Introduction

The study of higher fullerenes is quite complicated due to their very low abundance in the fullerene soot and the presence of hundreds of topologically possible isomers with rather lower symmetries that obey the isolated pentagon rule (IPR). IPR suggests that the fullerene molecule with abutting pentagons is unstable [1, 2]. Theoretical investigations of the structures of large fullerenes and their isomers become necessary to analyze experimental results, to determine the structure and stability of isomers, and to predict new structures. Among the other higher fullerenes Cn (n > 60), the theoretical information on C100 and C108 fullerenes is very limited.

For C100, 450 possible molecular geometries satisfy the IPR [3]. However, to date, there is no C100 isomers that have been isolated and characterized as an empty cage; they only have been registered by mass spectrometry. According to theoretical calculations, isomer 449 (D2) is the most stable one followed by isomers 18 (C2), 173 (C1), 174 (C2), 425 (C1), 440 (C2), 442 (C2), in different orders depending on the used theoretical level [4,5,6,7,8]. Besides, considerable temperature effects on the stability are found so that not only the ground-state 449 (D2) isomer but also several other structures (18 (C2), 426 (C1), 425 (C1), 442 (C2), 148 (C1)) are significantly populated within a wide temperature interval [5]. Nevertheless, several isomers of fullerene C100 have been identified as various derivatives (exohedral and endohedral).

For the first time, C100 was confirmed experimentally as an exohedral C100Cl12 with a molecule of unstable isomer 1 (D5d) with a nanotubular shape [9]. Isomers 18 (C2), 425 (C1), and 417 (C2v) of C100 were also obtained as chloro derivatives whose structures were identified by X-ray analysis [10]. Isomers 425 (C1) and 18 (C2) were also identified as C1–C100(425)Cl22 and C2–C100(18)Cl28/30 compounds, respectively, with retained IPR cage connectivities. In contrast, isomer 417 (C2v) has been isolated as Cs–C100Cl28 which loses a C2 fragment, resulting in the nonclassical C1–C98Cl26 with a heptagon in the carbon cage. The computations showed that this isomer originates presumably from isomer 603 (C1) of fullerene C102, which undergoes the skeletal transformations, including chloride formation, C2 elimination, and SW transformation [11]. A rather stable isomer 18 (C2) was also indirectly confirmed as the most probable starting fullerene, whose skeletal transformations lead to obtaining of the nonclassical exohedral C96Cl20 and C94Cl22 with three and one heptagons, respectively [12, 13].

In contrast to the empty C100 fullerenes, endohedral derivatives demonstrate completely different relative stabilities. Thus, IPR isomer 450 (D5) was theoretically predicted to be the most energetically advantageous C100 cage for the fullerenes with endohedral atoms able to donate up to six electrons to the carbon cage. [8, 14, 15] These theoretical findings were later confirmed when La2@C100(450) had been isolated and established by X-ray analysis [16]. It should be noted that this isomer is one of the most unstable and theoretical calculations predict for it extremely low HOMO-LUMO gap close to zero [5]. The isomer 450 (D5) was also found in the crystal structure of endohedral carbide cluster metallofullerene La2C2@C100 [17]. Yang and Dunsch reported the preparation of Dy2@C100 with two metal atoms inside the fullerene cage [18]. They suggested that the most possible hosts are presented by the most stable empty-cage isomers of C100 [18]. However, they ignored the charge transfer between the encapsulated metal atoms and the fullerene cage. The role of the latter has been emphasized by Valencia, Rodríguez-Fortea, and Poblet, who found that the isomer 450 (D5) was an unusually promising candidate [19]. It was also revealed that this cage provides maximal separation between the pentagons, representing the sites of negative charge localization in the fullerene anions [20, 21].

Even fewer studies are devoted to fullerene C108. For C108, 1799 possible molecular geometries satisfy the IPR [3]. Similar to fullerene C100, to date among them, no C108 isomers have been isolated and characterized as an empty cage. According to theoretical calculations, isomer 1771 (D2) is one of the most stable (with a significant HOMO−LUMO gap) among 1799 IPR isomers; thus, this isomer is likely to coexist in the soot and hard to separate experimentally [7]. The most stable isomer is followed by isomers 1643 (C1), 1687 (C1), 1644 (C2), 1648 (C1), 1686 (C2), 1025 (C1), 1646 (C1), 1779 (C2), 1769 (C1), 1062 (C1), 1078 (D3), 574 (C1), 1735 (S4), 1765 (C2), 206 (D3h), in different orders depending on the used theoretical level [6, 7].

Nevertheless, several isomers of fullerene C108 have been identified as various derivatives (exohedral and endohedral). The most stable isomer 1771 (D2) is identified as chloro derivative C108Cl12 by single-crystal X-ray analysis with the use of synchrotron radiation resulted in the structure determination and represents so far the largest pristine fullerenes with known cages [22]. Isomer 1660 (C1) has been very recently characterized by X-ray diffraction as endohedral carbide metallofullerenes Y2C2@C108 representing the largest metallofullerene that have been characterized by crystallography to date [23]. This giant species have been studied theoretically using the density functional theory (DFT), and even the MP2 technique, in order to supply further structure and stability data [24].

Thus, from 450 and 1799 IPR isomers for fullerenes C100 and C108, respectively, no one isomer has been isolated and characterized as an empty cage so far. Nevertheless, several isomers of them have been stabilized and identified as various exohedral and endohedral derivatives. In this report, we present the study of the molecular structures of IPR isomer 450 (D5) of C100 fullerene that is one of the most unstable isomers but stabilized as endohedral fullerenes La2@C100, La2C2@C100, and, presumably, Dy2@C100 and IPR isomer 1771 (D2) of C108 fullerene that is one of the most stable to establish their structures, reasons of instability, and their stabilization as various derivatives.

Methodology

Previously, we have developed an approach for studying the higher fullerenes that provides a complete structural formula of fullerene with the distribution of single, double, and π-delocalized bonds prior to the quantum-chemical calculations. We considered the substructures existing in the most stable fullerene molecules, for example, corannulene in C60 or indacene substructures in C70, as a factor which does not reduce the molecular stability of any fullerene. On the contrary, the presence of large substructures consisting only of hexagons is the reason for significant local strains in fullerene molecule, and radical substructures are the reasons of the instability of such fullerenes. In fact, the analysis of all IPR molecular structures of higher fullerenes, such as C72, C74, C76, C80, C82, C84, C86 [25,26,27,28,29 and references therein], and some small fullerenes, such as C40 and C50 [30,31,32], confirmed this assumption.

The distribution of bonds in researched IPR isomers was carried out in accordance with the following rules: (i) pentagons consist only of single bonds, while in hexagons there is an alternation of single and double bonds; (ii) according to symmetry requirement, the hexagon with delocalized pi-bonds can be implemented; (iii) the distribution of the bonds should not lower the fullerene molecular symmetry; (iv) corannulene and indacene substructures, characteristic for the most stable fullerene C60 and C70, are preferred.

The molecular structures of the investigated IPR isomers were fully optimized using DFT B3LYP functional [33, 34] with the 6-31G basis. At the first step, geometry optimization was performed without symmetry constraints. The calculations showed that in all cases, except for singlet and quintet configurations of isomer 450 (D5) of fullerene C100, the equilibrium geometries corresponded to the topological molecular symmetry of each isomer. Therefore, subsequent optimizations were carried out with the corresponding symmetry constraints. The standard keywords in the Gaussian package were used in optimization processes. To improve energies, geometry optimizations were followed by single-point calculations at the 6-31G* level. The calculations showed a good agreement between the results obtained for all used basis sets.

The isomer 450 (D5) of fullerene C100 was considered with the open-shell electronic structure; the quantum-chemical calculations were carried out in triplet, quintet, and septet configurations using unrestricted Kohn-Sham methodology. To ensure the calculated structures correspond to minima, vibrational analyses were performed at the same level of theory. The tests of the stability of wave functions were carried out. All calculations were performed using the GAUSSIAN16 program [35].

Results and discussion

Analysis of bond distribution in the isomer 450 (D5) of fullerene C100 (Fig. 1, left) revealed that it contains two substructures with five condensed phenalenyl-radicals at the poles of the molecule that we named as othrene (Fig. 1, right). Phenalenyl substructure is known as a radical because all three bonds that radiate from the central carbon atom of this substructure must be equivalent according to molecular symmetry. Therefore, the carbon atom located on the third-order symmetry axis is considered to have one unpaired electron. This situation is similar to radical-fullerene C74 (D3h) [27, 36] which has two phenalenyl-radical substructures on the C3 axis of a molecule passing through the central carbon atom. Interestingly, two similar substructures with five condensed phenalenyl-radicals (othrene substructures) at the poles of the molecule were previously found in the IPR isomer 31923 (D5h) of C80 fullerene [37]. It was shown that this IPR isomer cannot be isolated experimentally due to its radical nature [37]. Noteworthy is the presence of the equatorial belt from 20 condensed hexagons. So, according to an analysis of bond distribution, the isomer 450 (D5) of fullerene C100 has an open-shell electronic structure.

Fig. 1
figure 1

Schlegel diagram of the IPR isomer 450 (D5) of fullerene C100 (left) and substructure from five condensed phenalenyl-radicals—othrene (right)

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The isomer 1771 (D2) of fullerene C108 (Fig. 2, left) contains four corannulene and four s-indacene substructures that are stable like in C70 (D5h) fullerene. Additionally, analysis of bond distribution allowed to reveal two substructures with two condensed coronene at the equator of the molecule (Fig. 2, right). The pair of coronene substructures of C72 is the reason for high local strains because this substructure tends to be planar, whereas a fullerene cage is close to spherical [25, 26]. However, it seems quite evident that the presence of coronene substructures becomes less critical when molecular size increases. Nevertheless, in our opinion, the presence of a flat substructure with two condensed coronenes should introduce local strain even into a large molecule. In addition, both substructures of the “fused” coronenes are connected by two pairs of hexagons forming a belt consisting of 28 hexagons, which also introduces significant local strain. Thus, an analysis of bond distribution shows that this isomer has a closed shell without any radical centers with unpaired electrons, and the instability of this isomer is supposedly associated with high local strain.

Fig. 2
figure 2

Schlegel diagram of the IPR isomer 1771 (D2) of the fullerene C108 (left) and substructure from two condensed coronenes (right)

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DFT calculations predict the lowest energy for the triplet configuration of isomer 450 (D5) of fullerene C100, whereas isomer 1771 (D2) of the fullerene C108 has a closed electronic shell (Table 1). It agrees with our tentative structural estimations. A symmetry-lowering distortion for singlet and quintet electronic configurations of isomer 450 (D5) of fullerene C100 was founded. For triplet and septet configurations with 2 and 6 unpaired electrons, respectively, the D5 symmetry is maintained; therefore, further, only these configurations were analyzed.

Table 1 Relative energies (ΔE, kcal/mol) and HOMO-LUMO gaps (eV) of isomer 450 (D5) of fullerene C100 and isomer 1771 (D2) of fullerene C108
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The open shell nature of isomer 450 (D5) of fullerene C100 is additionally confirmed by the tests of the wave function stability. For the singlet states, computations predict RHF-to-UHF instability, suggesting the existence of state with lower energy. The wave functions of triplet, quintet, and septet states are stable under the considered perturbations.

The preliminary assumed bond distributions according to the developed approach are confirmed in DFT calculations: calculated bond lengths correspond to single, double, and delocalized pi-bonds plotted on Schlegel diagrams and are also in agreement with the well-known experimental values for most stable C60 and C70 fullerenes [38, 39] (Table 2 and Table S1 in Supporting Information).

Table 2 The shortest and the longest bond lengths (Å) of isomer 450 (D5) of fullerene C100 and isomer 1771 (D2) of fullerene C108
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However, the calculations revealed some deviations of the bond lengths from those expected during the bond distribution analysis by the developed approach: in particular, instead of some double bonds, the calculation shows the bond lengths, which should be classified as single (see max values of double bonds in Table 2 and Table S1 in Supporting Information). The analysis shows these discrepancies concern the bonds in the equatorial belt from 20 and 28 condensed hexagons in isomer 450 (D5) of fullerene C100 and isomer 1771 (D2) of fullerene C108, respectively. It should be mentioned that such bond length distribution was found for the nanotubes [40, 41]. This circumstance should probably be considered when analyzing large fullerene molecules with fragments where pentagons are completely absent.

Previously, it has been shown that high distortion of fullerene cage, i.e., high nonplanarity of hexagons and pentagons causing a local strain in a fullerene molecule, is directly connected with its thermodynamic instability [26, 37, 42,43,44,45,46]. The most stable fullerene C60 molecule contains all planar hexagons and pentagons [38], whereas in the C70 molecule some distortions of hexagons and pentagons appear, with highest nonplanarity in hexagons with delocalized π-bond [47]. Such distortions of higher fullerene molecules are probably caused by the compensation of π-delocalization of hexagons embedded in a sphere-like fullerene cage. The analysis of molecular geometries of the fullerenes under study shows that there are no flat hexagons and pentagons in their structures (Fig. 3, Table 3). In contrast to C70, maximum distortions are observed in hexagons with bond alternation (see Table 3). This suggests the presence of substantial local molecular strains in both studied fullerene molecules. The analysis of the localization of the most distorted cycles (with the maximum dihedral angles in a cycle) shows that they contain the elongated double bonds (see Fig. 3, depicted by bold).

Fig. 3
figure 3

Schlegel diagrams with maximal dihedral angles (degree) in cycles in isomer 450 (D5) of fullerene C100 (left) and isomer 1771 (D2) of the fullerene C108 (right)

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Table 3 Maximal dihedral angles (degree) in isomer 450 (D5) of fullerene C100 and isomer 1771 (D2) of fullerene C108
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A comparative analysis of dihedral angles between cycles (between hexagons, between pentagon and hexagon) in studied fullerenes and strained molecules of isomers 1 (D2), 2 (C2), and 20 (Td) of fullerene C84 [42, 44, 48] shows that dihedral angles in these molecules are comparable (Table 4) that also indicates the presence of significant local strains in the molecules of studied fullerenes.

Table 4 Dihedral angles between hexagons (HH) and between pentagon and hexagon (PH) in isomer 450 (D5) of fullerene C100 and isomer 1771 (D2) of fullerene C108 and locally strained molecules of isomers 1 (D2), 2 (C2), and 20 (Td) of fullerene C84
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The spin density in the triplet and septet configurations of isomer 450 (D5) of fullerene C100 is mainly concentrated on atoms of central pentagons of othrene radical substructures (Fig. 4 and Table S2 in Supporting Information). Analysis of DFT results revealed that configuration with 6 unpaired electrons (septet) is most preferred that is rationalized by the fact that part of the spin density in triplet configuration is outside of the othrene substructures, whereas in septet configuration the spin density is concentrated in the othrene substructures. The spin density distribution predetermines the position of the metal atom(s) inside the possibly synthesized in the future endohedral derivative or the order of radical addition in reactions of synthesis of exohedral derivatives. Really, DFT calculations of dimetallofullerenes M2@C100 (M = La, Y, Sc, Dy) show that metal atoms in these endohedral dimetallofullerenes are located near poles of molecules [14, 15, 17]. Authors explain such metal atom’s positions by the longest metal-metal distances to minimize the electrostatic repulsion between them [15]. However, in our opinion, the positions of metal atoms are determined by the initial structure of the fullerene with the presence of radical centers at the poles of the molecule and the corresponding distribution of spin density.

Fig. 4
figure 4

The spin density distribution (marked by bold circles in accordance with their values from 0.03 to 0.14) in the triplet (left) and septet (right) configurations of isomer 450 (D5) of fullerene C100; regions with maximum spin densities are depicted by yellow

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Analysis of the chlorination pattern of C108-1771(C2)Cl12 shows that 12 chlorine atoms are unequally distributed on the C108 cage [22]. Usually, the most stable addition patterns of the derivatives with 12 atoms or groups are characterized by their uniform distribution on the fullerene cages [22]. Authors explain such deviations from the general rule by the formation of stabilizing substructures on the carbon cages such as benzenoid rings or isolated C=C bonds. Our combined analysis of the initial molecule structure and experimental data on the chlorine addend distribution [22] reveals that 8 of 12 addends attached by pairs to all four hexagons with delocalized π-bonds (Fig. 5). Such an attachment is most advantageous from the point of view of minimal rearrangement of the entire electronic system of the molecule. So, analysis of chlorination pattern confirmed our earlier conclusion [49] about the preference of hexagons with delocalized π-bonds as the most feasible positions of addend in radiсal addition reaction.

Fig. 5
figure 5

The initial structure of isomer 1771 (D2) of fullerene C108 together with the position of addends in C108-1771(C2)Cl12 according to [22] (black circles denote the positions of attached Cl atoms)

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Thus, the instabilities of the studied fullerenes are caused by the significant local strains due to the presence of substructures consisting of condensed hexagons for both researched fullerenes and additionally by open-shell electronic structure for isomer 450 (D5) of fullerene C100. It means they should be unstable and probably could not be obtained as empty molecules. However, they can be stabilized in polymeric form and as exohedral or endohedral derivatives. It is shown that application of the developed approach in combination with quantum chemical calculations can be successfully used for determining the structure of molecules of higher fullerenes with the number of carbon atoms greater than 90 which is useful to obtain such information by experimental methods.

Conclusions

An analysis of the molecular structures of IPR isomer 450 (D5) of fullerene C100 and IPR isomer 1771 (D2) of fullerene C108 was carried out. The data about the distributions of single, double, and delocalized π-bonds in studied isomer molecules as well as their molecular formulas are presented for the first time. It is revealed that isomer 450 (D5) of fullerene C100 contains two substructures from condensed phenalenyl-radicals at the poles of the molecule (i.e., has an open electronic shell), whereas isomer 1771 (D2) of fullerene C108 has a closed electronic shell and contains a substructure from condensed coronenes at the equator of the molecule. Their stabilities are evaluated in accordance with local strains in the molecules and/or the presence of radical substructures. The most probable positions of addends in the products of radical addition are shown.