Accurate Thermochemical and Kinetic Stabilities of C84 Isomers

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A: Molecular Structure, Quantum Chemistry, and General Theory 84

Accurate Thermochemical and Kinetic Stabilities of C Isomers Simone Waite, Bun Chan, Amir Karton, and Alister J. Page J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b02404 • Publication Date (Web): 24 Apr 2018 Downloaded from http://pubs.acs.org on April 24, 2018

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The Journal of Physical Chemistry

Accurate Thermochemical and Kinetic Stabilities of C84 Isomers Simone L. Waite,a Bun Chan,b Amir Karton,c Alister J. Pagea* a

School of Environmental and Life Sciences, The University of Newcastle, Callaghan, NSW

2308, Australia b

Graduate School of Engineering, Nagasaki University, Bunkyo 1-14, Nagasaki 852-8521,

Japan c

School of Molecular Sciences, The University of Western Australia, Perth, WA 6009,

Australia

Abstract Accurate double-hybrid density functional theory and isodesmic-type reaction schemes are utilized to report accurate estimates to the heats of formation (∆fH) for all 24 isolatedpentagon-rule isomers of the 3rd most abundant fullerene, C84. Kinetic stabilities of these C84 isomers are also reported considered via C-C bond cleavage rates (Pcleav) calculated using density functional theory. Our results show that the relative abundance of C84 fullerene isomers observed in arc-discharge synthesis is the result of both thermochemical and kinetic factors. This provides timely insight regarding the characterisation of several C84 isomers that have been obtained experimentally to date. For instance, the established assignments of C84 isomers (using the Fowler-Manolopoulos numbering scheme) 22 [D2(IV)], 23 [D2d(II)], 19 [D3d], 24 [D6h], 11 [C2(IV)], and 4 [D2d(I)] are consistent with the relative ∆fH and Pcleav values for these structures. However, our thermochemical and kinetic stabilities of Cs isomers

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14, 15 and 16 indicate that the two experimentally isolated Cs isomers are 15 and 16, contrary to the some previous assignments. Of the remaining isolated isomers of symmetry C2 and D2, definitive assignment was not possible with consideration of only ∆fH and Pcleav. 1. Introduction Carbon-based nanomaterials such as fullerenes, nanotubes and graphene are a cornerstone of nanomaterials science due to their remarkable physical and chemical properties. From the point of view of chemistry however, the fullerenes are arguably the most important member of the nanocarbon family. The potential of C60 alone in many diverse chemistries is now well established – e.g. nucleophilic/electrophilic additions, electrocyclic reactions, radical additions, metal coordination and supramolecular complexation – as highlighted in numerous reviews1-7. The unique chemistry of (Ih)C60 is a result of its structure; (Ih)C60 consists of 12 hexagonal and 5 pentagonal rings, formed from a non-overlapping, octahedral arrangement of six individual pyracylene units. Adjacent pyracylene units are connected by a single (reactive) C=C double bond at their centre, which are trapped by the pentagonal rings in the fullerene cage. It is now more than 30 years since Kroto and co-workers8 identified this chemical structure of (Ih)C60. However, despite the wealth of investigation into this remarkable molecule, perhaps its most fundamental chemical property - its heat of formation - remains surprisingly uncertain. NIST currently recommends a value of 2560 kJ mol−19, but experimental values10-15 deviate from this value by up to ±100 kJ mol−1. The most reliable theoretical value of ∆fH for C60 (2511.7 kJ mol−1) was derived from isodesmic-type reaction schemes in conjunction with highly accurate composite first-principles thermochemical energy calculations16-18. This value is in good agreement with a previous theoretical prediction of ∆fH = 2520.0 kJ mol−1 derived from isodesmic-type reaction schemes based on


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double-hybrid

density

functional

theory

and

composite

first-principles

methods

calculations.19-20

Figure 1. Schlegel diagrams of 24 IPR isomers of C84. Isomers of the same point group are numbered consistent with reference 21.


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The thermodynamic and kinetic stability of many larger fullerenes remains largely unexplored. Chan et al.22 have reported first-principles ∆fH values for some giant fullerenes, but further exploration is complicated by the existence of myriad structural isomers for a given cage size. For instance, the third most abundant fullerene cage (after C60 and C70) formed in arc-discharge soot23, C84, has 24 unique structural isomers satisfying the isolated pentagon rule (IPR), shown in Figure 1 (numbered according to the Fowler-Manolopoulos scheme). Of these however, only 10 have been experimentally isolated and definitively characterised since the early 1990s.24 There are two major isomers formed in a ~2:1 ratio via arc-discharge are 22 [D2(IV)] and 23 [D2d(II)], both of which were assigned via independent NMR experiments.25-27 Avent et. al.28 first attempted to identify and characterise the minor isomers of C84 employing

13

C NMR on a partially separated C84 fullerene mixture, which

posed great difficulty in identification. The so-called minor isomers of C84 that have been characterised experimentally to date include 4 [D2d(I)], 5 [D2(II)], 11 [C2(IV)], 14 [Cs(III)], 16 [Cs(V)], 19 [D3d] and 24 [D6h].29-31 30, 32 However, the experimental assignment of the C84 minor isomers has continually proved problematic. For instance, the two Cs isomers (originally obtained and named Cs(a) and Cs (b) by Dennis et al.29) could not be assigned with one-dimensional

13

C NMR data. This

investigation also reported an additional isomer that remains unidentified to date. Subsequent attempts to assign these two Cs isomers have produced conflicting results; a combined electrochemical and density functional theory (DFT, BLYP/6-31G) study33 assigned Cs(b) to be isomer 16 [Cs(V)] and Cs(a) to be isomer 14 [Cs(III)], while a B3LYP/6-31G

13

C NMR

spectrum34 indicates the reverse assignment, consistent with single crystal XRD experiments35 and a number of independent theoretical and semi-empirical investigations.36-40 Moreover, while two C2 symmetry isomers have been isolated,29, 31 to date only one of them has been characterised (i.e. 11 [C2(IV)]) by confirming experimental NMR chemical shifts29


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with those calculated using DFT (B3LYP/6-31G*).34 Additional DFT studies32-33,

41-42

confirmed 11 [C2(IV)] to be the lowest energy C2 isomer. It is possible that the other C2 isomer isolated experimentally31 is 13 [C2(IV)], as B3LYP/6-31G energies42 indicate it to be the next most stable. However, this has never been proven experimentally. It is evident therefore that definitive thermochemical stabilities would assist the structural assignment of the of the C84 minor isomers. However, as yet thermodynamic stabilities of IPR C84 isomers are limited to predicted equilibrium mole percentages of each isomer at various charges.43 Neutral C84 isomers 22 [D2(IV)] and 23 [D2d(II)] where shown to be the most stable from their predicted equilibrium mole percentages, and for the minor isomers the most stable 11 [C2(IV)], 16 [Cs(V)], 19 [D3d] and 24 [D6h]. Moreover, the relative kinetic stabilities44 of the 24 IPR C84 isomers have not been considered whatsoever, despite the correlation observed between kinetic stability and fullerene abundance in arc-discharge synthesis. In the present study, we address these shortcomings by using isodesmic-type reaction schemes in conjunction with double-hybrid DFT energies (DHDFT) to provide the best theoretical estimate to date for heats of formation for all 24 IPR C84 isomers. We utilize these heats of formation to assist in the identification of experimentally isolated isomers. We also present the first analysis of the relative kinetic stabilities of the 24 IPR C84 isomers based on DFT calculations. These results indicate that the relative abundance of the fullerene isomers results from both kinetic and thermodynamic grounds.

2. Computational Methods The approach used in refs. 20 and 22 is used in this study to determine heats of formation of C84. Initial C84 isomer geometries were from the Yoshida database45 and Schlegel diagrams were made using Fullerene46 program. Briefly, geometries and harmonic vibrational


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frequencies of all structures were optimised using B3LYP/6-31G(d) as implemented in Gaussian 0947. Zero-point vibrational energies (ZPVE) and enthalpic temperature corrections (∆H298−0) to the total electronic energy were scaled using literature scaling factors48 (0.9826 for ZPVE and 1.0004 for the ∆H298−0 correction). If available, ZPVE, ∆H298−0, benchmark total energies, and heats of formation ∆fH for molecular fragments used in isodesmic-type reaction schemes (see below) were taken from refs 20 and 22. Double-hybrid DFT (DHDFT) was carried out using Orca 4.0.049-50. DHDFT calculations employed the DSD-PBEP8651 functional in conjunction with the def2-QZVPP52 basis set. Empirical D3 dispersion corrections53 with Becke-Johnson54-56 damping were applied throughout.

3. Results and Discussion 3.1. Overview of Isodesmic-Type Reaction Schemes for Fullerene Heats of Formation For many decades chemical reaction schemes have been employed in efforts to accurately predict thermochemical properties of organic molecules. This approach was first introduced by Pople and co-workers in the 1970s57-58, who demonstrated how appropriately balancing reaction energies yielded accurate molecular energies via efficient error-cancelation. While this was introduced for the sake of expediency at the time (contemporary computer technology was then not capable of accurate electron correlation methods), this approach has since evolved into a canonical hierarchy of methods59 that potentially affords thermochemical properties with sub-chemical accuracy (i.e. errors below the 1 kcal mol-1 mark). This hierarchy includes isogyric57, 60 (in which the number of electron pairs in reactants/products is conserved), isodesmic57-58,

61

(those in which the number and types of bonds in the

reactants/products are conserved), hypohomodesmotic59 (in which the number of carbon atoms in their various states of hybridization, and the number of carbon atoms regardless of


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hybridization state with zero, one, two and three hydrogen atoms attached in the reactants/products are conserved), homodesmotic59, 62 (in which the number of each type of carbon-carbon bond (eg. Csp3-Csp3), and number of each type of carbon atom (sp3, sp2, and sp) with zero, one, two, and three hydrogen atoms attached in reactants/products are conserved), hyperhomodesmotic63 (in which the number of carbon-carbon bond types [eg. H3C-CH2], and number of each type of carbon atom (sp3, sp2, and sp) with zero, one, two, and three hydrogen atoms attached in reactants/products are conserved) reactions. We note a number of other schemes outside of this hierarchy have also been reported in the literature (e.g. quasihomodesmotic64-65, isoplesiotic66, s-homodesmotic67-69). In this context, the generalized connectivity-based hierarchical (CBH-n)70-71 approach, which overcomes many limitations of the traditional schemes, must also be mentioned. In general however, once a reaction scheme is chosen for a molecule, its heat of formation is calculated by firstly performing electronic structure calculations on all molecules in the scheme to get the reaction energy and then applying Hess’ law using this reaction energy in conjunction with accurate enthalpies of formation on the reference molecules (obtained from experiment or high-level theoretical calculations). Recent investigations16,

20, 72

have employed isodesmic-type reactions to determine the

heats of formation for small polyaromatic hydrocarbons (PAHs) and C60, from which

∆fH°298[C60(g)] can be determined. For the calculation of ∆fH for corannulene (C20H10), ref. 20 recommended the following isodesmic-type reaction be used, C20H10 + 30 CH4 → 15 C2H6 + 10 C2H4

(1)

C20H10 + 5C2H4 → 5C6H6

(2)

as it minimises uncertainties in ∆fH derived from experimental heats of formation for the smaller fragments. Equation (1) additionally conserves the key molecular fragments (six-


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membered rings) between reactants and products. ∆fH°298[C20H10(g)] using this equation was calculated to be 485.2 ± 8 kJ mol−1 with the accurate W1h thermochemical method. A revised uncertainty value of 4.4 kJ mol−1 22 for corannulene was later proposed based on analysis of the results for a range of additional DFT and DHDFT methods. It was suggested that the currently accepted experimental value (∆fH°298[C20H10(g)] = 458.5 ± 9.2 kJ mol−1)73 be reexamined due to the significant difference between the two values. A later study16 reported

∆fH°298 of sumanene (C21H12) to be 535.3 ± 9 kJ mol−1, using accurate W1h energies in conjunction with the equation,

C21H12+ 3CH4 → 4C6H6

(3)

The heat of formation of C60 fullerene was subsequently calculated using DSDPBEP86/cc-pVQZ, via a series of isodesmic-type reactions involving only corannulene and planar polyacenes.20 Corannulene was chosen so that the curvature of the C60 π-conjugated carbon network, a product of its IPR structure, was conserved. For the choice of the planar polyacenes benzene and naphthalene were used, giving the isodesmic reactions,

C60 + 10C6H6 → 6C20H10

(4)

C60 + 10C10H8 → 8C20H10

(5)

Reaction (4) was considered an appropriate candidate for predicting the most reliable heat of formation, since a highly accurate ∆fH°298 value for benzene is available via the ATcT (Active Thermochemical Tables74-75). Additionally, reaction (4) employs a limited number of corannulene molecules which incur an uncertainty of 4.4 kJ mol−1 in the final ∆fH°298 for C60. Reaction (5) was also considered as it conserves larger molecular fragments on both sides of the reaction, meaning that errors in the final ∆fH°298[C60(g)] value derived from bond cleavage is minimised. Ultimately the best predicted DSD-PBEP86/cc-pVQZ value was achieved by averaging the heats of formation obtained with reactions (4) and (5), giving ∆fH°298[C60(g)] = 2521.6 ± 14 kJ mol−1. This value is in good agreement with the high-level G4(MP2) value of


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2511.7 kJ mol−1.16 The latter value was based on ∆fH°298 of sumanene, to account for the strain energy associated with hexagons surrounded by three hexagons and three pentagons. Isodesmic-type reactions have also been employed to predict accurate ∆fH°298 values for larger fullerene cages, notably the most stable isomers of C70, C76, C78, and C84.22 The strategy employed in this case was to rely on the accurate value for ∆fH°298[C60(g)] and use isodesmic reactions schemes which (i) preserved the chemical features between fullerenes of similar sizes and (ii) minimized the total number of species involved. This lead to the following series of homologous reactions,

C70 + 5C4H6 → C60 + 5C6H6

(6)

C70 + 5C8H8 → C60 + 5C10H8

(7)

C76 + C14H10 → C70 + C20H10

(8)

C78 + C4H6 → C76 + C6H6

(9)

C78 + C8H8 → C76 + C10H8

(10)

C84+ C14H10 → C78 + C20H10

(11)

in which the smaller molecule combinations are trans-butadiene(C4H6)/benzene(C6H6), styrene(C8H8)/naphthalene(C10H8), and phenanthrene(C14H10)/corannulene(C20H10). A larger set of reactions for each fullerene was considered to assess the uncertainty in calculated

∆fH°298 values induced by the use of smaller molecules. For example, for C84 two such additional reactions were, C84 + 3C4H6 → C78 + 3C6H6

(12)

C84 + 3C8H8 → C78 + 3C10H8

(13)

The predicted DSD-PBEPBE/cc-pVQZ ∆fH°298[C84(g)] values using reactions (11)-(13) were 2946.4, 2956.2(12), and 2939.9 kJ mol−1, respectively, constituting a range 16.3 kJ mol-1. ∆fH values obtained with reactions using phenanthrene/corannulene pair (reaction (11)) and the average ∆fH values obtained with the trans-butadiene/benzene (reaction (12)) and


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styrene/naphthalene (reaction (13)) combinations are comparable. Considering these results, reactions (11)-(13) provide a reliable basis for the calculation of ∆fH values of all 24 IPR C84 isomers. 3.2. Heats of Formation of C84 IPR Isomers DSD-PBEP86/def2-QZVPP ∆fH°298[C84(g)] values for the 24 IPR isomers of C84 (Figure 1) are presented in Table 1. Together with reactions (11)-(13), we consider also fragmentation schemes with sumanene (C21H12) instead of corannulene, C84 + 2 C18H12 → C78 + 2 C21H12

(14)

C84 + 3 C20H10 + 3 C6H6 → C78 + 4 C21H12

(15)

To conserve the chemical environments of the reactants/products to a large extent, C84 is broken down into its fundamental bowl-shaped aromatic fragments, corannulene and sumanene (Figure 2). Including sumanene in the reaction scheme to obtain the heat of formation of C84 is important, as both corannulene and sumanene represent the IPR structure but account for different strain energy.

Figure 2. B3LYP/6-31G(d) optimized structures of an IPR C84 isomer and its two basic structural units corannulene (C20H10), in which a central isolated pentagon is surrounded by hexagons, and sumanene (C21H12) in which a central hexagon is surrounded by isolated pentagons.


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Table 1 shows that the overall order of predicted heats of formation for C84 isomers is independent of the choice of reaction scheme, since the same isodesmic-type reactions are used for all C84 isomers. However, the choice of isodesmic reaction influences the accuracy of the ∆fH value, hence we provide an assessment of reactions (11) – (15). Overall, the predicted ∆fH for reactions (11) – (15) are within a range of 28.3 kJ mol-1, with reaction (13) yielding the highest value and (14) the lowest. This is within the range of the expected uncertainty for C84, since it relies on the ∆fH°298[C60(g)], which has an estimated uncertainty of ±20.7 kJ mol−1 22. The ∆fH for reactions (12) and (13), which both use smaller planar polyacenes in the isodesmic reaction scheme, differ by no more than 0.5 kJ mol-1 for all 24 IPR isomers. The similarity in the predicted ∆fH values using these two reactions is attributed to their use of highly accurate ATcT heats of formation,74 as well as their exclusive use of six-membered ring molecules and the inclusion of alkene chains (butadiene and styrene). By contrast, reactions (11) and (14), which both feature a combination of five- and sixmembered ring molecules results in a lower estimate of ∆fH. The variation when corannulene (reaction (11)) or sumanene (reaction (14)) is considered as a fragment molecule leads to a slightly larger variation of ~5 kJ mol-1 to the estimated ∆fH for the 24 IPR isomers. Reactions (11) and (14) have the advantage of balancing the strain energy on both sides of the reaction, since corannulene or sumanene better represent the curvature of C84 compared to the transbutadiene/benzene and styrene/naphthalene combinations. We note however, that the ∆fH of sumanene has a relatively large uncertainty of ±9 kJ mol−1, when compared to the revised value for corannulene (±4 kJ mol−1)22. The uncertainties associated with reactions (14) and (15) which contain sumanene therefore have a relatively large propagated error and the uncertainty of reaction (14) is ±21.5 kJ mol−1. The final reaction considered, reaction (15), separates C84 into both corannulene and sumanene while conserving the numbers of each formal bond type and the numbers of carbon atoms in each hybridization state. The larger


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fragmentation scheme in reaction (15) best represents the overall curvature and strain energy of the C84 isomers and can be assumed to be a reasonable estimate of ∆fH. However, reaction (15) has a large propagated error of ±34.3 kJ mol-1 so cannot reliably give the ∆fH of C84. With consideration of reactions (11) – (15) the final proposed values of the calculated heats of formation of the 24 IPR C84 isomers (shown in Figure 3) are taken as the average of reactions (11)–(13). These reactions afford efficient cancelation of errors in the calculated reaction energies. As previously discussed the associated uncertainties of reactions (14) and (15) are too large to be considered reliable. Reaction (11) should be considered in the final proposed value as it preserves the curved structure of C84 and minimizes the number of species involved in the reaction. While reactions (12) and (13) are included due to the use of highly accurate ATcT heats of formation associated with these reactions. Reaction (15) is within 3.8 kJ mol-1 of the final proposed heats of formation, further validating the proposed value. We note here that heats of formation determined from topological indicators have been shown to estimate B3LYP/6-31G* ∆fH values for various fullerenes, as detailed by Schwerdtfeger et al.76 There is reasonable correlation between our DSD-PBEP86/def2QZVPP ∆fH values (Table 1) for all 24 IPR isomers of C84 and that predicted using the approximate scheme proposed by Cioslowski et al,37 based on 30 structural motifs (central six-member rings surrounded by their first and second neighbours) and a curvature term. However, our recommended ∆fH values in Table 1 have limited correlation with those estimated using the local topological scheme of Alcami et al.,77 which is based on nine structural motifs (C-C bond with four incident rings). The latter scheme was designed to represent non-IPR fullerene structures and does not distinguish between fullerene isomers, while Cioslowski et al’s37 scheme represents only IPR fullerene structures and various


12

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fullerene isomers are considered. These comparisons are presented in Table S1 in Supporting Information. The two major C84 isomers, 22 [D2(IV)] and 23 [D2d(II)], are the most stable at the DSDPBEP86/def2-QZVPP level of theory and have comparable ∆fH (differing by ~10 kJ mol-1). These results are consistent with experimentally observed D2:D2d abundance ratio, ~2:1.26 For instance, assuming Boltzmann statistics at 2,000 K (a representative arc discharge temperature), the proposed ∆fH values provided in Table 1 predict the % abundances of 22 [D2(IV)] and 23 [D2d(II)] isomers to be ~64% and 25%, a ratio of ~2.5:1. A number of earlier theoretical studies25-27 have shown these isomers to be the most stable. However, B3LYP/631G41 results have indicated a smaller energy difference between these isomers than that reported here (less than 1.1 kJ mol-1). The minor isomers of C84 characterised experimentally to date are 4 [D2d(I)], 5 [D2(II)], 11 [C2(IV)], 14 [Cs(III)], 16 [Cs(V)], 19 [D3d] and 24 [D6h].29-32 Unambiguous experimental assignments have only been made for isomers 24, 19 and 4. Table 1 shows that isomer 24 (D6h) is the most stable of these isomers (∆fH = 2986.2 kJ mol-1) at the DSD-PBEP86/def2QZVPP level of theory. Isomer 19 (D3d) is only slightly less stable by comparison, with ∆fH = 2993.5 kJ mol-1. Isomer 4 [D2d(I)] is the 11th most thermochemically stable isomer according to ∆fH data provided in Table 1. All of these isomers have been previously isolated and characterised29-30, 32 and our DHDFT heats of formation corroborate these assignments. Isomer 11 [C2(IV)] has been characterised and isolated by Dennis et al.29 and assigned on the basis of B3LYP/6-31G* NMR spectra and relative energies.34


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Table 1. DSD-PBEP86/def2-QZVPP relative energies, zero point vibrational energies (ZPVE), thermal corrections for enthalpies (∆H298-0) and ∆fH°298 for 24 IPR C84 isomers (Figure 1). Energies are reported relative to the most stable D2 isomer (22); ∆fH°298 are predicted using reaction (11)-(15). All values reported in kJ mol−1. ∆fH°298[C84(g)] Isomer

Relative Energy

ZPVE

∆H298-0

Reaction (11)

(12)

(13)

(14)

(15)

Proposed value

80.6

3146.1

3169.7

3170.2

3142.0

3158.2

3162.0

1357.7

80.8

3052.0

3075.5

3076.1

3047.8

3064.0

3067.8

141.6

1353.3

81.6

3067.0

3090.5

3091.1

3062.8

3079.0

3082.8

4 [D2d(I)]

74.0

1358.6

80.8

3004.0

3027.6

3028.1

2999.9

3016.1

3019.9

5 [D2(II)]

65.2

1358.2

80.8

2994.8

3018.4

3018.9

2990.7

3006.9

3010.7

6 [C2v(I)]

89.1

1357.7

80.9

3018.3

3041.8

3042.3

3014.1

3030.3

3034.1

7 [C2v(II)]

118.1

1355.2

81.3

3045.2

3068.7

3069.3

3041.0

3057.2

3061.1

8 [C2(II)]

93.3

1354.9

81.2

3020.0

3043.5

3044.1

3015.8

3032.0

3035.9

9 [C2(III)]

107.2

1354.2

81.4

3033.4

3056.9

3057.4

3029.2

3045.4

3049.2

10 [Cs(II)]

122.2

1351.2

82.0

3045.9

3069.4

3070.0

3041.7

3057.9

3061.8

11 [C2(IV)]

40.0

1359.0

80.8

2970.4

2993.9

2994.5

2966.2

2982.4

2986.2

12 [C1(I)]

69.0

1357.4

81.0

2997.9

3021.5

3022.0

2993.8

3009.9

3013.8

13 [C2(V)]

105.6

1355.2

81.3

3032.6

3056.1

3056.7

3028.5

3044.6

3048.5

14 [Cs(III)]

86.0

1358.5

80.8

3015.8

3039.4

3039.9

3011.7

3027.9

3031.7

15 [Cs(IV)]

54.9

1357.3

81.0

2983.8

3007.3

3007.9

2979.6

2995.8

2999.7

16 [Cs(V)]

47.6

1359.2

80.7

2978.1

3001.6

3002.2

2974.0

2990.1

2994.0

17 [C2v(III)]

90.2

1357.2

81.0

3019.0

3042.5

3043.1

3014.9

3031.0

3034.9

18 [C2v(IV)]

78.7

1359.2

80.7

3009.3

3032.8

3033.3

3005.1

3021.3

3025.1

19 [D3d]

47.8

1358.4

80.9

2977.7

3001.2

3001.7

2973.5

2989.7

2993.5

20 [Td]

129.1

1359.0

80.6

3059.3

3082.9

3083.4

3055.2

3071.3

3075.2

21 [D2(III)]

67.8

1355.9

81.3

2995.5

3019.0

3019.6

2991.3

3007.5

3011.4

22 [D2(IV)]

0.0

1358.8

80.9

2930.3

2953.9

2954.4

2926.2

2942.4

2946.2

23 [D2d(II)]

10.7

1359.8

80.7

2941.8

2965.3

2965.9

2937.6

2953.8

2957.7

24 [D6h]

39.6

1359.4

80.7

2970.3

2993.8

2994.4

2966.1

2982.3

2986.2

1 [D2(I)]

217.6

1357.9

2 [C2(I)]

122.9

3 [Cs(I)]


14

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The assignment of the two Cs isomers obtained experimentally by Dennis et al.29 (Cs(a) and Cs(b)) has proved contentious. A number of investigations propose that these isomers are, respectively 14 and 16,32 and this is contradicted by other investigations proposing the reverse assignment.35 In either case, these NMR assignments are apparently made in consideration of relative stabilities of isomers 14 and 16 calculated using an earlier tightbinding model.34, 38 Isomer 16 [Cs(V)] is essentially isoenthalpic to 19 [D3d], according to our proposed ∆fH values of 2994.0 and 2993.5 kJ mol-1, respectively, and so abundances of these two isomers should be comparable on thermodynamic grounds. Moreover, our DHDFT heats of formation indicate that 16 [Cs(V)] and 15 [Cs(IV)] are both more thermochemically stable than 14 [Cs(III)]. Table 1 shows that, isomers 15 and 16 are nearly isoenthalpic (∆fH = 2999.7 and 2994.0 kJ mol-1, respectively), whereas isomer 14 is some ~30 kJ mol-1 less stable by comparison. Assuming that NMR peak intensity correlates with synthetic abundance (hence the heat of formation), our results contradict the prevailing assignment of these two Cs isomers. However, we note that they are consistent with previously reported semi-empirical and DFT relative energies36-40. Several C84 isomers have also not been definitively assigned to date. For instance, the experimentally isolated D2 isomer29, which could possibly be isomers D2(I)-(III), has been suggested to be isomer 5 [D2(II)] based upon the comparison of B3LYP/6-31G* NMR spectra, HOMO-LUMO gaps and relative energies.34 However, these authors also noted that the difference between the calculated NMR spectra of 5 [D2(II)] and 21 [D2(III)] was limited and suggested that 21 [D2(III)] was less likely due to a lower HOMO-LUMO gap. DHDFT ∆fH values reported in Table 1 are consistent with the B3LYP/6-31G* relative energies of Sun et al., with 21 [D2(III)] being essentially isoenthalpic to isomer 5 (∆fH = 3011.4 and 3010.7 kJ mol-1, respectively).


15


Page 16 of 33

Dennis et al.29 also reported an as-yet unidentified C84 isomer via HPLC. It is likely that the abundance of this isomer was comparable to those of both isomers 4 [D2d(I)] and 5 [D2(II)]. In light of the accurate ∆fH°298 values in Table 1, the unidentified isomer is likely either isomer 12 [C1(I)] or 21 [D2(III)], since all four isomers have heats of formation within 10 kJ mol-1 of each other. Abundances of these isomers, based on Boltzmann statistics (2,000 K) using DHDFT ∆fH values reported in Table 1 are 0.2, 0.4, 0.3 and 0.4%, respectively. Tagmatarchis et al.31 also isolated a C2 C84 isomer, which remains experimentally unassigned to our knowledge. It is difficult to assign the identity of this isomer solely on the basis of thermochemical stability; C2 isomers 8, 9 and 13 have ∆fH within ~14 kJ mol-1 (3035.9, 3049.2, and 3048.5 kJ mol-1, respectively), and therefore comparable abundances assuming thermochemical equilibrium (0.05, 0.02 and 0.02% at 1,500 K). However, while isomer 8 is the most thermochemically stable of these isomers observed here, it potentially exists in an open-shell triplet state due to radical substructures, depending on the level of theory employed.42 This is also the case for isomer 9 (we note however that Hückel molecular orbital theory predicts all 24 IPR C84 isomers to be closed shell). Isomers 8 and 9 may therefore be more susceptible to radical attack (i.e. recombination reactions) during the synthesis process, leading to decreased lifetime and synthetic abundances compared to isomer 13 (C2). By comparison, ∆fH for the only alternative C2 isomer (2 [C2(I)]) is 3067.8 kJ mol-1 and therefore significantly higher than those for isomers 8, 9 and 13.

3.3. Kinetic Stabilities of C84 IPR Isomers Fedorov et al.44 have previously shown that the abundance of fullerenes (up to C84) produced during arc discharge correlates with the kinetic stability of the fullerene structure. The kinetic stability, and therefore the synthetic abundance, of a fullerene cage is limited by


16

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the probability with which its weakest C-C bond breaks due to natural vibrational motion. By projecting the time-dependent vibrational motion of each normal mode along each distinct CC bond in the fullerene structure, the central limit theorem can be applied to determine the probability distribution of C-C bond displacements. If this probability exceeds some threshold value, the C-C bond is deemed to have broken. Our analysis of the kinetic stability here adopts a threshold bond length of 1.95 Å, consistent with Fedorov et al.44 The C-C bond cleavage rates (Pcleav) calculated using B3LYP/6-31G* harmonic vibrational frequencies are reported in Table 2 for the 24 IPR C84 isomers. B3LYP/6-31G* is used here based on the demonstrated accuracy of the B3LYP functional as a cost-effective means for obtaining scaled frequencies78-79. ∆fH and Pcleav for the 24 IPR C84 isomers are compared in Figure 3. The calculated cleavage probabilities for the 24 IPR isomers of C84 differ dramatically, consistent with previous reports for cages of varying sizes.44 For example, the most kinetically stable IPR isomer of C84 is isomer 20 [Td], for which Pcleav is 0.18 × 10-16. The corresponding value for the least kinetically stable isomer of C84 (isomer 10 [Cs(II)]) is ca. 3 orders of magnitude larger by comparison (30.13 × 10-16). The high kinetic stability of isomer 20 here serves to illustrate the difference between kinetic and thermochemical stabilities, since this isomer is the third least thermochemical C84 IPR isomer (∆fH = 3075.2 kJ mol−1, Table 1). This low thermochemical stability of isomer 20 is attributed to the high degree of geometrical strain in this tetrahedral fullerene cage, due to the presence of 4 coronene substructures.24 While a similar argument might suggest isomer 20 should have low kinetic stability as well (since strained C-C bonds will be more likely to break), we note here that the C-C bond corresponding to Pcleav in isomer 20 is located at the centre of one of these substructures, i.e. in a region of the cage that has minimal strain. The high kinetic stability of isomer 20 also indicates that the synthetic abundance of a particular fullerene isomer is due to both thermochemical and kinetic factors, i.e. while it is the most kinetically stable, its heat of


17


Page 18 of 33

formation is sufficiently large to prevent its formation during synthesis in appreciable quantities. We note that similar trends are observed for isomers 1 [D2(I)] and 2 [C2(I)] i.e. each of these isomers exhibit a high degree of geometrical strain, leading to the highest relative heats of formation but relatively low Pcleav values.

Figure 3. Comparison of B3LYP/6-31G(d) cleavage rates (Pcleav) at 1,500 K (Table 2) and DSD-PBEP86/def2-QZVPP ∆fH°298 values (Table 1) for 24 IPR C84 isomers. The red line denotes Pcleav(D2(IV)) (i.e. Pcleav of the most thermochemically stable isomer) for comparison.

As anticipated, the maximal C-C cleavage rates for all isolated minor isomers of C84 are significantly higher than those of the two major isomers. This indicates that the correlation between fullerene abundance and kinetic stability, established by Fedorov et al.44 for different fullerene cage sizes (e.g. C36, C60, C70, C84), is also evident for different structural isomers of the same fullerene. For isomers 22 [D2(IV)] and 23 [D2d(II)], Table 2 shows that Pcleav are comparable, being 0.30 × 10-16 and 0.37 × 10-16. The fact that isomer 23 is marginally more


18

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kinetically stable than isomer 22 leads us to attribute the ~2:1 abundance ratio26 of these two major isomers to their relative heats of formation, as opposed to their kinetic stabilities. Table 2. B3LYP/6-31G* maximal cleavage rates (Pcleav) and cleavage rates relative to the most thermochemically stable isomer (Pcleav(D2(IV))/Pcleav) at 1,500 K for 24 IPR C84 isomers. Isomer 1 [D2(I)] 2 [C2(I)] 3 [Cs(I)] 4 [D2d(I)] 5 [D2(II)] 6 [C2v(I)] 7 [C2v(II)] 8 [C2(II)] 9 [C2(III)] 10 [Cs(II)] 11 [C2(IV)] 12 [C1(I)] 13 [C2(V)] 14 [Cs(III)] 15 [Cs(IV)] 16 [Cs(V)] 17 [C2v(III)] 18 [C2v(IV)] 19 [D3d] 20 [Td] 21 [D2(III)] 22 [D2(IV)] 23 [D2d(II)] 24 [D6h]

Pcleav (/10-16) 0.59 0.47 6.09 1.51 0.61 0.88 1.07 1.26 0.94 30.13 0.80 0.64 1.07 0.96 0.80 0.50 1.18 0.53 0.59 0.18 0.61 0.37 0.30 0.33

Pcleav(D2(IV))/Pcleav 0.62 0.78 0.06 0.24 0.60 0.42 0.34 0.29 0.39 0.01 0.46 0.58 0.34 0.38 0.46 0.73 0.31 0.70 0.63 2.10 0.61 1.00 1.22 1.12

On the other hand, the abundances of some minor isomers are attributed to their relative kinetic stabilities. Isomers 19 [D3d] and 24 [D6h], for instance, are essentially isoenthalpic (Table 1) but produced in a ~ 2:3 ratio, consistent with their relative Pcleav values; Table 2 shows that the maximal bond cleavage rate for isomer 24 (0.33 × 10-16) is approximately half that of isomer 19 (0.59 × 10-16). We note also that the relatively low Pcleav values for isomers 19 and 24 correlates with them being the two most abundant minor isomers. Kinetic stability


19


Page 20 of 33

data presented in Table 2 and Figure 3 for isomers 14, 15 and 16 support our previous proposal that the two isolated Cs isomers (Cs(a) and (b)) are in fact isomers 15 and 16, rather than 14 and 16 as previously reported.32, 34-35, 38 For instance, B3LYP/6-31G* Pcleav values for isomers 14, 15 and 16 are, respectively, 0.96 × 10-16, 0.80 × 10-16 and 0.50 × 10-16. Thus, isomer 15 is marginally more stable than isomer 14, both kinetically and thermochemically, and therefore more likely to be synthesised in greater abundance. The established assignment of 11 [C2(IV)]

29

is consistent with this isomer’s Pcleav value

in Table 2 (0.80 × 10-16), which is the lowest for all C2 isomers. From the discussion in the previous section, it was not possible to assign the C2 isomer reported by Tagmatarchis31 definitively solely on the basis of thermochemical heats of formation. The kinetic stability results presented here do not provide further assistance either. Of the remaining C2 isomers, isomer 2 has the highest kinetic stability (Pcleav = 0.47 × 10-16), but the least favourable heat of formation (see above). Conversely, the most thermochemically stable C2 isomer (isomer 8, ∆fH = 3035.9 kJ mol-1) is the least kinetically stable (Pcleav = 1.26 × 10-16).The kinetic stability results reported here do not resolve the identity of the experimentally unassigned D229 isomer. For instance, the two candidate isomers (isomers 5 [D2(II)] and 21 [(D2(III)]) have equal Pcleav values (0.61 × 10-16), as well as effectively the same heats of formation (within 0.5 kJ mol-1). Similarly, definitive identification of the unassigned isomer observed by Dennis et al. is not aided by kinetic stability data here; Pcleav values for isomers 12 and 21 are effectively the same (0.64 × 10-16 and 0.61 × 10-16), which leads us to conclude that these isomers are present in comparable abundances, both on kinetic and thermodynamic grounds.

4. Conclusions


20

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In the present study, we provided the best theoretical estimate to date for heats of formation for all 24 IPR C84 isomers based on accurate double-hybrid DFT energies (DSDPBEP86/def2-QZVPP). An average of the isodesmic-type reactions (11)–(13) for the final proposed ∆fH°298[C84(g)] values was deemed to give efficient cancelation of errors in the calculated reaction energies. For the first time, the relative kinetic stabilities of the 24 IPR C84 isomers were calculated via C-C bond cleavage rates using B3LYP/6-31G* harmonic vibrational frequencies. With the consideration of both the ∆fH and Pcleav values we assist in the identification of several experimentally isolated isomers. The established assignments of isomers 22 [D2(IV)], 23 [D2d(II)], 19 [D3d], 24 [D6h], 11 [C2(IV)], and 4 [D2d(I)] were consistent with ∆fH and Pcleav values and indicated that the relative abundance of the fullerene isomers affords from both kinetic and thermodynamic grounds. Notably, our thermochemical and kinetic stabilities of isomers 14, 15 and 16 indicate that the two isolated Cs isomers (Cs(a) and (b)) are 15 and 16, contrary to some assignments reported previously in the literature. Of the remaining isolated isomers of symmetry C2 and D2, definitive assignment was not possible with consideration of only ∆fH and Pcleav.

Corresponding Author *[email protected] Acknowledgement AJP acknowledges support from the Australian Research Council (INTERSECT, LE170100032). SLW acknowledges a University of Newcastle Postgraduate Scholarship award. BC acknowledges financial support from Japan Society for the Promotion of Science (Grant number 16H07074). This research was undertaken with the assistance of resources ACS Paragon Plus Environment

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provided at the NCI National Facility systems at the Australian National University and INTERSECT systems, through the National Computational Merit Allocation Scheme supported by the Australian Government, and computational resources from RIKEN Advanced Center for Computing and Communication, Japan. The authors are grateful to Prof. Alex Fedorov (Russian Academy of Sciences), Prof. Henyk Witek (National Chaio Tung University) and Prof. Stephan Irle (Oakridge National Laboratory) for code and discussions regarding the kinetic stability of C84 isomers.

Supporting Information Comparison of C84 heats of formation calculated using DSD-PBEP86/def2-QZVPP and those obtained using topological indicator schemes. This material is available free of charge via the Internet at http://pubs.acs.org

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TOC Graphic


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Figure 1. Schlegel diagrams of 24 IPR isomers of C84. Isomers of the same point group are numbered consistent with reference 21 95x137mm (300 x 300 DPI)



Figure 2. B3LYP/6-31G(d) optimized structures of an IPR C84 isomer and its two basic structural units (a) corannulene (C20H10), in which a central isolated pentagon is surrounded by hexagons, and (b) sumanene (C21H12) in which a central hexagon is surrounded by isolated pentagons. 80x69mm (300 x 300 DPI)


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Figure 3. Comparison of B3LYP/6-31G(d) cleavage rates (Pcleav) at 1,500 K (Table 2) and DSD-PBEP86/def2QZVPP ∆fH°298 values (Table 1) for 24 IPR C84 isomers. The red line denotes Pcleav(D2(IV)) (i.e. Pcleav of the most thermochemically stable isomer) for comparison. 80x103mm (300 x 300 DPI)


Accurate Thermochemical and Kinetic Stabilities of C84 Isomers

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