brackets denote the vibrational partition functions of the species inside. The left-hand side ratio is from the transition states having the same reactive moiety C-H-H, and the other ratio is from the reactants having the same reactive


Page 16 of 35

Page 17 of 35

moiety component C-H (superscript ǂ refers to TS and R to reactants (superscripts) , subscript r refers to the reference reaction whereas a to arbitrary reaction within a given family).

Figure 4 shows the percent mean absolute deviations (MAD’s) of explicit values of fQ (eq.6) and its SAR (i.e. value averaged over the representative set of reactions) approximation (i.e. for different reaction types at 300, 500, 1000 and 2000 K.

MAD (%)

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


35 30 25 20 15 10 5 0

1000K 300K

300K

500K

1000K

2000K

Figure 4. Percent mean absolute deviations resulted from approximation of the partition function factor by simplified expressions at T=300, 500, 1000 and 2000 K for different reaction types.

The results show that the error in the partition function factor is nearly independent of the temperature. In all cases, i.e. all reaction types, this error is less than 30%. It was found smaller for H-abstraction reaction type and largest in the H-migration type.

2.2.3. Hindered rotation factor For some reactions, contributions from some low frequency motions need to be treated as hindered (internal) rotors, as described in Section 3.1. For simplicity, the influence of these hindered rotation factors can be ignored in the RC-TST approach. If the hindered rotations deem to be important, its effects can be explicitly considered in a factor, called Hindered Rotation Factor (fHR). This reaction class factor is a measure of the substituent effects on the rate constants from the hindered rotors, relative to that of the reference reaction. Results from applications to different reaction classes show that relative contribution of hindered rotations from alkyl groups larger than –CH3 is small due to the cancellation of the treatment within the 17 ACS Paragon Plus Environment


RC-TST framework3, 48-51, 62, 63, 73, 74 and thus can be effectively ignored. However, for some reactions, especially those with cyclic transition state structures such as H-shift reactions, the hindered rotor effects may be larger than those suggested by the averaged value of the fHR factor. Such situations can be properly treated by the full (explicit) version of the RC-TST. However, further study is certainly needed on this issue.

MAD (%)

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 18 of 35

14 12 10 8 6 4 2 0

1000K 300K

300K

500K

1000K

2000K

Figure 5. Percent mean absolute deviations (MAD’s) resulted from approximation of the hindered rotations (HR) factor by simplified expressions for T=300, 500, 1000 and 2000K for different reaction types.

Figure 5 shows the percent mean absolute deviation (i.e. difference between eq.8 and averaged value over a representative set of reactions) of the fHR factors for different reaction types plotted at 300, 500, 1000 and 2000 K. It can be seen that this error is rather small, i.e. less than 10% for most reaction types except for the β scission reaction type where it is HCHO + Alkyl Reaction Class: An Application of the Reaction Class Transition State Theory. Theor. Chem. Acc. 2007, 120, 107-118.


Page 32 of 35

Page 33 of 35

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


63. Huynh, L. K.; Zhang, S.; Truong, T. N. Kinetic of Hydrogen Abstraction O(3P) + Alkane → OH + Alkyl Radical Reaction Class: An Application of the Reaction Class Transition State Theory. Combust. Flame 2008, 152, 177-185. 64. Chase, M. W., Jr. NIST-JANAF Themochemical Tables, Fourth Edition. J. Phys. Chem. Ref. Data, Monograph 9 1998, 1-1951. 65. Baulch, D. L.; Cobos, C. J.; Cox, R. A.; Esser, C.; Frank, P.; Just, T.; Kerr, J. A.; Pilling, M. J.; Troe, J.; Walker, R. W. et al. Evaluated Kinetic Data for Combustion Modelling. J. Phys. Chem. Ref. Data 1992, 21, 411-734. 66. Kungwan, N.; Truong, 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. 67. Ratkiewicz, A.; Huynh, L. K.; Pham, Q. B.; Truong, T. N. Kinetics of the Hydrogen Abstraction ·C2H5 + Alkane → C2H6 + Alkyl Reaction Class: An Application of the Reaction Class Transition State Theory. Theor. Chem. Acc. 2013, 132, 1-17. 68. Corchodo, J. C.; Espinosa-Garcia, J.; Roberto-Neto, O.; Chuang, Y. Y.; Truhlar, D. G. J. Phys. Chem. A 1998, 102, 4899–4910. 69. Liu, G.-x.; Li, Z.-s.; Xiao, J.-f.; Liu, J.-y.; Fu, Q.; Huang, X.-R.; Sun, C.-C.; Tang, A.C. Calculations of the Rate Constants for the Hydrogen Abstraction Reactions C2H3+CH4→C2H4+CH3 and C2H3+C2H6→C2H4+C2H5. ChemPhysChem 2002, 3, 625-629. 70. Piansawan, T.; Kungwan, N.; Jungsuttiwong, S. Application of the Reaction Class Transition State Theory to the Kinetics of Hydrogen Abstraction Reactions of Alkanes by Atomic Chlorine. Comp. Theor. Chem. 2013, 1011, 65-74. 71. Fernández-Ramos, A.; Martı́nez-Núñez, E.; Marques, J. M. C.; Vázquez, S. A. Dynamics Calculations for the Cl+C2H6 Abstraction Reaction: Thermal Rate Constants and Kinetic Isotope Effects. J. Chem. Phys. 2003, 118, 6280-6288. 72. Violi, A.; Truong, T. N.; Sarofim, A. F. Kinetics of Hydrogen Abstraction Reactions from Polycyclic Aromatic Hydrocarbons by H Atoms. J. Phys. Chem. A 2004, 108, 48464852. 73. Huynh, L. K.; Panasewicz, S.; Ratkiewicz, A.; Truong, T. N. Ab Initio Study on the Kinetics of Hydrogen Abstraction for the H + Alkene -> H2+ Alkenyl Reaction Class. J. Phys. Chem. A 2007, 111, 2035-2252. 74. Huynh, L. K.; Barriger, K.; Violi, A. Kinetics Study of the OH+ Alkene --> H2O + Alkenyl Reaction Class. J. Phys. Chem. A 2008, 112, 1436-1444. 75. Park, J.; Zhu, R. S.; Lin, M. C. Thermal Decomposition of Ethanol. I. Ab Initio Molecular Orbital/Rice–Ramsperger–Kassel–Marcus Prediction of Rate Constant and Product Branching Ratios. J. Chem. Phys. 2002, 117, 3224-3231. 76. Ratkiewicz, A.; Bankiewicz, B. Kinetics of 1,5-Hydrogen Migration in Alkyl Radical Reaction Class. J. Phys. Chem. A 2012, 116, 242-254. 77. Ratkiewicz, A. Kinetics of 1,6-Hydrogen Migration in Alkyl Radical Reaction Class. React. Kinet. Mech. Catal. 2013, 108, 545-564. 78. Ratkiewicz, A.; Truong, T. N. Kinetics of the C-C Bond Beta Scission Reactions in Alkyl Radical Reaction Class. J. Phys. Chem. A 2012, 116, 6643-6654. 79. Zhao, Y.; Truhlar, D. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theor. Chem. Acc. 2008, 120, 215-241. 80. Duncan, W. T.; Bell, R. L.; Truong, T. N. Therate: Program for Ab Initio Direct Dynamics Calculations of Thermal and Vibrational-State-Selected Rate Constants. J. Comp. Chem. 1998, 19, 1039-1052.



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

81. Zheng, J.; Zhang, S.; Lynch, B. J.; Corchado, J. C.; Chuang, Y.-Y.; Fast, P. L.; Hu, W.-P.; Liu, Y.-P.; Lynch, G. C.; Nguyen, K. A. et al. POLYRATE 2010-A: Computer Program for the Calculation of Chemical Reaction Rates for Polyatomics 2010. 82. Ayala, P. Y.; Schlegel, H. B. Identification and Treatment of Internal Rotation in Normal Mode Vibrational Analysis J. Chem. Phys. 1998, 108, 2314-2325. 83. Orrego, J. F.; Truong, T. N.; Mondragon, F. A Linear Energy Relationship between Activation Energy and Absolute Hardness: A Case Study with the O(3P) Atom-Addition Reaction to Polyaromatic Hydrocarbons. J. Phys. Chem. Lett. 2008, 112, 8205-8207. 84. Gómez-Balderas, R.; Coote, M. L.; Henry, D. J.; Radom, L. Reliable Theoretical Procedures for Calculating the Rate of Methyl Radical Addition to Carbon−Carbon Double and Triple Bonds. J. Phys. Chem. A 2004, 108, 2874-2883. 85. Goldsmith, C. F.; Ismail, H.; Green, W. H. Pressure and Temperature Dependence of the Reaction of Vinyl Radical with Alkenes III: Measured Rates and Predicted Product Distributions for Vinyl + Butene. J. Phys. Chem. A 2009, 113, 13357-13371. 86. 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. 87. Tsang, W.; McGivern, W. S.; Manion, J. A. Multichannel Decomposition and Isomerization of Octyl Radicals. Proc. Combust. Inst. 2009, 32, 131-138. 88. Tsang, W.; Walker, J. A.; Manion, J. A. The Decomposition of Normal Hexyl Radicals. Proc. Combust. Inst. 2007, 31, 141-148. 89. McGivern, W. S.; Awan, I. A.; Tsang, W.; Manion, J. A. Isomerization and Decomposition Reactions in the Pyrolysis of Branched Hydrocarbons: 4-Methyl-1-Pentyl Radical. J. Phys. Chem. A 2008, 112, 6908-6917. 90. Ratkiewicz, A. First-Principles Kinetics of N-Octyl Radicals. Prog. React. Kinet. Mech. 2013, 34, 323-341. 91. Sarathy, S. M.; Vranckx, S.; Yasunaga, K.; Mehl, M.; Oßwald, P.; Metcalfe, W. K.; Westbrook, C. K.; Pitz, W. J.; Kohse-Höinghaus, K.; Fernandes, R. X. et al. A Comprehensive Chemical Kinetic Combustion Model for the Four Butanol Isomers. Combust. Flame 2012, 159, 2028-2055. 92. Kim, Y.; Choi, S.; Kim, W. Y. Efficient Basin-Hopping Sampling of Reaction Intermediates through Molecular Fragmentation and Graph Theory. J. Chem. Theor. Comp. 2014, 10, 2419-2426. 93. Ratkiewicz, A. A Generic Interface Between Automated Mechanism Generator and Quantum Chemistry Based Tools for Obtaining Rates Constants. In COST Action CM901: Detailed Chemical Kinetic Models for Cleaner Combustion: 1st Annual meeting, ENSIC: Nancy, 2010, 28-29. 94. Broadbelt, L .J.; Stark, S. M.; Klein, M. T. Computer Generated Pyrolysis Modeling: Onthe-Fly Generation of Species, Reactions, and Rates Ind. Eng. Chem. Res. 1994, 33, 790–799 95. Magoon, G. R.; Green, W. H. Design And Implementation of a Next-Generation Software Interface For On-The-Fly Quantum and Force Field Calculations in Automated Reaction Mechanism Generation, Comp. Chem. Eng. 2013, 52, 35-45 96. Paraskevas, P. D.; Sabbe, M. K.; Reyniers, M.-F.; Papayannakos, N. G.; Marin, G. B. Group Additive Kinetic Modeling for Carbon-centered Radical Addition to Oxygenates and β-Scission of Oxygenates, AIChe J. 2016 in press, doi:10.1002/aic.15139


Page 34 of 35

Page 35 of 35

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


Graphical Abstract (TOC)


Performance of First-Principles-Based Reaction Class Transition State

Recommend Documents