Search for lowest-energy fullerenes: C98 to C110

Kekulé Counts, Clar Numbers, and ZZ Polynomials for All Isomers of (5,6)-Fullerenes C52-C70

We report an extensive tabulation of several important topological invariants for all the isomers of carbon (5,6)-fullerenes Cn with n = 52-70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported in the form of the corresponding Zhang-Zhang (ZZ) polynomials. The ZZ polynomials appear to be distinct for each isomer cage, providing a unique label that allows for differentiation between various isomers. Several chemical applications of the computed invariants are reported. The results suggest rather weak correlation between the Kekulé count, Clar count, Clar number invariants, and isomer stability, calling into doubt the predictive power of these topological invariants in discriminating the most stable isomer of a given fullerene. The only exception is the Clar count/Kekulé count ratio, which seems to be the most important diagnostic discovered from our analysis. Stronger correlations are detected between Pauling bond orders computed from Kekulé structures (or Clar covers) and the corresponding equilibrium bond lengths determined from the optimized DFTB geometries of all 30,579 isomers of C20-C70.

Electron-vibrational renormalization in fullerenes through ab initio and machine learning methods

The effect of nuclear vibrations on the electronic eigenvalues and the HOMO-LUMO gap is known for several kinds of carbon-based materials, like diamond, diamondoids, carbon nanoclusters, carbon nanotubes and others, like hydrogen-terminated oligoynes and polyyne. However, it has not been widely analysed in another remarkable kind which presents both theoretical and technological interest: fullerenes. In this article we present the study of the HOMO, LUMO and gap renormalizations due to zero-point motion of a relatively large number (163) of fullerenes and fullerene derivatives. We have calculated this renormalization using density-functional theory with the frozen-phonon method, finding that it is non-negligible (above 0.1 eV) for systems with relevant technological applications in photovoltaics and that the strength of the renormalization increases with the size of the gap. In addition, we have applied machine learning methods for classification and regression of the renormalizations, finding that they can be approximately predicted using the output of a computationally cheap ground state calculation. Our conclusions are supported by recent research in other systems.

Vibrational anatomy of C90, C96, and C100 fullertubes: probing Frankenstein's skeletal structures of fullerene head endcaps and nanotube belt midsection

Fullertubes are tubular fullerenes with nanotube-like middle section and fullerene-like endcaps. To understand how this intermediate form between spherical fullerenes and nanotubes is reflected in the vibrational modes, we performed comprehensive studies of IR and Raman spectra of fullertubes C₉₀-D_5h, C₉₆-D_3d, and C₁₀₀-D_5d. An excellent agreement between experimental and DFT-computed spectra enabled a detailed vibrational assignment and allowed an analysis of the localization degree of the vibrational modes in different parts of fullertubes. Projection analysis was performed to establish an exact numerical correspondence between vibrations of the belt midsection and fullerene headcaps to the modes of nanotubes and fullerene C₆₀-I_h. As a result, we could not only identify fullerene-like and CNT-like vibrations of fullertubes, but also trace their origin in specific vibrational modes of CNT and C₆₀-I_h. IR spectra were found to be dominated by vibrations of fullerene-like caps resembling IR-active modes of C₆₀-I_h, whereas in Raman spectra both caps and belt vibrations are found to be equally active. Unlike the resonance Raman spectra of CNTs, in which only two single-phonon bands are detected, the Raman spectra of fullertubes exhibit several CNT-like vibrations and thus provide additional information on nanotube phonons.

Two Isolated-Pentagon-Rule Isomers C98(110) and C98(111) Isolated as Trifluoromethylfullerenes C98(CF3)22

The fullerene C₉₈ has 259 topologically possible isomers that obey the isolated-pentagon rule (IPR). In this work, a family of experimentally confirmed IPR isomers of C₉₈ fullerene is extended by the high-performance liquid chromatography isolation and X-ray structural characterization of two trifluoromethyl derivatives, C₁-C₉₈(110)(CF₃)₂₂ and C₁-C₉₈(111)(CF₃)₂₂. The carbon cages of isomers C_s-C₉₈(110) and C_s-C₉₈(111) differ by a Stone-Wales rotation of only one C-C bond, which results in very similar addition patterns of 22 CF₃ groups in the C₉₈(CF₃)₂₂ molecules. The stabilizing substructures in both C₉₈(CF₃)₂₂ molecules include six benzenoid rings and four isolated C═C bonds. Both C_s-C₉₈(110) and C_s-C₉₈(111) belong to the isomers of moderate relative stability among altogether seven IPR isomers of C₉₈ fullerene with experimentally confirmed cage structures.

Substructural Approach for Assessing the Stability of Higher Fullerenes

This review describes the most significant published results devoted to the study of the nature of the higher fullerenes stability, revealing of correlations between the structural features of higher fullerene molecules and the possibility of their producing. A formalization of the substructure approach to assessing the stability of higher fullerenes is proposed, which is based on a detailed analysis of the main structural features of fullerene molecules. The developed substructure approach, together with the stability of the substructures constituting the fullerene molecule, helps to understand deeper the features of the electronic structure of fullerenes.

Three Isolated-Pentagon-Rule Isomers of C96 Fullerene Isolated as Trifluoromethyl Derivatives

The family of experimentally confirmed isolated-pentagon-rule (IPR) isomers of C₉₆ fullerene is extended by trifluoromethylation of a C₉₆ fraction of the fullerene soot, high-performance liquid chromatography separation of CF₃ derivatives, and a single-crystal X-ray diffraction study of C₉₆(CF₃)_n compounds with the use of synchrotron radiation. New cage isomers were revealed in C₉₆(94)(CF₃)_18/20 and C₉₆(182)(CF₃)₁₈ compounds, whereas isomer C₉₆(181), previously known in the adduct with nickel porphyrinate, was confirmed in C₉₆(181)(CF₃)_18/20 derivatives. Common and special features of the addition patterns of CF₃ groups on C₉₆ carbon cages are discussed in more detail. The investigated isomers belong to the most stable C₂-C₉₆(181) and slightly less stable C₁-C₉₆(94) and C₂-C₉₆(182) among the altogether 15 experimentally confirmed IPR isomers of C₉₆ fullerene.

Structural Studies of Giant Empty and Endohedral Fullerenes

Structure elucidations of giant fullerenes composed of 100 or more carbon atoms are severely hampered by their extremely low yield, poor solubility and huge numbers of possible cage isomers. High-temperature exohedral chlorination followed by X-ray single crystal diffraction studies of the chloro derivatives offers a practical solution for structure elucidations of giant fullerenes. Various isomers of giant fullerenes have been determined by this method, specially, non-classical giant fullerenes containing heptagons generated by the skeletal transformations of carbon cages. Alternatively, giant fullerenes can be also stabilized by encapsulating metal atoms or clusters through intramolecular electron transfer from the encapsulated species to the outer fullerene cage. In this review, we present a comprehensive overview on synthesis, separation and structural elucidation of giant fullerenes. The isomer structures, chlorination patterns of a series of giant fullerenes C_2n (2n = 100-108) and heptagon-containing non-classical fullerenes derived from giant fullerenes are summarized. On the other hand, giant endohedral fullerenes bearing different endohedral species are also discussed. At the end, we propose an outlook on the future development of giant fullerenes.

Trifluoromethyl Derivatives of Elusive Fullerene C98

Data concerning the isomeric composition of C₉₈ and the chemistry of C₉₈ derivatives are scarce due to very low abundance of C₉₈ in the fullerene soot. Trifluoromethylation of C₉₈ -containing mixtures followed by HPLC separation of CF₃ derivatives and single crystal X-ray diffraction study resulted in structural characterization of four compounds C₉₈ (248)(CF₃ )_18/20 , C₉₈ (116)(CF₃ )₁₈ , and C₉₈ (120)(CF₃ )₂₀ . To date, these compounds represent the largest fullerenes isolated as CF₃ derivatives with experimentally determined molecular structures. The addition patterns of C₉₈ (CF₃ )_18/20 are discussed in detail revealing the stabilizing factors, such as isolated double C=C bonds and benzenoid rings on C₉₈ fullerene cages. A detailed comparison with the addition patterns of the known C₉₈ Cl_n allowed us to contribute to the better understanding the chemistry of elusive C₉₈ fullerene.

Chlorination-Promoted Cage Transformation of IPR C92 Discovered via Trifluoromethylation under Formation of Non-classical C92 (NC)(CF3 )22

High-temperature trifluoromethylation of isolated-pentagon-rule (IPR) fullerene C₉₂ chlorination products followed by HPLC separation of C₉₂ (CF₃ )_n derivatives resulted in the isolation and X-ray structural characterization of IPR C₉₂ (38)(CF₃ )₁₈ and non-classical C₉₂ (NC)(CF₃ )₂₂ . The formation of C₉₂ (38)(CF₃ )₁₈ as the highest CF₃ derivative of the known isomer C₉₂ (38) can be expected. The formation of C₉₂ (NC)(CF₃ )₂₂ was interpreted as chlorination-promoted cage transformation of C₉₂ (38) followed by trifluoromethylation of non-classical C₉₂ (NC) chloride. Noticeably, C₉₂ (NC)(CF₃ )₂₂ shows the highest degree of trifluoromethylation among all known CF₃ derivatives of fullerenes. The addition patterns of C₉₂ (38)(CF₃ )₁₈ and C₉₂ (NC)(CF₃ )₂₂ are discussed and compared to the chlorination patterns of C₉₂ (38)Cl_n compounds.

Hybrid ADFT Study of the C104 and C106 IPR Isomers

This work presents a hybrid auxiliary density functional theory (ADFT) study of the neutral and hexaanionic C₁₀₄ and C₁₀₆ fullerenes with the aim to determine their ground state structures. To this end, all C₁₀₄ and C₁₀₆ fullerene structures that obey the isolated pentagon rule (IPR) were optimized with the Perdew-Burke-Ernzerhof generalized gradient approximation followed by a single-point energy calculation with the PBE0 hybrid functional. Our studies show that this composite approach yields relative energies of giant fullerenes that are accurate to around 1 kcal/mol. As a result, the ground states of C₁₀₄, C₁₀₄^6-, and C₁₀₆^6- can be assigned to the isomers 234:C_s, 821:D₂, and 891:C_s, respectively. On the other hand, the energetically lowest lying IPR isomers of C₁₀₆, 331:C_s, 1194:C₂, 534:C₁ are separated by less than 1 kcal/mol which makes an unequivocal ground state assignment by hybrid DFT methods impossible. To guide future experiments, we also report the simulated IR and Raman spectra of the most stable neutral and hexaanionic C₁₀₄ and C₁₀₆ fullerenes.

Assessment of electronic transitions involving intermolecular charge transfer in complexes formed by fullerenes and donor–acceptor nanohoops

Inserting an anthraquinone or tetracyanoanthraquinone unit in cycloparaphenylene nanohoops facilitates intermolecular electron transfer to a fullerene guest.

New Isolated-Pentagon-Rule Isomers of Fullerene C98 Captured as Chloro Derivatives

Fullerene C₉₈ possesses 259 isomers obeying the isolated pentagon rule (IPR), from which two, nos. 116 and 248, have been confirmed earlier as chloro derivatives. High-temperature chlorination of C₉₈-containing mixtures afforded crystals of several chloro derivatives, and their structure elucidation by X-ray crystallography revealed the presence of new isomers, nos. 107, 109, and 120, in the fullerene soot. Evidence for an isomer of no. 111 is also presented. In addition, a new chloride of the known isomer 248 has been isolated and structurally studied. The chlorination patterns of the chlorides are discussed in terms of the formation of isolated C═C bonds and aromatic substructures on the fullerene cages.

Chloro Derivatives of Isomers of a Giant Fullerene C104 : C104 (234)Cl16/18 , C104 (812)Cl12/24 , and C104 (811)Cl28

The chemistry of a giant fullerene, C₁₀₄ , has been extended by the synthesis and structural study of several chloro derivatives of three isolated pentagon rule (IPR) isomers of C₁₀₄ nos. 234, 812, and 811. In the structure of C₁₀₄ (234)Cl_16/18 , two molecules with 16 and 18 attached Cl atoms occupy the same crystallographic site with an occupancy ratio of 61/39. The structures of C₁₀₄ (812)Cl₁₂ and C₁₀₄ (812)Cl₂₄ demonstrate substructure relationships of their chlorination patterns with single and double Cl attachments to 12 cage pentagons. The structure of C₁₀₄ (811)Cl₂₈ is compared with the known C₁₀₄ (811)Cl₂₄ thus revealing dramatic changes in the chlorination pattern, which occur with relatively small increases in the degree of chlorination.

Generalized structural motif model for studying the thermodynamic stability of fullerenes: from C60to graphene passing through giant fullerenes

A generalized motif model to describe the stability of neutral fullerenes, covering the full range of cage sizes, starting from C₆₀, going through giant fullerenes, and ultimately leading to graphene.

New Giant Fullerenes Identified as Chloro Derivatives: Isolated-Pentagon-Rule C108(1771)Cl12 and C106(1155)Cl24 as well as Nonclassical C104Cl24

High temperature chlorination of HPLC fractions of higher fullerenes followed by single crystal X-ray diffraction with the use of synchrotron radiation resulted in the structure determination of IPR C106(1155)Cl24 and IPR C108(1771)Cl12. C106(1155)Cl24 is cocrystallized with C104Cl24, a chloride of the nonclassical isomer of C104. The moderately stable isomer C106(1155) and the most stable C108(1771) represent so far the largest pristine fullerenes with known cages.

The First Experimentally Confirmed Isolated Pentagon Rule (IPR) Isomers of Higher Fullerene C98 Captured as Chlorides, C98(248)Cl22 and C98(116)Cl20

High-temperature chlorination of pristine C98 fullerene isomers separated by HPLC from the fullerene soot afforded crystals of C98Cl22 and C98Cl20. An X-ray structure elucidation revealed, respectively, the presence of carbon cages of the most stable C2-C98(248) and rather unstable C1-C98(116), which represent the first isolated pentagon rule (IPR) isomers of fullerene C98 confirmed experimentally. The chlorination patterns of the chlorides are discussed in terms of the formation of isolated C=C bonds and aromatic substructures on the fullerene cages.

New Isolated-Pentagon-Rule and Skeletally Transformed Isomers of C100 Fullerene Identified by Structure Elucidation of their Chloro Derivatives

High-temperature chlorination of C100 fullerene followed by X-ray structure determination of the chloro derivatives enabled the identification of three isomers of C100 from the fullerene soot, specifically numbers 18, 425, and 417, which obey the isolated pentagon rule (IPR). Among them, isomers C1-C100 (425) and C2-C100 (18) afforded C1-C100 (425)Cl22 and C2-C100 (18)Cl28/30 compounds, respectively, which retain their IPR cage connectivities. In contrast, isomer C2v -C100 (417) gives Cs -C100 (417)Cl28 which undergoes a skeletal transformation by the loss of a C2  fragment, resulting in the formation of a nonclassical (NC) C1-C98 (NC)Cl26 with a heptagon in the carbon cage. Most probably, two nonclassical C1-C100 (NC)Cl18/22 chloro derivatives originate from the IPR isomer C1-C100 (382), although both C1-C100 (344) and even nonclassical C1-C100 (NC) can be also considered as the starting isomers.

Capturing an unstable C100 fullerene as chloride, C100(1)Cl12, with a nanotubular carbon cage

The chlorination of a HPLC C100 fraction afforded C100(1)Cl12 with an unprecedented nanotubular carbon cage of a highly unstable D5d-C100 fullerene.

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96-145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website.

The most stable isomers of giant fullerenes C102 and C104 captured as chlorides, C102(603)Cl18/20 and C104(234)Cl16/18/20/22

The chlorination of HPLC fractions with pristine giant fullerenes, C102 and C104, followed by X-ray crystallographic study of chlorides, C102(603)Cl18/20 and C104(234)Cl16-22, confirmed the presence of the most stable IPR (IPR = Isolated Pentagon Rule) isomers, C102(603) and C104(234), in the fullerene soot. The discussion concerns the chlorination patterns of polychlorides and relative stability of pristine isomers of C102 and C104 fullerenes.

Chlorination of IPR C100 fullerene affords unconventional C96 Cl20 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.

The first structural confirmation of a C102 fullerene as C102Cl20 containing a non-IPR carbon cage

The chlorination of a pristine C102 fullerene separated by HPLC from fullerene soot afforded crystals of C102Cl20 with a non-IPR (IPR = isolated pentagon rule) cage containing two pairs of fused pentagons; structural reconstruction of a two-step Stone-Wales rearrangement revealed the starting IPR isomer (no. 19) of C102.

Search for lowest-energy nonclassical fullerenes III: C22

Density functional and second-order Møller-Plesset perturbation (MP2) methods were employed in the investigation of low-lying C22 isomers. All cage structures with four-, five-, six-, and seven-membered rings were examined with the monocyclic ring, bowl, and other noncage structures. Cage isomers were first identified via graph theoretical methods, and noncages were identified by basin-hopping methods. Initial isomer screenings were carried out at the PBE/DND level of theory. Low-lying isomers, within 0.6 eV of the predicted lowest-energy isomer, were further evaluated at the PBE1PBE/cc-pVTZ and MP2/cc-pVTZ levels. Our results confirm that the cage structures are more stable than the ring structure and the bowl structure. The lowest-energy structure for C22 is predicted to be the C22-1 cage containing one four-membered ring. Anion photoelectron and optical spectra of the six lowest-lying isomers are also computed.