Essential Factors for Control of the Equilibrium in the Reversible Rearrangement of M3N@Ih‐C80 Fulleropyrrolidines: Exohedral Functional Groups versus Endohedral Metal Clusters

The effects of exohedral moieties and endohedral metal clusters on the isomerization of M₃N@I_h‐C₈₀ products from the Prato reaction through [1,5]‐sigmatropic rearrangement were systematically investigated by using three types of fulleropyrrolidine derivatives and four different endohedral metal clusters. As a result, all types of derivatives provided the same ratios of the isomers for a given trimetallic nitride template (TNT) as the thermodynamic products, thus indicating that the size of the endohedral metal clusters inside C₈₀ was the single essential factor in determining the equilibrium between the [6,6]‐isomer (kinetic product) and the [5,6]‐isomer. In all the derivatives, the [6,6]‐ and [5,6]‐Prato adducts with larger metal clusters, such as Y₃N and Gd₃N, were equally stable, which is in good agreement with DFT calculations. The reaction rate of the rearrangement was dependent on both the substituent of exohedral functional groups and the endohedral metal‐cluster size. Further DFT calculations and ¹³C NMR spectroscopic studies were employed to rationalize the equilibrium in the rearrangement between the [6,6]‐ and [5,6]‐fulleropyrrolidines.     

Nonclassical fullerenes with a heptagon violating the pentagon adjacency penalty rule

Nonclassical fullerenes with heptagon(s) and their derivatives have attracted increasing attention, and the studies on them are performing to enrich the chemistry of carbon. Density functional theory calculations are performed on nonclassical fullerenes C_n (n = 46, 48, 50, and 52) to give insight into their structures and stability. The calculated results demonstrate that the classical isomers generally satisfy the pentagon adjacency penalty rule. However, the nonclassical isomers with a heptagon are more energetically favorable than the classical ones with the same number of pentagon–pentagon bonds (B₅₅ bonds), and many of them are even more stable than some classical isomers with fewer B₅₅ bonds. The nonclassical isomers with the lowest energy are higher in energy than the classical ones with the lowest energy, because they have more B₅₅ bonds. Generally, the HOMO–LUMO gaps of the former are larger than those of the latter. The sphericity and asphericity are unable to rationalize the unique stability of the nonclassical fullerenes with a heptagon. The pyramidization angles of the vertices shared by two pentagons and one heptagon are smaller than those of the vertices shared by two pentagons and one hexagon. It is concluded that the strain in the fused pentagons can be released by the adjacent heptagons partly, and consequently, it is a common phenomenon for nonclassical fullerenes to violate the pentagon adjacent penalty rule. These findings are heuristic and conducive to search energetically favorable isomers of C_n, especially as n is 62, 64, 66, and 68, respectively. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Structural connectivity and formation mechanism of monometallic cluster fullerenes YCN@Cn (n = 68–84)

Excited by the recently experimental reports of monometallic cluster fullerenes, we examined the electronic and geometrical properties of monometallic cluster fullerenes YCN@C_n with size from C₆₈ to C₈₄ by density functional theory and statistical thermodynamic calculations. The calculations demonstrate that the thermodynamically favored isomers of YCN@C_n are in good agreement with available experimental results. Morphology analysis shows that the lowest‐energy YCN@C_n species are structurally connected by C₂ insertion/extrusion and Stone–Wales rotation, which can be promoted under high temperature; enthalpy–entropy interplay can change the relative abundances of low‐energy isomers significantly at high temperature. All the results suggest that there is a structural evolution among these metallic cluster fullerenes in discharge condition, and thus, can rationalize their structural diversity in the soot and partly disclose their formation mechanism. The geometrical structures, electronic properties of these endohedral fullerene were discussed in detail.     

Bingel–Hirsch Addition on Endohedral Metallofullerenes: Kinetic Versus Thermodynamic Control

An extensive theoretical study of the Bingel–Hirsch addition of bromomalonate on scandium nitride endohedral fullerenes has been carried out. The prototypical and highly symmetrical Sc₃N@I_h‐C₈₀, with a structure that satisfies the isolated pentagon rule (IPR), and the non‐IPR Sc₃N@D₃(6140)‐C₆₈ fullerene show analogous reaction paths despite the distinct topology of the carbon networks and different rotation freedom of the internal nitride cluster. For the two metallofullerenes, our results predict that the reaction takes place under kinetic control yielding open‐cage fulleroids on [6,6] bonds, which is in good agreement with experimental data. The theoretical studies also show that predicting the reactivity of endohedral metallofullerenes is not straightforward and often an accurate analysis of the potential energy surface is required.     

Purification of endohedral trimetallic nitride fullerenes in a single, facile step

A major hurdle hampering the development of fullerenes, endohedral metallofullerenes, and nanotubes has been the difficulty of obtaining high purity samples. Soots prepared in the usual manner via a Kr?tschmer-Huffman electric-arc generator consist of mixtures of insoluble carbonaceous materials and soluble fullerenes: C60, C70, C76, C78, C84, etc. When metals are introduced as endohedral species the complexity of the resultant soot is even greater because of the presence of multiple isomers of both the empty fullerenes and the endohedral metallofullerenes. Here, for the first time, we report that lanthanide trimetallic nitride endohedral metallofullerenes, A3N@C80 (A = lanthanide atom, e.g., Er, Gd, Ho, Lu, Sc, Tb, Tm, Y), can be obtained in pure form directly from as-prepared soots in a single facile step by taking advantage of their extraordinary kinetic chemical stability with respect to the other fullerenes in Diels-Alder reactions with a cyclopentadiene-functionalized resin. We show that careful control of conditions (stoichiometry, time, temperature) allows separation of fullerenes with different cage sizes, as well as isomeric species. Furthermore, the Diels-Alder reaction is thermally reversible, and we demonstrated that the bound empty-cage fullerenes and classical endohedral metallofullerenes can be recovered by displacement with maleic anhydride.     

Chlorination of IPR C100 Fullerene Affords Unconventional C96Cl20 with a Nonclassical Cage Containing Three Heptagons

Chlorination of C₁₀₀ fullerene with a mixture of VCl₄ and SbCl₅ afforded C₉₆Cl₂₀ with a strongly unconventional structure. In contrast to the classical fullerenes containing only hexagonal and pentagonal rings, the C₉₆ cage contains three heptagonal rings and, therefore, should be classified as a fullerene with a nonclassical cage (NCC). There are several types of pentagon fusions in the C₉₆ cage including pentagon pairs and pentagon triples. The three‐step pathway from isolated‐pentagon‐rule (IPR) C₁₀₀ to C₉₆(NCC‐3hp) includes two C₂ losses, which create two cage heptagons, and one Stone–Wales rotation under formation of the third heptagon. Structural reconstruction established C₁₀₀ isomer no. 18 from 450 topologically possible IPR isomers as the starting C₁₀₀ fullerene. Until now, no pristine C₁₀₀ isomers have been confirmed based on the experimental results.     

2‐Aminoethanol Extraction as a Method for Purifying Sc3N@C80 and for Differentiating Classes of Endohedral Fullerenes on the Basis of Reactivity

Extraction with 2‐aminoethanol is an inexpensive method for removing empty cage fullerenes from the soluble extract from electric‐arc‐generated fullerene soot that contains endohedral metallofullerenes of the type Sc₃N@C_2n (n=34, 39, 40). Our method of separation exploits the fact that C₆₀, C₇₀, and other larger, empty cage fullerenes are more susceptible to nucleophilic attack than endohedral fullerenes and that these adducts can be readily extracted into 2‐aminoethanol. This methodology has also been employed to examine the reactivity of the mixture of soluble endohedral fullerenes that result from doping graphite rods used in the Krätschmer–Huffman electric‐arc generator with the oxides of Y, Lu, Dy, Tb, and Gd. For example, with Y₂O₃, we were able to detect by mass spectrometry several new families of endohedral fullerenes, namely Y₃C₁₀₈ to Y₃C₁₂₆, Y₃C₁₀₇ to Y₃C₁₂₅, Y₄C₁₂₈ to Y₄C₁₄₆, that resisted reactivity with 2‐aminoethanol more than the empty cage fullerenes and the mono‐ and dimetallo fullerenes. The discovery of the family Y₃C₁₀₇ to Y₃C₁₂₅ with odd numbers of carbon atoms is remarkable, since fullerene cages must involve even numbers of carbon atoms. The newly discovered families of endohedral fullerenes with the composition M₄C_2n (M=Y, Lu, Dy, Tb, and Gd) are unusually resistant to reaction with 2‐aminoethanol. Additionally, the individual endohedrals, Y₃C₁₁₂ and M₃C₁₀₂ (M=Lu, Dy, Tb and Gd), were remarkably less reactive toward 2‐aminoethanol.     

Singly Bonded Monoadduct rather than Methanofullerene: Manipulating the Addition Pattern of Trimetallic Nitride Clusterfullerene through One Endohedral Metal Atom Substitution

Bingel–Hirsch reactions of trimetallic nitride clusterfullerenes (NCFs) generally yield methanofullerene (cyclopropane) adducts instead of singly bonded derivatives, which have been reported for monometallofullerenes. Herein, we report the synthesis and characterization of the Bingel–Hirsch derivative of a mixed metal nitride clusterfullerene (MMNCF) TiY₂N@I_h‐C₈₀. Surprisingly, in contrast to the reported Bingel–Hirsch cyclopropane adducts of the analogous NCF Y₃N@I_h‐C₈₀, the Bingel–Hirsch derivative of TiY₂N@I_h‐C₈₀ is the first singly bonded monoadduct (labeled as TiY₂N@C₈₀‐Mono) to be reported, which was determined unambiguously by single‐crystal X‐ray crystallography. Besides, the reactivity of TiY₂N@I_h‐C₈₀ was found to be significantly improved relative to that of Y₃N@I_h‐C₈₀. Upon substituting one endohedral yttrium (Y) atom of Y₃N@I_h‐C₈₀ with titanium (Ti), the Bingel–Hirsch derivative changes from the cyclopropane to the singly bonded monoadduct, revealing that not only the reactivity but also the addition pattern of NCFs can be manipulated simultaneously through one endohedral metal atom substitution.     

The Exohedral Diels–Alder Reactivity of the Titanium Carbide Endohedral Metallofullerene Ti2C2@D3h‐C78: Comparison with D3h‐C78 and M3N@D3h‐C78 (M=Sc and Y) Reactivity

The chemical functionalization of endohedral (metallo)fullerenes has become a main focus of research in the last few years. It has been found that the reactivity of endohedral (metallo)fullerenes may be quite different from that of the empty fullerenes. Encapsulated species have an enormous influence on the thermodynamics, kinetics, and regiochemistry of the exohedral addition reactions undergone by these species. A detailed understanding of the changes in chemical reactivity due to incarceration of atoms or clusters of atoms is essential to assist the synthesis of new functionalized endohedral fullerenes with specific properties. Herein, we report the study of the Diels–Alder cycloaddition between 1,3‐butadiene and all nonequivalent bonds of the Ti₂C₂@D_3h‐C₇₈ metallic carbide endohedral metallofullerene (EMF) at the BP86/TZP//BP86/DZP level of theory. The results obtained are compared with those found by some of us at the same level of theory for the D_3h‐C₇₈ free cage and the M₃N@D_3h‐C₇₈ (M=Sc and Y) metallic nitride EMFs. It is found that the free cage is more reactive than the Ti₂C₂@D_3h‐C₇₈ EMF and this, in turn, has a higher reactivity than M₃N@D_3h‐C₇₈. The results indicate that, for Ti₂C₂@D_3h‐C₇₈, the corannulene‐type [5, 6] bonds c and f , and the type B [6, 6] bond 3 are those thermodynamically and kinetically preferred. In contrast, the D_3h‐C₇₈ free cage has a preference for addition to the [6, 6] 1 and 6 bonds and the [5, 6] b bond, whereas M₃N@D_3h‐C₇₈ favors additions to the [6, 6] 6 (M=Sc) and [5, 6] d (M=Y) bonds. The reasons for the regioselectivity found in Ti₂C₂@D_3h‐C₇₈ are discussed.     

Geometrical and Electronic Rules in Fullerene‐Based Compounds

The discovery of buckminsterfullerene C₆₀ opened up a new scientific area and stimulated the development of nanoscience and nanotechnology directly. Fullerene science has since emerged to include fullerenes, endohedral fullerenes (mainly metallofullerenes), exofullerenes, and carbon nanotubes as well. Herein, we look back at the development of fullerene science from the perspective of epistemology by highlighting the proposed main rules or criteria for understanding and predicting the structures and stability of fullerene‐based compounds. We also point out that a rule or criterion may contribute significantly to the corresponding discipline and suggest that two unsolved issues in fullerene science are the addition patterns of fullerene derivatives and the structures and stability of nonclassical fullerenes.     

From C58 to C62 and back: Stability,structural similarity,and ring current

An increasing number of observations show that non‐classical isomers may play an important role in the formation of fullerenes and their exo‐ and endo‐derivatives. A quantum‐mechanical study of all classical isomers of C₅₈, C₆₀, and C₆₂, and all non‐classical isomers with at most one square or heptagonal face, was carried out. Calculations at the B3LYP/6‐31G* level show that the favored isomers of C₅₈, C₆₀, and C₆₂ have closely related structures and suggest plausible inter‐conversion and growth pathways among low‐energy isomers. Similarity of the favored structures is reinforced by comparison of calculated ring currents induced on faces of these polyhedral cages by radial external magnetic fields, implying patterns of magnetic response similar to those of the stable, isolated‐pentagon C₆₀ molecule. © 2016 Wiley Periodicals, Inc.     

Endohedral chemical shifts in higher fullerenes with 72–86 carbon atoms

For all isolated pentagon isomers of the fullerenes C₆₀–C₈₆ with nonzero HOMO–LUMO gap and for one nonclassical C₇₂ isomer (C_{2 v} ), endohedral chemical shifts have been computed at the GIAO-SCF/3-21G level using B3LYP/6-31G* optimized structures. The experimental ³He NMR signals are reproduced reasonably well in cases where assignments are unambiguous (e.g. C₆₀, C₇₀ and C₇₆). On the basis of the calculated thermodynamic stability order and the comparison between the computed and experimental ³He chemical shifts, the assignments of the observed ³He NMR spectra are discussed for all higher fullerenes, and new assignments are proposed for one C₈₂ and one C₈₆ isomer (C₈₂:3 and C₈₆:17). The calculated helium chemical shifts also suggest the reassignment of the δ(³He) resonances of two C₇₈ isomers. Received: 26 March 2001 / Accepted: 10 May 2001 / Published online: 11 October 2001     

Stable Non‐IPR C60 and C70 Fullerenes Containing a Uniform Distribution of Pyrenes and Adjacent Pentagons

The most abundant fullerenes, C₆₀ and C₇₀, and all the pure carbon fullerenes larger than C₇₀, follow the isolated‐pentagon rule (IPR). Non‐IPR fullerenes containing adjacent pentagons (APs) have been stabilized experimentally in cases where, according to Euler’s theorem, it is topologically impossible to isolate all the pentagons from each other. Surprisingly, recent experiments have shown that a few endohedral fullerenes, for which IPR structures are possible, are stabilized in non‐IPR cages. We show that, apart from strain, the physical property that governs the relative stabilities of fullerenes is the charge distribution in the cage. This charge distribution is controlled by the number and location of APs and pyrene motifs. We show that, when these motifs are uniformly distributed in the cage and well‐separated from one other, stabilization of non‐IPR endohedral and exohedral derivatives, as well as pure carbon fullerene anions and cations, is the rule, rather than the exception. This suggests that non‐IPR derivatives might be even more common than IPR ones.     

Synthesis,Isolation, and Trifluoromethylation of Two Isomers of C84‐Based Monometallic Cyanide Clusterfullerenes: Interplay between the Endohedral Cluster and the Exohedral Addends

As an emerging member of endohedral fullerenes, metal cyanide clusterfullerenes (CYCF) are unique in terms of the encapsulation of a monometallic cluster. To date the reported carbon cages of CYCFs are limited to C₈₂ and C₇₆, and little is known about the chemical reactivity of CYCFs. Herein, two isomers of the first C₈₄‐based CYCFs, YCN@C₈₄, were isolated as trifluoromethyl derivatives, including YCN@C₈₄(23)(CF₃)₁₈ and three isomers of YCN@C₈₄(13)(CF₃)₁₆, which are based on a unique chiral C ₂‐C₈₄(13) cage. As a common feature of the CF₃ addition patterns, the YCN@C₈₄(CF₃)_16/18 compounds are stabilized by the formation of isolated C=C bonds and benzenoid rings on the carbon cages. The interplay between the endohedral YCN cluster and the exhohedral CF₃ addends was unveiled according to single‐crystal X‐ray diffraction studies, thus offering new insight into the chemical reactivity of CYCFs.     

Synthesis,Isolation, and Trifluoromethylation of Two Isomers of C84‐Based Monometallic Cyanide Clusterfullerenes: Interplay between the Endohedral Cluster and the Exohedral Addends

As an emerging member of endohedral fullerenes, metal cyanide clusterfullerenes (CYCF) are unique in terms of the encapsulation of a monometallic cluster. To date the reported carbon cages of CYCFs are limited to C₈₂ and C₇₆, and little is known about the chemical reactivity of CYCFs. Herein, two isomers of the first C₈₄‐based CYCFs, YCN@C₈₄, were isolated as trifluoromethyl derivatives, including YCN@C₈₄(23)(CF₃)₁₈ and three isomers of YCN@C₈₄(13)(CF₃)₁₆, which are based on a unique chiral C ₂‐C₈₄(13) cage. As a common feature of the CF₃ addition patterns, the YCN@C₈₄(CF₃)_16/18 compounds are stabilized by the formation of isolated C=C bonds and benzenoid rings on the carbon cages. The interplay between the endohedral YCN cluster and the exhohedral CF₃ addends was unveiled according to single‐crystal X‐ray diffraction studies, thus offering new insight into the chemical reactivity of CYCFs.     

Expanding the Number of Stable Isomeric Structures of the C80 Cage: A New Fullerene Dy3N@C80

The production, isolation, and spectroscopic characterization of a new Dy₃N@C₈₀ cluster fullerene that exhibits three isomers ( 1 – 3 ) is reported for the first time. In addition, the third isomer ( 3 ) forms a completely new C₈₀ cage structure that has not been reported in any endohedral fullerenes so far. The isomeric structures of the Dy₃N@C₈₀ cluster fullerene were analyzed by studying HPLC retention behavior, laser desorption time‐of‐flight (LD‐TOF) mass spectrometry, and UV‐Vis‐NIR and FTIR spectroscopy. The three isomers of Dy₃N@C₈₀ were all large band‐gap (1.51, 1.33, and 1.31 eV for 1 – 3 , respectively) materials, and could be classified as very stable fullerenes. According to results of FTIR spectroscopy, the Dy₃N@C₈₀ (I) ( 1 ) was assigned to the fullerene cage C₈₀:7 (I_h), whereas Dy₃N@C₈₀ (II) ( 2 ) had the cage structure of C₈₀:6 (D_5h). The most probable cage structure of Dy₃N@C₈₀ (III) ( 3 ) was proposed to be C₈₀:1 (D_5d). The significant differences between Dy₃N@C₈₀ and other reported M₃N@C₈₀ (M=Sc, Y, Gd, Tb, Ho, Er, Tm) cluster fullerenes are discussed in detail, and the strong influence of the metal on the nitride cluster fullerene formation is concluded.     

Synthesis and Isolation of the Titanium–Scandium Endohedral Fullerenes—Sc2TiC@Ih‐C80, Sc2TiC@D5h‐C80 and Sc2TiC2@Ih‐C80: Metal Size Tuning of the TiIV/TiIII Redox Potentials

The formation of endohedral metallofullerenes (EMFs) in an electric arc is reported for the mixed‐metal Sc–Ti system utilizing methane as a reactive gas. Comparison of these results with those from the Sc/CH₄ and Ti/CH₄ systems as well as syntheses without methane revealed a strong mutual influence of all key components on the product distribution. Whereas a methane atmosphere alone suppresses the formation of empty cage fullerenes, the Ti/CH₄ system forms mainly empty cage fullerenes. In contrast, the main fullerene products in the Sc/CH₄ system are Sc₄C₂@C₈₀ (the most abundant EMF from this synthesis), Sc₃C₂@C₈₀, isomers of Sc₂C₂@C₈₂, and the family Sc₂C_{2 n} (2 n=74, 76, 82, 86, 90, etc.), as well as Sc₃CH@C₈₀. The Sc–Ti/CH₄ system produces the mixed‐metal Sc₂TiC@C_{2 n} (2 n=68, 78, 80) and Sc₂TiC₂@C_{2 n} (2 n=80) clusterfullerene families. The molecular structures of the new, transition‐metal‐containing endohedral fullerenes, Sc₂TiC@I_h‐C₈₀, Sc₂TiC@D_5h‐C₈₀, and Sc₂TiC₂@I_h‐C₈₀, were characterized by NMR spectroscopy. The structure of Sc₂TiC@I_h‐C₈₀ was also determined by single‐crystal X‐ray diffraction, which demonstrated the presence of a short Ti=C double bond. Both Sc₂TiC‐ and Sc₂TiC₂‐containing clusterfullerenes have Ti‐localized LUMOs. Encapsulation of the redox‐active Ti ion inside the fullerene cage enables analysis of the cluster–cage strain in the endohedral fullerenes through electrochemical measurements.     

Synthesis and Electrochemical Studies of Bingel–Hirsch Derivatives of M3N@Ih‐C80 (M=Sc,Lu)

Bingel–Hirsch derivatives of the trimetallic nitride template endohedral metallofullerenes (TNT‐EMFs) Sc₃N@I_h‐C₈₀ and Lu₃N@I_h‐C₈₀ were prepared by reacting these compounds with 2‐bromodiethyl malonate, 2‐bromo‐1,3‐dipyrrolidin‐1‐ylpropane‐1,3‐dionate bromide, and 9‐bromo fluorene. The mono‐adducts were isolated and their ¹H NMR spectra showed that the addition occurred with high regioselectivity at the [6,6] bonds of the I_h‐C₈₀ fullerene cage. Electrochemical analysis showed that the reductive electrochemistry behavior of these derivatives is irreversible at a scan rate of 100 mV s^?1, which is comparable to the behavior of the pristine fullerene species. The first reduction potential of each derivative is either cathodically or anodically shifted by a different value, depending on the attached addend. Bis‐adducts containing EtOOC‐C‐COOEt and HC‐COOEt addends were isolated by HPLC and in the case of Sc₃N@I_h‐C₈₀ the first reduction potential exhibits a larger shift towards negative potentials when compared to the mono‐adduct. This observation is important for designing acceptor materials for the construction of bulk heterojunction (BHJ) organic solar cells, since the polyfunctionalization not only increases the solubility of the fullerene species but also offers a promising approach for bringing the LUMO energy levels closer for the donor and the acceptor materials.     

Electronic structure and stability of C86 fullerene Isolated‐Pentagon‐Rule isomers

All 19 Isolated‐Pentagon‐Rule isomers of fullerene C₈₆ were investigated by Density Functional Theory (DFT) methods with B3LYP functional at 6‐31G, 6‐31G*, and 6‐31+G* levels. Preliminary distribution of single, double, and delocalized pi‐bonds in molecules of these isomers of fullerene C₈₆ is fulfilled. Obtained results are perfectly supported by DFT quantum–chemical calculations of electronic and geometrical structures of these isomers. The main reason of instability of isomers 1, 3–15, 18, and 19 are phenalenyl‐radical substructures. Thus, there is a possibility to obtain them only as endohedral metallofullerenes or exohedral derivatives. Isomer 2 (C₂) is unstable due to higher local molecular strain. It is shown that empty C₈₆ may be produced and extracted only as isomers 16 (C_s) and 17 (C₂). © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Structures of Chlorinated Fullerenes,IPR C96Cl20 and Non‐classical C94Cl28 and C92Cl32: Evidence of the Existence of Three New Isomers of C96

Chlorination of various HPLC fractions of C₉₆ with a mixture of VCl₄ and SbCl₅ at 340–360 °C and single‐crystal X‐ray diffraction study of the products led to the identification of three new IPR isomers of C₉₆. The C₉₆(175) isomer forms a stable chloride, C₉₆(175)Cl₂₀, while chlorides of two other new isomers, C₉₆(114) and C₉₆(80), undergo cage shrinkage yielding C₉₄(NC1)Cl₂₈ and C₉₆(NC2)Cl₃₂ with non‐classical (NC) cages. These two NC chlorides contain, respectively, one and two heptagons flanked by pairs of fused pentagons and are stabilized by chlorine attachment to the emerging pentagon–pentagon junctions. Thus, the number of the experimentally confirmed C₉₆ isomers has reached nine, which corroborates the empirical rule that the C_6n fullerenes exhibit particularly rich isomerism.