ARTICLE pubs.acs.org/JPCA
Electronic Structure and Stability of Fullerene C82 Isolated-Pentagon-Rule Isomers Ayrat R. Khamatgalimov and Valeri I. Kovalenko* A.E. Arbuzov Institute of Organic and Physical Chemistry of Kazan Scientific Center of Russian Academy of Sciences, Kazan 420088, Russian Federation
bS Supporting Information ABSTRACT: All nine isolated-pentagon-rule isomers of fullerene C82 were investigated by the DFT method with the B3LYP functional at the 6-31G, 6-31G*, and 6-31+G* levels. The distribution of single, double, and delocalized π-bonds in the molecules of these isomers is shown for the first time. The obtained results are fully supported by DFT quantum-chemical calculations of electronic and geometrical structures of these isomers. The molecules of isomers 7 (C3v), 8 (C3v), and 9 (C2v) contain some radical substructures (such as the phenalenyl-radical substructure), which indicates that they are unstable and cannot be obtained as empty molecules. Thus, there is a possibility of obtaining them only as endohedral metallofullerenes or exohedral derivatives. Isomers 1 (C2), 2 (Cs), 4 (Cs), 5 (C2), and 6 (Cs) with closed electronic shell are supposed to be stable, resembling isomer 3 (C2), which has just been extracted experimentally as an empty fullerene. We assume they can be produced as empty molecules.
I. INTRODUCTION Searching for the criteria of stability of fullerene molecules is an important task in view of the fundamental knowledge of an influence of fullerene structure on their stability and the prediction of a possibility of producing stable fullerenes and their prospective applications. Experimental studies of higher fullerenes are substantially limited by the difficulties of their production. There is a significant volume of structural information for the most accessible fullerenes C60 and C70 as well as for some higher fullerenes such as C84. As to fullerenes C82, most publications relate to their endohedral metallofullerenes; very few publications are devoted to structural investigations of C82 empty molecules due to the lack of produced and isolated species. According to the isolated pentagon rule (IPR) fullerene C82 is known to have 9 isomers: three isomers with C2 symmetries (isomers 1, 3, and 5), three isomers with Cs symmetries (isomers 2, 4, and 6), two isomers with C3v symmetries (isomers 7 and 8), and one isomer with C2v symmetry (isomer 9).1 Until now, only one out of nine isomers, i.e., the C2 isomer, has been extracted and tentatively characterized by 13C NMR.2 On the basis of quantum-chemical calculations, this isomer was determined as isomer 3 (C2).3 6 In addition, it was also produced as an exohedral derivative C82(CF3)12,18 together with isomer 5 (C2), which is a second representative of C82 produced as an exohedral derivative C82(CF3)12.7 9 Four isomers, isomer 5 (C2),10 12 isomer 6 (Cs),12 14 isomer 8 (C3v),15 21 and isomer 9 (C2v),12,22 28 were produced as various endohedral metallofullerenes. It has not been yet explained why some IPR isomers cannot be obtained as empty molecules. Generally, instability of fullerenes can be caused by the presence of radical substructures, or in other words unpaired electrons in a molecule (an open shell), and/or by a local strain caused by cage distortions.29 “Missing” fullerenes C74 and C72 are typical examples of these two types of instability, r 2011 American Chemical Society
respectively.29 32 The molecule of C74 is a biradical; two coronene substructures of C72 are the cause of higher local strain of the molecule: these fragments induce higher local sterical strain because these substructures tend to be planar, whereas a fullerene cage is close to spherical. None of these fullerenes is available in empty form yet, though they are regularly observed in mass spectra of a soot.31 The apparent discrepancy between higher stability and the strain (per one carbon atom) in the molecules of C60 and C70 that is higher than the strain in other higher fullerene molecules shows great influence of fullerene cage topology. More uniform distribution of the pentagons (C60, C70, isomers 23 and 22 of C84) defines their high stability, whereas nonuniform pentagon distribution due to many condensed hexagons in C84 isomers 1 (D2) and 20 (Td) leads to substantial local strains and consequently makes them unstable.33 In the present work we attempt to establish the connection between the molecular stability of fullerene C82 isomers and their electronic and geometrical structures, to explain why some IPR isomers cannot be obtained as empty molecules, and finally to predict the possibility of obtaining some of them. For this purpose a distribution of single, double, and delocalized π-bonds of nine IPR isomers of C82 was carried out a priori on the basis of criteria developed by us34 prior to fulfill quantum-chemical calculations. In fact, a fullerene molecule acquires a conventional structural formula with all types of chemical bonds instead of the molecular graph, which only reflects a topology of carbon atoms distribution, i.e., the relative position of hexagons and pentagons. According to this procedure recently we succeeded in disclosing radical substructures responsible for molecular instability of some Received: May 16, 2011 Revised: September 22, 2011 Published: September 30, 2011 12315
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The Journal of Physical Chemistry A higher fullerenes, e.g., C72, C74, C80, C84, and C86;29,30,33,35 39 the results showed to be in perfect agreement with further quantum-chemical calculations. The application of this approach leads to identifying a number of typical substructures named after their aromatic analogs, such as naphthalene, s-indacene, pyrene, perylene, corannulene, coronene, etc. According to our observations, these substructures seem to retain their identity in terms of electronic properties irrespective of the fullerene molecule they belong to. The obtained results are presented as Schlegel diagrams of all nine IPR isomers with the distribution of single, double, and delocalized π-bonds is shown (Figures 1 3).
Figure 1. Schlegel diagrams of stable (experimentally produced) IPR isomer 3 (C2) of fullerene C82. Hereinafter, all pentagons are marked gray; single and double bonds are marked with single and double lines, respectively; delocalized π-bonds are marked with circles.
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II. COMPUTATIONAL METHODS The molecular structures of all IPR C82 fullerene isomers were fully optimized using DFT B3LYP functional with 6-31G basis. In the first step, geometry optimization was performed without symmetry constraints. The calculations showed that in all cases (except singlet configuration of isomer 7 (C3v) and isomer 8 (C3v)) the equilibrium geometry was very close to the molecular symmetry of each isomer. In the case of isomers 7 (C3v) and 8 (C3v) the equilibrium geometries downgrade to Cs symmetry. Similar symmetry reduction of isomers 7, 8, and 9 was also revealed in refs 5 and 40 43 at different levels of the theory. It was expected that the neutral open-shell fullerenes would lift the degeneracy by Jahn Teller distortion to a structure of lower symmetry.44 Therefore, subsequent optimization was carried out with the corresponding symmetry constraint. To improve energies, geometry optimization was followed by a single-point calculation at 6-31G* and 6-31+G* levels. The calculations have shown a good agreement between the results obtained for these basis sets. For isomers assumed to be the radicals the quantum-chemical calculations were carried out in both singlet and triplet configurations using unrestricted Kohn Sham methodology. To ensure the calculated structures were indeed minima, vibrational analyses were performed using the same methods. The tests of stability of singlet and triplet wave functions were carried out. All calculations were performed using the GAUSSIAN03 program.45 The standard keywords in the Gaussian package were used in optimization processes. The exceptions were the high spin states for isomer 8 (C3v), where SCF procedure was performed with the DIIS keyword, which calls for the use of Pulay’s direct inversion in the iterative subspace extrapolation method because conventional method led to convergence failure. Table 1. Average Bond Lengths (Å) of IPR Isomers of C82 isomer no.
Figure 2. Schlegel diagrams of IPR isomers 1 (C2), 2 (Cs), 4 (Cs), 5 (C2), and 6 (Cs) of fullerene C82.
single
double
delocalized
1 (C2)
singlet
1.4562
1.4004
1.4239
2 (Cs)
singlet
1.4555
1.4004
1.4266
3 (C2)
singlet
1.4567
1.3983
1.4258
4 (Cs)
singlet
1.4571
1.3977
1.4276
5 (C2)
singlet
1.4578
1.3965
1.4282
6 (Cs) 7 (C3v)
singlet triplet
1.4571 1.4550
1.3949 1.4011
1.4285 1.4330
8 (C3v)
triplet
1.4597
1.3993
1.4309
9 (C2v)
triplet
1.4583
1.3911
1.4325
Figure 3. Schlegel diagrams of unstable open-shell IPR isomers 7 (C3v), 8 (C3v), and 9 (C2v) of fullerene C82. Radical substructures are marked with dotted lines. 12316
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Table 2. Relative Energies E (kcal/mol) and HOMO LUMO Gaps (eV) of IPR Isomers of C82 E substructuresa
isomer no.
6-31G*
6-31+G*
6-31G
6-31G*
6-31+G*
1 (C2)
singlet
C, I
7.40
6.67
7.31
1.31
1.26
1.24
2 (Cs)
singlet
C, I, C1
6.70
7.98
7.23
1.71
1.65
1.63
3 (C2)
singlet
C, I
0.00
0.00
0.00
1.67
1.63
1.61
4 (Cs)
singlet
C, I, P
4.20
1.14
3.16
1.58
1.55
1.54
17.63
15.81
21.64
0.41
0.43
0.42
8.73
2.99
7.37
1.29
1.28
1.27
15.29
7.32
14.57
0.72
0.73
1.12
triplet 5 (C2)
singlet
C, I, P
triplet 6 (Cs) 7 (C3v)
singlet triplet
C, I, P
12.97 15.48
10.80 17.81
11.40 15.08
1.10 0.89
1.10 0.90
1.09 0.89
singlet (Cs)
C, C1, Ph
26.59
26.19
26.08
0.85
0.85
0.83
22.54
21.93
22.20
1.14
1.16
1.27
28.36
30.70
32.17
0.72
0.74
0.73
28.44
25.42
27.61
0.95
0.95
0.95
19.24
17.65
18.14
0.73
0.73
0.73
17.69
12.05
16.72
0.86
0.87
0.86
triplet 8 (C3v)
singlet (Cs)
C, I, P, Ph
triplet 9 (C2v)
singlet triplet
a
6-31G
HOMO LUMO
I, P, Ph
Substructures designations: C, corannulene; I, s-indacene; C1, coronene; P, perylene; Ph, phenalenyl.
III. RESULTS AND DISCUSSION The characteristics of topological and electronic molecular structure of the stable and isolable isomer 3 (C2) of fullerene C82 (Figure 1) are similar to those of most fullerenes C60 and C70: it has a closed shell; its pentagons are rather uniformly distributed on the cage; it contains corannulene substructure (pentagon surrounded by five hexagons, like C60 and C70) and s-indacene substructure (hexagon with delocalized π-bonds, like C70) (Figure 1). The results of preliminary bond distributions in this and other isomers are completely supported by DFT calculations (Table 1 and Supporting Information): values of calculated bond lengths correspond to plotted single, double, and delocalized π-bonds at Schlegel diagrams. According to quantum-chemical calculations isomer 3 (C2) is the most energetically favorable (Table 2). This result agrees well with the results of other theoretical studies.5,6,41 43,46 49 The analysis of molecular structures of isomers that were not obtained yet, i.e., 1 (C2), 2 (Cs), and 4 (Cs), shows that in addition to corannulene substructures they have three indacene substructures. Furthemore, the isomer 2 (Cs) has one coronene substructure (hexagon surrounded by six hexagons) and isomer 4 (Cs) has one perylene substructure (hexagon with two pairs of oppositely attached hexagons with delocalized π-bonds) (Figure 2). The molecules of isomers 5 9 of C82 have more complex structures composed of condensed hexagons (Figures 2 and 3) with delocalized π-bonds. Besides corannulene substructures the isomer 5 (C2) has two indacene and perylene substructures; isomer 6 (Cs) have two indacene and three perylene substructures (Figure 2). Isomers 7 (C3v), 8 (C3v), and 9 (C2v) include substructures that are suspected to be radical ones (Figure 3); i.e., they contain some unpaired electrons (depicted by dots on Figure 3), similar to the IPR fullerene C7430 or some other openshell fullerenes.37,38 Accordingly, they may be kinetically unstable. The possibility of isolation of a fullerene in an individual form is governed by its stability (both thermodynamic and kinetic). We relate to fullerenes as stable compounds that can be synthesized (the thermodynamic stability) and that remain
Figure 4. Position of endohedral metal atom (depicted by black circle) in isomers of fullerene C82 (two views) and radical substructures (dotted lines).
unchanged for a long period of time under normal conditions, i.e., on air at room temperature (the kinetic stability).29 Undoubtedly, the more the number of unpaired electrons on the fullerene shell, the less stable is that structure. They cannot be produced as empty molecules; nevertheless, they might be obtained as endohedral metallofullerenes or exohedral derivatives. Quantum-chemical calculations show that the lowest energy wave functions of these open-shell isomers 7 (C3v), 8 (C3v), and 9 (C2v) are the triplet configurations, which is in agreement with our preliminary estimations (Table 2). Additional confirmation that triplet states of these isomers are more favorable than singlet ones was revealed by checking the wave function stability and showed that in the case of a singlet state of these isomers RHF-toUHF instability was observed. This means that the triplet is a lower-lying state than the singlet. The wave functions of triplet states are stable under the perturbations considered. We also explored some higher spin states with multiplicities of 5 and 7 for isomers 7 (C3v), 8 (C3v), and 9 (C2v) which have 4, 4, and 6 radical centers, respectively (Figure 4). The quantum-chemical calculations show these states are not the ground states (see Supporting Information). 12317
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Table 3. Maximum Dihedral Angles (deg) in IPR Isomers of C82 hexagons delocalized π-bonds
isomer no.
bond alternation
pentagons
1 (C2)
17.28 (2)*
14.77 (2)
9.91 (2)
2 (Cs) 3 (C2)
16.39 (2) 16.47 (2)
12.52 (2) 13.66 (2)
12.74 (2) 10.68 (2)
Stable Isomers
4 (Cs)
15.09 (2)
12.86 (2)
11.04 (2)
5 (C2)
13.77 (2)
12.94 (2)
10.09 (2)
6 (Cs)
11.07 (2)
12.81 (2)
9.27 (2)
7 (C3v)
11.94 (3)
12.27 (6)
8 (C3v)
10.39 (6)
10.63 (6)
8.59 (3)
9 (C2v)
7.83 (4)
10.38 (4)
6.88 (4)
Open-Shell Isomers 11.55 (6)
* Numbers in parentheses represent the number of equivalent cycles according to the isomer symmetry. Experimentally produced isomer 3 (C2) is indicated in bold.
Figure 5. Maximal spin densities in triplet configurations of isomers 7 (C3v), 8 (C3v), and 9 (C2v) of C82. Red and blue indicate the positive and negative values, respectively.
A detailed comparative analysis of bond lengths and bond angles did not display substantial deviations from these main geometrical parameters of fullerene molecules in question. All of them are close to typical values for most stable fullerenes C60 and C70. This observation is quite obvious because the structures of these isomers are close to spherical. However, the results of the analysis of dihedral angles inside hexagons and pentagons are more impressive. The high distortion of fullerene cage, i.e., high nonplanarity of hexagons and pentagons, reflects local strain of a fullerene molecule33 so it is directly connected with its thermodynamic instability. It is well-known1 that the molecule of the most stable fullerene C60 consists only of planar hexagons and pentagons; i.e., their dihedral angles are zeros. A distortion of hexagons and pentagons appears in fullerene C70; furthermore, maximum distortions (dihedral angles vary at 10 15°) appear in hexagons with delocalized π-bonds that do not affect this fullerene stability.50 An analysis of molecular geometry of studied IPR isomers of C82 shows that all of them have no high local strains. All hexagons and pentagons in isomers of fullerene C82 are distorted, but similarly to C70, maximum distortions are in hexagons with delocalized π-bonds (Table 3 and Supporting Information). The degree of deformations in other hexagons and pentagons does not exceed distortions in stable isomer 3 (C2). Thus, the instability of “empty” isomers 7 (C3v), 8 (C3v), and 9 (C2v) is governed by the presence of radical substructures. Production of
isomers 7 9 as empty molecules seems to be hardly probable, whereas the isomers 1 (C2), 2 (Cs), 4 (Cs), 5 (C2), and 6 (Cs) with closed electronic shell have to be stable (see Table 2). These isomers could be produced as empty molecules. The possibility of production of isomers 1 (C2), 2 (Cs), and 4 (Cs) as empty species was also supposed in refs 5 and 6. The position of the atom(s) inside the fullerene cage is one of the basic questions studied in investigations of endohedral metallofullerene structures because of its strong influence on electronic properties of fullerene. The metal atom(s) is(are) usually located at an off-center position of the fullerene cage; moreover, sometimes a dynamic motion of metal atoms within the cage is observed.23,51 Combining numerous experimental and theoretical data3,23,28,52 61 that concern metal ion positions in molecules of fullerene C82, one may see that endohedral metal ions are located close to condensed hexagon radical substructures in molecules of isomers 7 (C3v), 8 (C3v), and 9 (C2v) (Figure 4). It means that the addition of extra electrons to the unpaired ones creates a higher electron density, namely at these fragments of molecules, instead of being diffused on the fullerene sphere. Therefore, due to a higher electronic density of these substructures, metal cations shift to them. It should be noted that spin densities in triplet configurations of isomers 7 (C3v), 8 (C3v), and 9 (C2v) of C82 are mainly concentrated on radical substructures likewise the C74 biradical30 and spin density distributions in isomers 7 (C3v) and 8 (C3v) are quite similar (Figure 5).
IV. CONCLUSIONS As a result of an analysis of energy and geometrical parameters of molecules of nine C82 fullerene IPR isomers carried out on the basis of an early approach developed by us and followed by DFT calculations, their molecules obtain the form of a conventional structural formula with all types of chemical bonds instead of the molecular graph that reflects only the relative position of hexagons and pentagons. Molecules of isomers 7 (C3v), 8 (C3v), and 9 (C2v) are shown to contain radical substructures (like the phenalenyl-radical substructure). It means they are unstable and cannot be obtained as empty molecules. However, they become stable as exohedral derivatives or as endohedral metallofullerenes due to well-known “donating” electrons to the electron-deficient radical substructures of fullerene cage. By analysis of calculated and experimentally founded metal atom(s) positions it was revealed that endohedral metal ions are located near determined condensed hexagons identified as radical substructures. The spin densities in triplet configurations of isomers 7 (C3v), 8 (C3v), and 9 (C2v) of C82 are mainly concentrated on this radical substructures. Isomers 1 (C2), 2 (Cs), 4 (Cs), 5 (C2), and 6 (Cs) with closed electronic shell are supposed to be stable and may be produced as empty molecules. ’ ASSOCIATED CONTENT
bS
Supporting Information. Schlegel diagrams of all nine IPR isomers of fullerene C82 with plotted calculated bond lengths, and dihedral angle values in hexagons and pentagons. Relative energies and HOMO LUMO gaps of higher spin states for IPR isomers 7 (C3v), 8 (C3v), and 9 (C2v) of C82. This information is available free of charge via the Internet at http:// pubs.acs.org.
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’ AUTHOR INFORMATION Corresponding Author
*Fax: +7 (843) 2732253. Tel: +7 (843) 2732283. E-mail koval@ iopc.ru.
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