Extensive Theoretical Study of the Thermochemical Properties of

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An Extensive Theoretical Study of the Thermochemical Properties of Unsaturated Hydrocarbons, Allylic and Super-Allylic Radicals: The Development and Optimization of Group Additivity Values Yang Li, and Henry J. Curran J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b02912 • Publication Date (Web): 10 May 2018 Downloaded from http://pubs.acs.org on May 11, 2018

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The Journal of Physical Chemistry

An Extensive Theoretical Study of the Thermochemical Properties of Unsaturated Hydrocarbons, Allylic and Super-Allylic Radicals: The Development and Optimization of Group Additivity Values Yang Li∗, Henry J. Curran Combustion Chemistry Centre, School of Chemistry & Ryan Institute, National University of Ireland, Galway, Ireland Abstract In this study, the thermochemistry of C2 – C7 unsaturated hydrocarbons (22 alkene and 6 diene molecules), 16 allylic and 5 super-allylic radicals are determined using high-accuracy quantum chemistry calculations. In addition, the group additivity values (GAVs) of a total of 19 relevant groups are systematically optimized based on the calculated thermochemistry of species clusters. The M06-2X method using the 6-311++G(d,p) basis set is used for the geometry optimizations, vibrational frequency calculations and for the internal rotation scans for lower frequency modes. The composite compound methods: CBS-APNO, G3, and G4 are utilized to derive the average atomization formation enthalpies. The entropy and temperature-dependent heat capacity values of all species are calculated using statistical thermodynamics in MultiWell. These results are in good agreement with literature data. A GAVs optimization is performed based on a statistical analysis: a Bland-Altman plot, which is employed to visualize the agreement between the results from the quantum chemical calculations and the GA method. It is found that the 298 K entropies of the CD/C2, C/CD2/H2, C/C/CD2/H and C/CD3/H groups disagree by more than 5 cal K–1 mol–1 compared to existing values, while the values for the ALLYLS and ALLYLT radical groups also differ by ~ 2.4 and 4.1 cal K–1 mol–1, respectively. The 298 K formation enthalpies of the C/CD2/H2, C/C/CD2/H, C/CD3/H and ALLYLT groups are modified by more than 1 kcal mol–1, compared to existing values. The updated GAVs can be used with increased confidence to estimate the thermochemical properties of combustion relevant unsaturated hydrocarbon molecules and their radicals which are critical for the development of accurate chemical kinetic models describing the pyrolysis and oxidation of hydrocarbon and oxygenated hydrocarbon fuels.

∗

Corresponding author: [email protected] 1

ACS Paragon Plus Environment

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1. Introduction The accurate determination of thermochemical properties of combustion-relevant species plays a significant role in chemical kinetic model development. Typically, in a kinetic model, rate constants of elementary reactions are included in one direction only, with the rate constants in the alternate direction being calculated via the equilibrium constant determined from the thermodynamic properties of the reactants and products involved in the reaction. There are a number of common databases available in the literature that provide thermodynamic properties of a large number of compounds. These include NIST1, Baulch et al.2, PrIMe3, the Active Thermochemical Tables (ATcT)4-6, the Third Millennium Thermodynamic Database (TMTD)7, and the seminal text “Thermochemical Data of Organic Compounds” by Pedley et al.8 Most recently, Goldsmith et al.9 calculated the thermochemistry for 219 small molecules including many radical, biradical,

and

triplet species

using

high-level

quantum chemistry.

The

RQCISD(T)/cc-

PV∞QZ//B3LYP/6-311++G(d,p) method was used to compute the electronic energies, and a bond additivity correction for this method was developed to remove systematic errors in the enthalpy calculations, using values in the ATcT as a reference. The group additivity (GA) method was developed by Benson10, and can be used to estimate the thermochemical properties of a molecule including the enthalpy of formation, entropy and heat capacities as function of temperature. Ritter and Bozzelli developed THERM (THermo Estimation for Radicals and Molecules)11 based on Benson’s GA method, which can be used as illustrated in Figure 1. Bozzelli and co-workers12-27 have also studied the thermochemistry of different classes of species including sulphurated, nitridated, fluorinated, oxygenated and saturated hydrocarbons, Table 1. Enthalpies of formation, entropies and heat capacities have been compiled for certain class of species using density functional theory (DFT) and composite compound methods, from which bond dissociation energies (BDEs) of similar type of bonds and GAVs were systematically reported and compared. Group Contribution Database

Molecule/Radical Description THERMAKE.xlsm

THERM Thermo Property List File

THERMLST

Molecule/Radical Description File

THERMFIT

Reaction Description

NASA Format Polynomial File

THERMRXN

Input to Kinetic Model

Thermodynamic Analysis for Reactions

Figure 1. Functional diagram for the THERM ensemble of programs as developed by Ritter and Bozzelli. 2


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Table 1. Previous thermochemistry studies published by Bozzelli and co-workers since 2010. No.

Class of species

Reference

1

Methyl Sulfinic Acid, Methyl Sulfinic Acid Ester and Their Radicals

Gunturu et al.12

2

Nitrocarbonyls, Nitroolefins, Corresponding Nitrites, and Their Carbon Centered Radicals

Snitsiriwat et al.13

3

Disulfide Oxygen (S-S-O)-Bridged CH3SSOH and CH3SS(=O)H and Radicals

Pillai et al.14

4

C1 – C4 Fluorinated Hydrocarbons

Wang et al.15

5

C3–C5 Cycloalkyl Hydroperoxides and Peroxy Radicals

Auzmendi-Murua et al.16

6

Methyl-Substituted Cyclic Alkyl Ethers and Radicals for Oxiranes, Oxetanes, and Oxolanes


7

Ketones

Hudzik et al.18

8

3- to 5-Member Ring Cyclic Ether Hydroperoxides, Alcohols, and Peroxy Radicals


9

RC(=O)–OOH, RC(=O)O–OH, R(C=O)OO–H, RC–OOH, RCO–OH and RCOO–H (R is phenyl, vinyl and alkyl groups)

Sebbar et al.20

10

Hydroxycyclohexadienyl Peroxy Isomers

Chen et al.21

11

C1 – C4 Hydroperoxides and Peroxy Radicals

Wang et al.22

12

Alcohols, Hydroperoxides, and Vinylic, Alkoxy, and Alkylperoxy Radicals

Yommee et al.23

13

Hydroxyl and Hydroperoxide Substituted Furan, Methylfuran, and Methoxyfuran

Hudzik et al.24

14

Isooctane and Carbon Radicals

Snitsiriwat et al.25

15

Thermochemistry of C7H16 to C10H22 Alkane Isomers

Hudzik et al.26

16

Ring-Opened Diradicals and Carbenes of exo-Tricyclo[5.2.1.02,6]decane

Hudzik et al.27

Burke et al.28 compiled and evaluated literature values for enthalpies of formation and molar entropies for C1 – C4 alkanes, alkenes, alcohols, hydroperoxides, and their associated radicals. Based on this compilation, the GAVs of a significant number of relevant groups present in THERM were updated hierarchically. With these updated group values, Bugler et al.29 illustrated the influence of this thermochemistry on important low-temperature reaction classes for n-pentane oxidation, which led to a significant decrease in reactivity (ignition delay time predicts are ~ an order of magnitude slower at 800 K). This clearly indicates the importance of thermochemical parameters for the critical species included in oxidation mechanisms.

3



In recently developed propene30-31, isobutene32, 2-butene33, 1-butene34 and 1,3-butadiene35 oxidation mechanisms allylic radicals (allyl, 2-methylallyl and 2-propenyl,1-methyl) were found to be the most important intermediate species influencing the prediction of ignition delay times, species profiles as a function of time/temperature and flame speed experimental results. In determining the thermochemistry of 2-propenyl,1-methyl (Ċ4H71-3) radical, values from three different resources were employed for comparison, namely (i) the Third Millennium Thermodynamic Database (TMTD)7 (in black), (ii) the Group Additivity (GA) method (in red) and (iii) Quantum Chemistry (QC) calculations (in blue), Figure 2. The theory employed in the QC calculations will be discussed in the Computational Methods Section later. In Figure 2, the solid lines and dotted lines are the enthalpy of formation and heat capacities at constant pressure, respectively, which show good agreement among the three sources. However, a discrepancy of 3 cal K–1 mol–1 in entropy (dashed lines) is observed throughout the temperature range shown, with the result from the GA method being consistently lower than that determined using the TMTD and QC calculations.

140 160

S TMTD GA method QC calculation

-1

140

100

∆ fH

80

TMTD GA method QC calculation

60 Cp

40

TMTD GA method QC calculation

20 400

800

1200

1600

2000

80 60 40

-1

120

100

Cp & S / cal K mol

-1

120

∆ fH / kcal mol

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 4 of 35

20

2400

T/K

Figure 2. Thermochemistry comparison of 2-propenyl,1-methyl radical (298.15 – 2500 K). Solid line: enthalpy of formation, dash line: entropy, dot line: heat capacities at constant pressure. Black: Third Millennium Thermodynamic Database (TMTD)7, red: Group Additivity (GA) method, blue: Quantum Chemistry (QC) calculation. Figure 3 shows the effect of this difference in entropy on calculated ignition delay times (IDTs) for 2-butene in the temperature range of ~ 700 – 1400 K, where the solid line represents the model predictions using the QC entropy value, while the dashed line represents the predictions using the values generated from THERM using the GA method. The difference in IDT predictions at 900 K is about an order of magnitude. Using the higher entropy value form the QC calculations, the 4


Page 5 of 35

equilibrium shifts to form this radical which promotes reactivity, resulting in shorter predicted IDTs and better agreement with the experimental data. Further investigation found that, using the GA method, there are two different ways in which the thermochemistry of 2-propenyl,1-methyl radical can be calculated, as shown in Figure 4 and Table 2. Note that, the GAVs listed in Table 2 are current values. It can be formed either using 1-butene as the parent and “ALLYLS” as the radical group or by using 2-butene as the parent and “ALLYLP” radical group. In THERM the GAV for the “ALLYLP” group was recently updated by Burke et al.

28

, but no data existed to update the

“ALLYLS” group and thus its reliability is questionable.

T/K 14001300 1200 1100

1000

900

800

2-Butene Ignition 100

IDT / ms

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


10

GA method QC calculation -1 -1 ∆S = 3 cal K mol

1

0.1 0.7

0.8

0.9

1.0

1.1

1.2

1.3

1000 K / T Figure 3. Entropy of 2-propenyl,1-methyl radical effect on 2-butene oxidation.

Figure 4. Two different approaches to estimate the thermochemistry of 2-propenyl,1-methyl radical using the GA method in THERM.

5


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Table 2 The thermochemistry calculation of 2-propenyl,1-methyl radical by current GAVs using 1- and 2-butene as the parent molecules. (A symmetry number of 3 was used for 2-propenyl,1-methyl radicals, units: kcal mol–1 for formation enthalpy, cal K–1 mol–1 for entropy and heat capacity, the 298 K formation enthalpy of H-atom is 52.10 kcal mol–1.) ∆fHӨ Group

Approach 1 (Parent: 2-butene)

(Parent: 1-butene)

Cp

Number 298.15 K 298.15 K 300 K 400 K 500 K 600 K 800 K 1000 K 1500 K

C/CD/H3

2

–10.01

30.29

6.22

7.74

9.24

10.62

12.84

14.59

17.35

CD/C/H

2

8.65

7.83

4.31

5.22

6.03

6.74

7.87

8.67

9.87

ALLYLP

1

87.9

–1.77

–0.8

–0.93

–1.24

–1.59

–1.88

–2.57

–3.64

33.08

72.29

20.26

24.99

29.30

33.13

39.54

43.95

50.80

2-propenyl,1-methyl radical

Approach 2

SӨ

CD/H2

1

6.28

27.59

5.10

6.31

7.44

8.42

9.99

11.20

13.17

CD/C/H

1

8.65

7.83

4.31

5.22

6.03

6.74

7.87

8.67

9.87

C/C/CD/H2

1

–4.93

9.55

5.00

6.54

7.91

9.06

10.86

12.19

14.11

C/C/H3

1

–10.01

30.29

6.22

7.74

9.24

10.62

12.84

14.59

17.35

ALLYLS

1

85.50

–3.81

–1.54

–1.82

–2.08

–2.32

–2.75

–3.14

–3.85

33.39

69.27

19.09

23.99

28.54

32.52

38.81

43.51

50.65

2-propenyl,1-methyl radical

6


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In light of the above, this study has three main objectives. The first us to develop an accurate database for the thermodynamic properties of 36 C2 – C6 alkene molecules and their related allylic radicals as shown in Figure 5 using consistent QC calculations. There are five different types of allylic radicals, namely PP, PS, PT, SS and ST. Taking the PS group as an example, the thermodynamic properties of the radicals in this group can be estimated based on two different “parents” associated with either “ALLYLP” or “ALLYLS” radical groups using the GA method, as for the 2-propenyl,1-methyl radical in Table 2 above. Parents

Allylic Radicals PP

PS

PT

SS

ST

Figure 5. C2–C6 alkene molecules and their related allylic radicals. Secondly, to use the database so developed as a reference to optimize the relevant GAVs including 6 groups containing a C=C double bond (CD/H2, CD/C/H, CD/C2, C/CD/H3, C/C/CD/H2 and C/C2/CD/H) and 3 allylic radical groups (ALLYLP, ALLYLS and ALLYLT), Figure 6. The ideology behind this work is to generate thermochemistry estimations of allylic radicals as accurately as possible by employing different parents and the alternate radical groups (ALLYLP, ALLYLS and ALLYLT). In using the GA method one should calculate the same thermochemistry for the radical species (e.g. 2-propenyl,1-methyl) regardless of which parent is used, either 1- or 2-butene.

7



CD/H2

CD/C/H

C/CD/H3

C/C/CD/H2

ALLYLP

Page 8 of 35

CD/C2

C/C2/CD/H

ALLYLS

ALLYLT

Figure 6. Target groups to be optimized. Thirdly, to calculate the thermodynamic properties of six diene molecules with different bond structures (conjugated and skipped) and the relevant super-allylic radicals, as depicted in Figure 7. The three diene molecules in the top left corner highlighted in red have a conjugated bond structure, and, based on their thermodynamic parameters (enthalpies of formation, entropies and heat capacities), different super-allylic radical groups (SUALLYLP, SUALLYLS and SUALLYLT) have been developed. The two diene molecules highlighted in blue, have a ‘skipped’ or isolated bond structure, losing the hydrogen atom on the carbon adjacent to two double bonds leading to the double-stabilization of electrons, which makes this C–H bond even weaker than the C–H bond in the above conjugated diene molecules. Therefore, two double super-allylic radical groups (DSUALLYLS and DSUALLYLT) have been developed. Furthermore, for triple-skipped dienes highlighted in green, there is no conjugation in the parent but extensive cross conjugation in the corresponding super-allylic radical, which is reflected in a very low bond dissociation energy (BDE), and this a triple super-allylic radical group (TSUALLYLT) has also been developed.

8


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Parents

SUALLYLP SUALLYLS

Super Allylic Radicals

SUALLYLT Parents DSUALLYLS Double Super Allylic Radicals DSUALLYLT Parents TSUALLYLT Triple Super Allylic Radicals

Figure 7. Diene molecules and super allylic radicals to be calculated, and target groups to be developed.

2. Computational Methods 2.1. Ab initio calculations In this study, all density functional theory (DFT) and composite compound method calculations were performed using the Gaussian 09 software package36. The M06-2X37 method with the 6311++G(d,p) basis set was used in the geometry optimizations, vibrational frequency calculations and in the hinder rotation treatments for lower frequency modes. All frequencies were scaled by 0.983, with zero point vibrational energies (ZPVEs) scaled by 0.9698, as recommended for the M062X functional by Zhao and Truhlar37. The zero Kelvin energies (ZKEs) were obtained using combined compound methods: CBS-APNO, G3, and G438-40. This combination of methods has been

found to yield results rivaling “chemical accuracy” (arbitrarily, ∼4 kJ mol−1 or 1 kcal mol−1) when

benchmarked against enthalpy of formation values in the Active Thermochemical Tables (ATcT)4-6, 41-42

.

2.2. Chemically-balanced reaction schemes Both atomization and isodesmic methods were utilized to derive the enthalpies of formation at 0 K. In the atomization method, a molecule or radical is divided into its component atoms via the reaction:

→ + in which the theoretical atomization energy at 0 K (TAE0) can be calculated by: 9



= + − where H0 is the enthalpy of formation at 0 K calculated using each compound method. Thereafter, the enthalpy of formation of the species (∆fH0) can be calculated knowing the TAE0 and the standard formation enthalpies of the component atoms in their gaseous state from the ATcT4-6, shown in Table 3, via:

Δ = Δ + Δ − Table 3. Standard gaseous atomic formation enthalpies (kJ mol–1). T/K 0

C (3P)

H (2S1/2)

711.38 216.034

In the isodesmic method, for a given molecule or radical, the reaction is balanced so that the numbers of chemical bonds of each formal type are same on both sides of the equation. The enthalpy of formation (∆fH0) is then determined based on the heat of reaction and the standard formation enthalpies of the reactants and products involved in the reaction. The 298.15 K formation enthalpies, entropies and temperature-dependent heat capacities were then calculated from traditional statistical thermodynamics using the MultiWell program suite43. A series of automation Perl codes from Somers44 were utilized.

2.3. Conformer distribution Most of the alkene species and allylic radicals presented in this study are conformationally complex, with more than one conformer contributing to the thermodynamic parameters. All conformer searches were carried out using the Spartan 10 software package45 at the B3LYP/631G(d,p)46 level of theory. The Boltzmann distribution is applied to determine the contribution of each conformer, xi, taking due account of Gibbs free energies, ∆G⊖, and degeneracies, σ, via: (

= −Δ !"/$ " %& −Δ ! "/$ "' )*

The thermochemistry of each molecule was then derived based on the value and contribution of each conformer xi, via:

(

$+," = %& $+! "' )*

$+," and $+! " represents the enthalpy of formation, entropy and heat capacity of each

molecule and its corresponding conformers respectively. 10


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2.4. Statistical comparison and GAVs optimization To compare the thermochemical properties calculated using the quantum chemistry (QC) and group additivity (GA) methods, and to visualize the agreement between the two datasets, a BlandAltman plot47 was employed, which is also identical to the Tukey mean-difference plot. Simmie et al.48 employed this statistical analysis in a thermochemistry study in which it was shown to be effective. Table 4 lists the key parameters used in the Bland-Altman plot. For the two sets of thermochemistry data: quantum chemistry (QC) calculation results (-* , - , … , -( ) and group

additivity (GA) results ( * , , … , ( ), their average values (Xi) and differences (di) are set as x- and y-axes, respectively. The bias and 95% limits of agreement show the overall trend of the comparison and the dispersion of two datasets, respectively. Therefore, the objective of the GAV optimization is to result in a bias being as close as possible to zero, and to compress the 95% limits of agreement to shrink the dispersion of results among the two datasets. Based on this ideology, optimization is performed to minimize the “|Bias| + Sd” value by varying the existing GAVs of certain classes of groups using Microsoft Excel’s solver add-in49, and a “two stage optimization and development” is practically applied, as shown in Figure 8:

•

The GAVs optimization of 10 groups containing C=C double bond (CD/H2, CD/C/H, CD/C2, C/CD/H3, C/C/CD/H2, C/C2/CD/H, CD/CD/H, C/CD2/H2, C/C/CD2/H and C/CD3/H) based on the thermochemistry of alkene and diene molecules (parents)

•

The GAVs optimization of three allylic radical groups (ALLYLP, ALLYLS and ALLYLT) based on the thermochemistry of allylic radicals

•

The GAVs development of six super allylic radical groups (SUALLYLP, SUALLYLS, SUALLYLT,

DSUALLYLS,

DSUALLYLT

and

TSUALLYLT)

based

on

the

thermochemistry of super allylic radicals and their corresponding diene molecules Table 4. Key parameters of the Bland-Altman plot. Key parameters

Definition

n -* , - , … , -( 234 * , , … , (

Number of data points

5 =3−1

Degree of freedom

7 = 85, 9:;
Average values

4 = - − "

Differences

(

(

4̅ = %- − "/3 = %4 "/3

The mean difference or bias

11



AB = C

Sample standard deviation

Page 12 of 35

∑4 − 4̅ 3 − 1"

4̅ ± 1.96AB

95% Limits of agreement

7 × AB /√3

Error bars for bias

1.709 × 7 × AB /√3

Error bars for limits of agreement

• Two Stage Optimization and Development:

Optimize

CD/C2 CD/H2 CD/C/H C/CD/H3 C/C/CD/H2 C/C2/CD/H

C/CD2/H2

CD/CD/H

C/C/CD2/H

C/CD3/H

Optimize ALLYLP

Develop

ALLYLS

ALLYLT

SUALLYLP

SUALLYLS

DSUALLYLS

DSUALLYLT

SUALLYLT

TSUALLYLT

Figure 8. GAVs optimization process.

3. Results and Discussion 3.1. Comparison of the atomization and isodesmic method To demonstrate the reliability of the atomization method we compare it to the isodesmic method for two selected species; the smallest allylic radical, allyl radical (C3H5-a), and the biggest molecule, 5-methyl-1,3-hexadiene (C7H12). Figure 9 shows the isodesmic reaction schemes employed for these two species. The standard enthalpies of formation for the reference species used in the reaction are provided in Table 5.

+ CH4 →

+

+ CH4 → (a)

+

(b)

Figure 9. Isodesmic reaction schemes of (a) allylic radical and (b) 5-methyl-1,3-hexadiene

12


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Table 5. Standard enthalpies of formation for reference species in isodesmic work reaction / kJ mol–1. Species Enthalpy of formation at 0 K CH4

–66.55

C2H4

60.96

C2H5

130.92

C4H6

125.05

iC4H10

–106.17

Table 6 compares the enthalpies of formation calculated from both the atomization and isodesmic methods using each compound method and their average values. It is found that the isodemic method gives more consistent results using three compound methods, especially for the bigger molecule (C7H12). However, by taking the average value of the three compound methods, the atomization method also produces reliable results, which are within 0.2 kcal mol–1 of isodesmic method results. In another word, in this case these two methods are similar for systems that only contain hydrogen and carbon. Therefore, the atomization method has been consistently applied for the calculation of enthalpies of formation for all of the species in this study. Table 6. 0 K Enthalpies of formation comparison of atomization and isodesmic method / kcal mol–1. Allylic radical Method

5-Methyl-1,3-hexadiene

Atomization Isodesmic Atomization Isodesmic

CBS-APNO

42.09

42.16

12.58

14.01

G3

42.38

42.24

15.02

14.05

G4

42.73

42.87

15.00

13.92

Average

42.40

42.42

14.20

13.99

3.2. Conformer contribution In this study, 49 C2–C7 species have been selected and a conformer search has been carried out. In total, 107 conformers were located, and their geometries are provided as Supplementary Material. Here, 1-hexene is selected as a representative, which has the largest number of conformers. By rotating the four C–C single bonds, 14 conformers can be located, as shown in Figure 10. The atom and bond highlighted in gray are the C=C double bond. Corresponding to this figure, the contribution from each conformer of 1-hexene can be calculated based on the method proposed in Section 2.3, as shown in Table 7. The first conformer, C1, is the dominant one but this dominance decreases with increasing temperature. The thermochemistry of 1-hexene is then derived from the value and weighted contribution of each conformer.

13



Page 14 of 35

Figure 10. The structures of 14 conformers of 1-hexene. Table 7. The contribution from each conformer of 1-hexene. T/K

C1

C2

C3

C4

C5

C6

C7

C8

C9

C10

C11

C12

C13

C14

298

32%

16%

17%

10%

6%

6%

4%

4%

2%

2%

1%

0%

0%

0%

300

32%

16%

17%

10%

6%

6%

4%

4%

2%

2%

1%

0%

0%

0%

400

26%

15%

16%

11%

7%

7%

4%

6%

3%

3%

2%

1%

0%

0%

500

22%

13%

15%

11%

7%

7%

5%

7%

4%

3%

3%

1%

1%

0%

600

19%

12%

15%

11%

7%

8%

5%

8%

4%

4%

4%

2%

1%

0%

800

16%

11%

14%

10%

7%

8%

6%

9%

5%

4%

5%

3%

1%

1%

1000

14%

10%

13%

10%

7%

8%

6%

9%

6%

5%

6%

4%

2%

2%

1100

13%

10%

12%

10%

7%

8%

6%

9%

6%

5%

6%

4%

2%

2%

1200

13%

10%

12%

9%

7%

8%

6%

9%

6%

5%

6%

4%

3%

2%

1300

12%

9%

12%

9%

7%

8%

6%

9%

6%

5%

7%

4%

3%

2%

1400

12%

9%

12%

9%

7%

8%

6%

9%

6%

5%

7%

5%

3%

2%

1500

11%

9%

12%

9%

7%

8%

6%

9%

6%

5%

7%

5%

3%

3%

1600

11%

9%

11%

9%

7%

7%

6%

9%

6%

5%

7%

5%

3%

3%

1700

11%

9%

11%

9%

7%

7%

6%

9%

6%

5%

7%

5%

4%

3%

1800

10%

9%

11%

8%

7%

7%

6%

9%

6%

5%

7%

5%

4%

3%

1900

10%

9%

11%

8%

7%

7%

6%

9%

7%

5%

7%

6%

4%

3%

2000

10%

8%

11%

8%

7%

7%

6%

9%

7%

5%

7%

6%

4%

3%

14


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


3.3. Thermochemistry comparison against literature data In order to demonstrate the reliability of the calculated thermochemical values, three thermochemistry data bases have been selected from the literature for comparison:

•

Current study: CBS-APNO/G3/G4//M06-2X/6-311++G(d,p)

•

Thermochemical Data of Organic Compounds (TDOC) by Pedley et al.8: Experiments

•

Active Thermochemical Tables (ATcT): refs.4-6

•

Goldsmith et al.9: RQCISD(T)/cc-pVT,QZ//B3LYP/6-311++G(d,p), with bond additivity correction

Table 8 shows a comparison for C2, C3 and C4 alkenes calculated in this study. Excellent agreement is obtained for the 298 K enthalpies of formation among the four different sources, and the 298 K entropies and heat capacities at selected temperatures calculated in this study agree well with the results from Goldsmith et al.9 All of the values are within the uncertainty limits. For the C5 and C6 alkenes, only the 298 K enthalpies of formation exist in Thermochemical Data of Organic Compounds (TDOC)8, and they are compared in Table 9. For all of the 19 species shown a maximum 0.3 kcal mol–1 discrepancy is observed. Such agreement again confirms the accuracy and reliability of the computational method employed here. Moreover, compared to the ab initio method used in the study by Goldsmith’s et al., the calculations performed here are significantly cheaper. For allylic and super-allylic radicals, only C3 and C4 allylic radicals exist in the ATcTs and the study by Goldsmith et al., they are compared in Table 10. Excellent agreement is obtained for the entropies and heat capacities comparing to those calculated by Goldsmith et al. However, the enthalpies of formation calculated here are consistently about 1.0 kcal mol–1 lower than the literature values. We believe this is probably due to the relatively low zero kelvin energies (ZKEs) calculated using the CBS-APNO compound method. Therefore, the enthalpies of formation of all of the allylic and super-allylic radicals have been manually increased by 1.0 kcal mol–1 on the basis of quantum chemistry calculations.

15



Page 16 of 35

Table 8. Thermochemistry comparison for C2, C3 and C4 alkenes calculated in this study. (Units: kcal mol–1 for ΔfHӨ, cal K–1 mol–1 for SӨ and Cp) ∆fHӨ Species

SӨ

Cp

Resource

298.15 K

298.15 K 300 K

400 K

500 K

600 K

TDOC

12.5±0.1

–

–

–

–

–

ATcT

12.5±0.0

–

–

–

–

–

800 K

1000 K

1500 K

–

–

–

–

–

–

Goldsmith et al. 12.5±0.1

52.3±0.4

10.2±0.7 12.5±0.9 14.7±1.0 16.7±1.1 19.8±1.1 22.2±1.1 26.1±0.8

Current study

12.4

52.3

10.2

12.5

14.8

16.7

19.8

22.2

26.1

TDOC

4.8±0.2

–

–

–

–

–

–

–

–

ATcT

4.8±0.1

–

–

–

–

–

–

–

–


63.6±0.9

15.4±1.0 19.1±1.4 22.6±1.5 25.6±1.6 30.5±1.7 34.2±1.6 40.2±1.2

Current study

4.6

63.7

15.4

19.1

22.6

25.6

30.5

34.2

40.1

TDOC

0.0±0.2

–

–

–

–

–

–

–

–

ATcT

0.0±0.0

–

–

–

–

–

–

–

–


73.2±1.5

20.8±1.5 26.1±1.9 30.9±2.0 35.0±2.1 41.5±2.1 46.5±2.0 54.4±1.6

Current study

–0.1

73.4

20.6

26.0

30.9

35.1

41.6

46.6

54.4

TDOC

–2.7±0.2

–

–

–

–

–

–

–

–

ATcT

–2.7±0.1

–

–

–

–

–

–

–

–

Goldsmith et al. –2.7±0.2

70.7±1.5

21.1±1.3 26.0±1.7 30.5±2.0 34.6±2.1 41.2±2.2 46.2±2.1 54.2±1.6

Current study

–2.9

70.7

21.0

25.8

30.4

34.5

41.1

46.1

54.2

TDOC

–4.0±0.2

–

–

–

–

–

–

–

–

ATcT

–4.1±0.1

–

–

–

–

–

–

–

–

Goldsmith et al. –4.1±0.2

70.2±1.5

21.1±1.4 26.1±1.8 30.8±2.0 34.8±2.1 41.3±2.2 46.3±2.1 54.3±1.6

Current study

70.1

21.1

–4.3

26.2

30.8

16


34.8

41.3

46.3

54.2

Page 17 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


Table 9. Thermochemistry comparison for C5 and C6 alkenes calculated in this study. (Units: kcal mol–1 for ΔfHӨ). ∆fHӨ Species

Resource

298.15 K

TDOC

∆fHӨ Species

Resource

298.15 K

–7.6±0.3

TDOC

Current study

–7.7

TDOC

∆fHӨ Resource

298.15 K

–12.9±0.4

TDOC

–13.4±0.4

Current study

–12.7

Current study

–13.2

–5.1±0.2

TDOC

–10.4±0.4

TDOC

–11.8±0.4

Current study

–5.1

Current study

–9.9

Current study

–11.6

TDOC

10.0±0.3

TDOC

–16.0±0.4

TDOC

–16.3±0.3

Current study

–10.4

Current study

–15.3

Current study

–16.3

TDOC

–8.4±0.2

TDOC

–14.2±0.3

TDOC

–15.0±0.3

Current study

–8.5

Current study

–13.7

Current study

–15.0

TDOC

–6.6±0.2

TDOC

–14.7±0.3

TDOC

–13.0±0.3

Current study

–7.1

Current study

–14.7

Current study

–12.5

TDOC

18.2±0.2

TDOC

–12.3±0.4

Current study

18.7

Current study

–12.1

TDOC

25.2±0.3

TDOC

–15.1±0.3

Current study

25.4

Current study

–15.0

17


Species


Page 18 of 35

Table 10. Thermochemistry comparison for C3 and C4 allylic radicals calculated in this study. (Units: kcal mol–1 for ∆fHӨ, cal K–1 mol–1 for SӨ and Cp). ∆fHӨ

Species

SӨ

Cp

Resource

298.15 K 298.15 K 300 K

ATcT

40.1±0.1

–

400 K

–

500 K

–

–

600 K

–

800 K

–

1000 K

1500 K

–

–


61.5±0.8

14.9±1.2 18.7±1.4 22.0±1.5 24.6±1.5 28.7±1.4 31.8±1.3 36.6±1.0

Current study

61.6

14.9


72.0±1.3

19.9±1.4 24.8±1.9 29.3±2.0 33.1±2.1 39.1±2.0 43.6±1.9 50.6±1.5

Current study

72.1

19.7


70.2±1.2

19.6±1.7 24.9±2.0 29.5±2.1 33.3±2.1 39.2±2.0 43.6±1.9 50.6±1.5

Current study

70.3

19.7

39.6

31.9

31.8

18.8

24.8

25.1

22.1

29.3

29.7

18


24.8

33.1

33.4

28.8

39.1

39.3

31.8

43.6

43.7

36.6

50.6

50.6

Page 19 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


3.4. GAVs optimization and development 3.4.1. First stage optimization: C=C double bond groups In our first stage optimization, the GAVs for all of the C=C double bond groups, Figure 8, have been optimized. Figure 11 shows the Bland-Altman plot of the 298 K formation enthalpies of the alkenes and dienes containing these groups, and the key parameters of this statistical comparison are listed in Table 11. Before the optimization, a positive bias is observed in Figure 11 (a), showing that the GA method tends to under-estimate the enthalpies of formation compared to the QC calculations. In addition, 2,3-dimethyl-2-butene is found to be an outlier, which shows a 4.4 kcal mol–1 difference between the two methods. Moreover, the formation enthalpy values of 2,3-dimethyl-but-1-ene, 3methyl-2-pentene and 2-methyl-2-pentene are also under-estimated by about 1.5 kcal mol–1 using the GA method. The common point of these four molecules is that they all contain the two groups: CD/C2 and C/CD/H3, which are highlighted in red and blue, respectively. It is worth noting that the GAVs of these two groups were updated by Burke et al.28 based on isobutene, and we do see that the formation enthalpy of isobutene is in good agreement among the two methods. This indicates that the GA method is probably not able to produce 100% accurate values for all species, however, it is still a reliable and affordable method which can produce consistent and reasonably good results for thermochemistry computation, and this will be shown in the following sections. Therefore, the optimization of the group values used in the GA method becomes critical, and such an optimization needs to be performed statistically, in order to “satisfy” an appreciable number of species containing the targeting groups. After optimization, Figure 11 (b) shows that the bias turned into zero, and the 95% confidence intervals decreased from 4.09 kcal mol–1 to 1.89 kcal mol–1. The bias of the four species mentioned above (2,3-dimethyl-2-butene 2,3-dimethyl-1-butene, 3-methyl-2-pentene and 2methyl-2-pentene) have also dropped close to zero by increasing the formation enthalpies in the CD/C2 and C/CD/H3 groups. At the same time, isobutene moved away from the zero line by 1.13 kcal mol–1, but still within the 95% confidence interval. Figure 12 shows the Bland-Altman plot of the 298 K entropies of the alkenes and dienes containing these groups, and the key parameters are listed in Table 12. After optimization, the bias dropped from 0.75 cal K–1 mol–1 to 0, and the 95% confidence intervals decreased from 3.64 cal K–1 mol–1 to 2.21 cal K–1 mol–1. For the temperature dependent heat capacities, 800 K is selected as a representative temperature. The optimization results are shown in Figure 13, with some key parameters shown in Table 13. Again, similar to the 298 K entropies, the bias hits the zero line, and the 95% confidence intervals are significantly compressed.

19



5

CD/C2 C/CD/H3

(QC calculation - GA method)

4 3 2 1 0 -1 -2 -3 -4 -5 -30

-20

-10

0

10

20

30

40

60

50

(QC calculation + GA method) / 2

(a) Before optimization (black points and lines) 5 4


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 20 of 35

3 2 1 0 -1 -2 -3 -4 -5 -30

-20

-10

0

10

20

30

40

50

60


(b) After optimization (red points and lines) Figure 11. 298 K Formation enthalpies of the alkenes and dienes calculated in this study / kcal mol–1 Table 11. The key parameters of the Bland-Altman plot shown in Figure 11. Key parameters

Before optimization After optimization


28

28

Degree of freedom

27

27

Student’s t-value

2.05

2.05


0.75

0.00


1.04

0.48

Error bars for bias

0.40

0.19


0.69

0.32

20


Page 21 of 35


4 3 2 1 0 -1 -2 -3 -4 50

60

70

80

90

100


Figure 12. 298 K Entropies of the alkenes and dienes calculated in this study / cal K–1 mol–1. Table 12. The key parameters of the Bland-Altman plot shown in Figure 12. Key parameters



28

28

Degree of freedom

27

27

Student’s t-value

2.05

2.05


0.61

0.00


0.93

0.56

Error bars for bias

0.36

0.22


0.62

0.37

3


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 1 0 -1 -2 -3 10

20

30

40

50

60

70

80


Figure 13. 800 K Heat capacities of the alkenes and dienes calculated in this study / cal K–1 mol–1. 21



Table 13. The key parameters of the Bland-Altman plot shown in Figure 13. Key parameters Before optimization After optimization Number of data points 28.00 28.00 Degree of freedom 27.00 27.00 Student’s t-value 2.05 2.05 The mean difference or bias –0.32 0.00 Sample standard deviation 0.55 0.20 Error bars for bias 0.21 0.08 Error bars for limits of agreement 0.37 0.13 3.4.2. Second stage optimization: allylic radical groups Based on the optimized GAVs of the C=C double bond groups shown in Figure 8, in this section the optimization of the three allylic radical groups (ALLYLP, ALLYLS and ALLYLT), will be discussed. Figure 14(a), (b) and (c) show the Bland-Altman plot of the 298 K formation enthalpies of the allylic radicals containing ALLYLP, ALLYLS and ALLYLT groups respectively, while Figure 14(d) shows the 298 K formation enthalpies of all of the allylic radicals, and the key parameters of Figure 14(d) are listed in Table 14. Before the optimization, positive biases are observed for all types of allylic radicals, indicating that the GA method underestimates the heats of formation compared to our QC calculations. Therefore, by reducing the 298 K formation enthalpies of the ALLYLP, ALLYLS and ALLYLT groups, it was possible for all of the biases of all four Bland-Altman plots to reach zero, and the 95% confidence intervals also decreased by 0.26, 0.34 and 1.55 cal K–1 mol–1 respectively. In the same way, Figure 15(a)–(d) show the 298 K entropies of different types of allylic radicals (containing different types of radical groups), and Table 15 lists the key parameters of Figure 15(d). The GAVs of the ALLYLP group were updated by Burke et al.28 based on the thermochemistry of the 1-buten-4-yl radical produced from the 2-butene parent. Thus, good agreement is observed between our QC calculation and the GA method for the allylic radicals containing this group, as shown in Figure 15(a). However, as the major motivation of this study, as shown in Figure 2, the 298 K entropies of ALLYLS group hasn’t been updated, and the GA method gives approximately a 3 cal K–1 mol–1 lower entropy value for the 2-propenyl,1-methyl radical compared to our QC calculation. Figure 15(b) shows that this is true, not only for 2-propenyl,1-methyl radical, but also for the other 10 allylic radicals containing this ALLYLS group, with the 298 K entropy values of all of these radicals being consistently under-predicted by about 3.3 cal K–1 mol–1. This under-prediction is also observed for the ALLYLT group, as shown in Figure 15(c). The current optimization has significantly improved the GAVs of these two groups. Figure 16 and Table 16 show the 800 K heat capacity optimizations of all of the allylic radicals and some key parameters of the Bland-Altman plot. 22


Page 22 of 35

Page 23 of 35


6 4 2 0 -2 -4 -6 10

15

20

25

30

35

40

45

50


(a) Allylic radicals containing ALLYLP group


6 4 2 0 -2 -4 -6 10

15

20

25

30

35

40

45

50


(b) Allylic radicals containing ALLYLS group 6


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


4 2 0 -2 -4 -6 10

15

20

25

30

35

40

45


(c) Allylic radicals containing ALLYLT group

23


50



6 4 2 0 -2 -4 -6 10

15

20

25

30

35

40

45

50

(QC calculation + QA method) / 2

(d) Allylic radicals containing all ALLYL groups (ALLYLP, ALLYLS and ALLYLT) Figure 14. 298 K Formation enthalpies of the allylic radicals calculated in this study / kcal mol–1. Table 14. The key parameters of the Bland-Altman plot shown in Figure 14 (d). Key parameters



27.00

27.00

Degree of freedom

26.00

26.00

Student’s t-value

1.71

1.71


0.44

0.00


1.07

0.70

Error bars for bias

0.35

0.23


0.60

0.39

10 8


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 24 of 35

6 4 2 0 -2 -4 -6 -8 -10 55

60

65

70

75

80

85

90

95


(a) Allylic radicals containing ALLYLP group 24


100

Page 25 of 35

10


8 6 4 2 0 -2 -4 -6 -8 -10 55

60

65

70

75

80

85

90

95

100


(b) Allylic radicals containing ALLYLS group 10


8 6 4 2 0 -2 -4 -6 -8 -10 55

60

65

70

75

80

85

90

95

100


(c) Allylic radicals containing ALLYLT group 10 8


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


6 4 2 0 -2 -4 -6 -8 -10 55

60

65

70

75

80

85

90

95

100


(d) Allylic radicals containing all ALLYL groups (ALLYLP, ALLYLS and ALLYLT) Figure 15. 298 K Entropies of the allylic radicals calculated in this study / cal K–1 mol–1. 25



Table 15. The key parameters of the Bland-Altman plot shown in Figure 15 (d). Key parameters



27

27

Degree of freedom

26

26

Student’s t-value

1.71

1.71


2.03

0.00


1.90

0.55

Error bars for bias

0.62

0.18


1.07

0.31

2


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 26 of 35

1

0

-1

-2 25

30

35

40

45

50

55

60

65

70


Figure 16. 800 K Heat capacities of various allylic radicals calculated in this study / cal K–1 mol–1. Table 16. The key parameters of the Bland-Altman plot shown in Figure 16. Key parameters



27.00

27.00

Degree of freedom

26.00

26.00

1.71

1.71

–0.32

0.00


0.41

0.29

Error bars for bias

0.13

0.10


0.23

0.16

Student’s t-value The mean difference or bias

26


Page 27 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


3.5. Verification of the optimized GAVs In order to verify the optimized GAVs, the 298 K formation enthalpies of four C7 and C8 unsaturated hydrocarbons are compared in Table 17. These have not been calculated using QC calculations but are compared to the values provided in the Thermochemical Data of Organic Compounds (TDOC)8. In Table 17, the values calculated using the optimized GAVs are consistently higher than those using the current GAVs by about 1.0 – 2.6 kcal mol–1, which is due to the significant increase/refinement of the CD/C2 group as discussed in Section 3.4.1. After the optimization, the formation enthalpies calculated using the GA method agree better with the experimental values from the TDOC, with an average error of approximately 0.5 kcal mol–1. Table 17. The comparison of the 298 K formation enthalpies of C7 and C8 unsaturated hydrocarbons / kcal mol–1. Species

TDOC (experiment) Optimized GAVs Current GAVs

–18.98±0.26

–20.25

–21.58

–20.03±0.33

–20.10

–21.11

–19.00±0.33

–19.37

–21.87

–23.97±0.31

–24.32

–26.94

3.6. Comparison of similar radical groups As shown in Figure 7, based on the disparity of thermochemical properties (formation enthalpy, entropy and heat capacity) between the super-allylic radical and its corresponding diene molecule (parent), various super-allylic radical groups including SUALLYLS, SUALLYLS, SUALLYLT, DSUALLYLS, DSUALLYLT and TSUALLYLT were systematically developed. These can be used to generate the thermochemistry of larger super-allylic radicals, and their use should depend on the structure of the parent, as discussed in Section 1. Ultimately, we can compare the 298 K formation enthalpies of the similar radical groups, as shown in Table 18. These values decrease in two dimensions:

•

primary → secondary → tertiary

•

alkyl → allylic → super-allylic → double super-allylic → triple super-allylic 27



Figure 17 visualizes such sequential decreases. Generally, the difference in heat of formation between primary and secondary radical types is approximately 2.2 – 3.1 kcal mol–1, while that between secondary and tertiary radicals is in the range 1.5 – 1.6 kcal mol–1. Table 18. A comparison of the 298 K formation enthalpies (kcal mol–1) of similar radical groups. P

S

T

101.15

98.07

96.52

ALLYLP

ALLYLS

ALLYLT

87.47

85.26

83.77

SUALLYLP SUALLYLS

SUALLYLT

81.40

78.46

76.78

–

DSUALLYLS DSUALLYLT

–

74.70

73.16

–

–

TSUALLYLT

–

–

71.19

101.15 98.07 96.52

100

85.26

81.40

80

83.77

78.46 76.78

74.70

60

73.16

40

71.19

, 298K

87.47

Hf

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 28 of 35

20 0

ALLYL SUALLYL DSUALLYL TSUALLYL

y ar im r P

ry da n co Se

ry tia r Te Figure 17. The comparison of the 298 K formation enthalpies (kcal mol–1) of similar radical groups. 28


Page 29 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


3.7. The optimized and developed GAVs and their comparison against current values Finally, the optimized GAVs of 19 groups are summarized in Table 19. These can be used to generate the thermochemistry of larger species where accurate high-level calculations are difficult and expensive to perform.

Table 19. Optimized GAVs. Units: kcal mol–1 for ∆fHӨ, and cal K–1 mol–1 for SӨ and Cp. ∆fHӨ

SӨ

Cp

T/K

298.15 298.15

300

400

500

600

800

C/CD/H3

–10.44

33.93

4.69

5.67

6.74

7.80

9.66 11.15 13.63

CD/H2

5.95

27.23

5.33

6.46

7.52

8.45

9.96 11.12 13.03

CD/C/H

9.19

4.51

5.55

7.03

8.29

9.29 10.78 11.84 13.41

C/C/CD/H2

–5.48

13.26

3.98

5.00

5.94

6.73

8.03

8.98 10.56

C/C2/CD/H

–2.29

–8.15

3.47

4.58

5.35

5.83

6.51

6.88

CD/CD/H

7.02

6.68

4.38

5.85

6.95

7.71

8.67

9.26 10.14

C/CD2/H2

–4.83

17.09

2.12

2.73

3.34

3.91

4.89

5.66

6.93

C/C/CD2/H

–1.30

–3.47

1.82

2.42

2.81

3.02

3.32

3.49

3.81

C/CD3/H

–0.15

–0.17

0.37

0.32

0.29

0.27

0.24

0.22

0.21

CD/C2

11.98 –18.21

5.33

7.35

8.92 10.04 11.56 12.52 13.76

ALLYLP

87.47

–2.95 –0.50 –0.42 –0.64 –1.00 –1.79 –2.48 –3.59

ALLYLS

85.26

–1.40 –1.35 –1.55 –1.82 –2.11 –2.66 –3.13 –3.92

ALLYLT

83.77

0.38 –1.66 –2.21 –2.62 –2.90 –3.28 –3.59 –4.11

SUALLYLP

81.40

–3.10 –1.51 –1.87 –2.18 –2.39 –2.65 –2.84 –3.23

SUALLYLS

78.46

–2.57 –1.70 –2.59 –3.09 –3.31 –3.40 –3.40 –3.50

SUALLYLT

76.78

–0.62 –2.69 –3.74 –4.27 –4.40 –4.26 –4.05 –3.85

DSUALLYLS

74.70

–5.44 –1.05 –0.71 –0.67 –0.81 –1.29 –1.78 –2.69

DSUALLYLT

73.16

–6.25 –2.27 –2.02 –1.94 –1.95 –2.13 –2.40 –2.98

TSUALLYLT

71.19

–6.09 –3.09 –2.72 –2.69 –2.83 –3.39 –4.03 –5.19

29


1000

1500

7.53


Page 30 of 35

Table 20 shows the differences between the optimized GAVs and previous THERM values, which are calculated as “optimized values – previous THERM values”. The majority of the GAVs of the groups (C/CD/H3, CD/H2, CD/C/H, C/C/CD/H2 and C/C2/CD/H) applicable to alkene molecules have been changed by only marginal amounts (less than 1.0 kcal mol–1 in enthalpy and less than 4.0 cal K–1 mol–1 changes in entropy and heat capacity). However, the GAVs of the three groups (C/CD2/H2, C/C/CD2/H and C/CD3/H) applicable to diene molecules have been changed significantly; by more than 1.0 kcal mol–1 for the enthalpy and more than 7.0 cal K–1 mol–1 for entropy and heat capacity values. The GAVs of CD/C2 have also been changed significantly; by more than 2.0 kcal mol–1 in enthalpy and more than 7.0 cal K–1 mol–1 in entropy and heat capacity. These updates are derived from the thermochemistry of the molecules shown in Figure 11. For the three allylic radical groups, the entropy values of ALLYLS and ALLYLT have been increased by 2.41 and 4.07 cal K–1 mol–1 respectively.

Table 20. The comparison between the optimized GAVs and the previous THERM values (optimized values – previous values). Units: kcal mol–1 for ∆fHӨ, and cal K–1 mol–1 for SӨ and Cp. ∆fHӨ T/K

SӨ

298.15 298.15

Cp 300

400

500

600

800

1000

1500

3.64 –1.53 –2.07 –2.50 –2.82 –3.18

–3.44

–3.72

–0.08

–0.14

2.91

3.17

3.54

C/CD/H3

–0.43

CD/H2

–0.33

–0.36

0.23

0.15

0.08

0.03 –0.03

CD/C/H

0.54

–3.32

1.24

1.81

2.26

2.55

C/C/CD/H2

–0.55

3.71 –1.02 –1.54 –1.97 –2.33 –2.83

–3.21

–3.55

C/C2/CD/H

0.56

3.54 –0.69 –1.33 –1.99 –2.36 –2.95

–3.31

–3.75

CD/CD/H

0.30

0.30 –0.08

0.32

0.15

0.05

C/CD2/H2

–0.07

7.29 –3.00 –4.13 –4.98 –5.58 –6.33

–6.82

–7.43

C/C/CD2/H

1.01

7.84 –1.96 –3.43 –4.70 –5.28 –6.29

–6.84

–7.58

C/CD3/H

1.99

10.76 –3.03 –5.44 –7.39 –8.17 –9.52 –10.25 –11.29

CD/C2

2.32

–5.94

1.90

2.97

3.85

4.54

5.50

6.16

7.10

ALLYLP

–0.43

–1.18

0.30

0.51

0.60

0.59

0.09

0.09

0.05

ALLYLS

–0.24

2.41

0.19

0.27

0.26

0.21

0.09

0.01

–0.07

ALLYLT

–1.23

4.07

0.13

0.17

0.12

0.07

0.00

–0.04

–0.04

0.06

0.20

0.29

30


Page 31 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


4. Conclusions This study is motivated by the need for accurate thermochemical properties of allylic radicals. These can be produced using the GA method; however reliable GAVs of the relevant groups need to be produced so that accurate thermodynamic parameters can be calculated for large alkenes, dienes and their radicals. Therefore, this study focused on the following two aspects:

•

Comprehensive theoretical calculations of the thermochemistry of C2–C7 unsaturated hydrocarbons, allylic radicals and super-allylic radicals using quantum chemical methods.

•

A systematic optimization and development of GAVs of 19 relevant groups based on a Bland-Altman statistical method.

A consistent ab initio method, namely CBS-APNO/G3/G4//M06-2X/6-311++G(d,p) has been employed for the electronic structure calculations, and the atomization method was utilized for the 0 K formation enthalpy calculations, statistic thermodynamics was applied for the temperaturedependent thermochemical properties (enthalpy, entropy and heat capacity) calculations, and conformer contribution was also taken into account based on the Boltzmann distribution. The theoretical results show good agreement with the literature databases (TDOC, ATcT and Goldsmith et al.). The GAVs of 10 C=C double bond groups (CD/H2, CD/C/H, CD/C2, C/CD/H3, C/C/CD/H2, C/C2/CD/H, CD/CD/H, C/CD2/H2, C/C/CD2/H and C/CD3/H) and three allylic radical groups (ALLYLP, ALLYLS and ALLYLT) were optimized based on the above theoretical results of alkenes, dienes and allylic radicals respectively. The optimization was performed statistically using a Bland-Altman plot. On the basis of the above 13 updated groups, six new super-allylic radical groups (SUALLYLP, SUALLYLS, SUALLYLT, DSUALLYLS, DSUALLYLT and TSUALLYLT) were developed based on different types of dienes and their corresponding super-allylic radicals. The adoption of these super-allylic radical groups will increase the predictive ability of the GA method for a wide range of super-allylic radicals which are crucial intermediate species in diene oxidation kinetics. The overall approach employed in this study has resulted not only in an improved agreement between the GA method and QC calculations for the C2–C7 unsaturated hydrocarbons, allylic radical and super-allylic radicals, but also in increased predictive power of the GA method in estimating the thermochemistry of large hydrocarbon species where accurate high-level calculations are difficult to perform.

31



Supplementary Material

•

The GA_DOC.txt file contains a list of each species with both the current and optimized GAVs used to describe that species.

•

The GA_LST.txt file lists each species together with its enthalpy, entropy and heat capacity values calculated by both current and optimized GAVs.

•

The GA_DAT.txt file contains a list of each species with its thermodynamic parameters expressed in NASA polynomial format calculated by both current and optimized GAVs.

•

The QC_LST.txt file lists each species together with its enthalpy, entropy and heat capacity values calculated by quantum chemical method.

•

The Species_Glossary.pdf file shows the glossary of all the species studied in this paper.

•

The geometries of all the species studied in this paper and the input and output files for MultiWell/Thermo codes are also provided.

Acknowledgments This work is supported by the FUELCOM project of Saudi Aramco. The authors wish to acknowledge the computational resources provided from the Irish Centre for High-End Computing (ICHEC), under project number ngche049c. Dr. Kuiwen Zhang is acknowledged for the provision of THERMAKE.xlsm Excel Sheet with VB code, and Prof. John Simmie is also acknowledged for the

helpful discussions.

32


Page 32 of 35

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Page 34 of 35

TOC Graphic 101.15 98.07 87.47

96.52

100

85.26

81.40

80

83.77

78.46 76.78

74.70

60

73.16

40

71.19

298K

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


Hf,

Page 35 of 35

20 0

ALLYL SUALLYL DSUALLYL TSUALLYL

y ar im Pr

y ar nd co Se

y ar rti Te

35


Extensive Theoretical Study of the Thermochemical Properties of

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