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     

Strain Release of Fused Pentagons in Fullerene Cages by Chemical Functionalization

According to the isolated pentagon rule (IPR), for stable fullerenes, the 12 pentagons should be isolated from one another by hexagons, otherwise the fused pentagons will result in an increase in the local steric strain of the fullerene cage. However, the successful isolation of more than 100 endohedral and exohedral fullerenes containing fused pentagons over the past 20 years has shown that strain release of fused pentagons in fullerene cages is feasible. Herein, we present a general overview on fused‐pentagon‐containing (i.e. non‐IPR) fullerenes through an exhaustive review of all the types of fused‐pentagon‐containing fullerenes reported to date. We clarify how the strain of fused pentagons can be released in different manners, and provide an in‐depth understanding of the role of fused pentagons in the stability, electronic properties, and chemical reactivity of fullerene cages.     

Structures,stabilities, and IR and 13C-NMR spectra of dihedral fullerenes: A density functional theory study

Dihedral fullerenes are thermodynamically stable molecules with D nd or D nh symmetry.Based on experimental findings,two series of dihedral fullerenes with five-fold(C5) and six-fold(C6) symmetry have been studied using density functional theory(DFT).The DFT calculations showed that for both series the stabilities increased with increasing fullerene size.Structural analyses indicated that the stabilities are related to specific local geometries.In the case of the more abundant C5 series,the presence of approximately planar pentagons and hexagons on the top bowl favors their formation.That is to say,those fullerenes with small dihedral angles within the polygons are readily formed,because planar hexagons lead to strengthened conjugation which lowers average bonding energies(ABE) and increases thermodynamic stabilities.Non-planar hexagons at equatorial positions in tube-shaped fullerenes have an adverse effect on the conjugation and inhibit their formation.Calculations also demonstrated that fullerenes in the two series,including C 50(D 5h),C 60(D 6h),C 80(D 5d),C 96(D 6d),C 110(D 5h),and C 120(D 5d),have thermodynamically stable triplet structures with strong conjugation.The calculated IR and 13 C NMR spectra of the fullerenes show some similarities and regular trends due to their homogenous structures.The electronic structures indicate that short double bonds in hexagons with high electron occupancies are readily attacked by electrophilic agents and can also be coordinated by transition metals.Mechanistic discussions suggested that C 2 additions and C 2 losses constitute reversible processes at high temperature and C 2 additions in pentagonal fusions are crucial to the kinetics of the curvature of structures.C 3 additions lead to the formation of large fullerenes of other types.     

Deformation and thermodynamic instability of a C84 fullerene cage

Quantum chemical calculations of electronic and geometric structures were performed for molecules of 24 isomers of C₈₄ fullerenes obeying the isolated pentagons rule. The reasons for the instability of isomers not obtained experimentally were established, and the possibility of obtaining some of them was proven. It was shown that the deformation of hexagons and pentagons is the most important geometric parameter directly connected with the thermodynamic instability of fullerenes having closed shells, reflecting the local strain of the molecules.     

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     

Unusual pentagon and hexagon geometry of three isomers (no 1, 20, and 23) of fullerene C84

Three isomers 23 (D_2d), 1 (D₂), and 20 (T_d) of fullerene C₈₄ have been investigated by PM3, HF/6‐31G*, and DFT methods with B3LYP functional at the 6‐31G and 6‐31G* levels. In this article we reveal for the first time that some distortion of hexagon (pentagon), measured as its maximal dihedral angles, caused by local molecular strains may serve as a new criterion of stability of fullerenes with closed shell. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Double Negatively Curved C70 Growth through a Heptagon‐Involving Pathway

All previously reported C₇₀ isomers have positive curvature and contain 12 pentagons in addition to hexagons. Herein, we report a new C₇₀ species with two negatively curved heptagon moieties and 14 pentagons. This unconventional heptafullerene[70] containing two symmetric heptagons, referred to as dihept‐C₇₀, grows in the carbon arc by a theoretically supported pathway in which the carbon cluster of a previously reported C₆₆ species undergoes successive C₂ insertion via a known heptafullerene[68] intermediate with low energy barriers. As identified by X‐ray crystallography, the occurrence of heptagons facilitates a reduction in the angle of the π‐orbital axis vector in the fused pentagons to stabilize dihept‐C₇₀. Chlorination at the intersection of a heptagon and two adjacent pentagons can greatly enlarge the HOMO–LUMO gap, which makes dihept‐C₇₀Cl₆ isolable by chromatography. The synthesis of dihept‐C₇₀Cl₆ offers precious clues with respect to the fullerene formation mechanism in the carbon‐clustering process.     

Structure and reactivity of C54q+ (q = 0, 1, 2 and 4) fullerenes

Using density functional theory we have studied the structural properties of eleven C54 isomers that appear in the C60 fragmentation. We have evaluated the relative stability of the different isomers with respect the most stable one, which corresponds to the structure with the minimum number (four) of adjacent pentagons. On average, the length of a bond shared by pentagons and/or hexagons increases in the order hexagon-hexagon, hexagon-pentagon and pentagon-pentagon. However, we have found that the central bond in the confluence of four hexagons, i.e. a pyrene substructure, is anomalously large, becoming in some cases the largest one. We have also evaluated the nucleus-independent chemical shifts (NICS) at the center of every individual ring in the most stable isomers. For the chlorine derivatives, our calculations show that the favorite position for chlorine addition are the bonds shared by pentagons.     

The structure of fullerene C<Subscript>66</Subscript>, which does not obey the rule of isolated pentagons,and endohedral metallofullerene Sc<Subscript>2</Subscript>@C<Subscript>66</Subscript>: Quantum-chemical calculations

The molecular structure of the isomer 4348 (C _2v ) of fullerene C₆₆, which does not obey the rule of isolated pentagons, is analyzed. Data on the distribution of single, double, and delocalized bonds in this molecule were obtained for the first time. Quantum-chemical (DFT) calculations showed that the hypothetical C₆₆ molecule was a biradical, which was the reason for its instability. In metallofullerene Sc₂@C₆₆, the main changes in bond lengths and electron density transfer from scandium atoms to the carbon shell of fullerene occurred in pentalene substructures and conjugated π bond chains.     

Electronic structures of some of C84 fullerene isomers and the structures of their perfluoroalkyl derivatives

The electronic structures of the pristine fullerene molecules have been shown for the first time to be is the most important factor affecting the distribution of addends in the addition reactions of perfluoroalkyl radicals R^F to C₈₄ fullerene, and most likely positions of addends on the fullerene core are hexagons with delocalized π-bonds.     

Further test of the isolated pentagon rule: Thermodynamic and kinetic stabilities of C84 fullerene isomers

Thermodynamic and kinetic stabilities of 73 C₈₄ fullerene isomers were estimated from the MM3 heats of formation and the recently defined bond resonance energies (BREs), respectively. The BRE represents the contribution of a given π bond in a molecule to the topological resonance energy (TRE). All π bonds shared by two pentagons turned out to be highly reactive without exceptions. C₈₄ fullerene isomers with such π bonds must be incapable of survival during harsh synthetic processes. Thus, the isolated pentagon rule (IPR) proved to be applicable to such large fullerene cages. For sufficiently large fullerenes like C₈₄, some isolated-pentagon isomers are also predicted to be very unstable with highly antiaromatic π bonds. © 1996 by John Wiley & Sons, Inc.     

Effects of Intrinsic Pentagon Defects on Electrochemical Reactivity of Carbon Nanomaterials

Theoretical calculations reveal that intrinsic pentagons in the basal plane can contribute to the local electronic redistribution and the contraction of band gap, making the carbon matrix possess superior binding affinity and electrochemical reactivity. To experimentally verify this, a pentagon‐defect‐rich carbon nanomaterial was constructed by means of in situ etching of fullerene molecules (C₆₀). The electrochemical tests show that, relative to hexagons, such a carbon‐based material with abundant intrinsic pentagon defects makes much greater contribution to the electrocatalytic oxygen reduction activity and electric double layer capacitance. It shows a four‐electron‐reaction mechanism similar to commercial Pt/C and other transition‐metal‐based catalysts, and a higher specific capacitance than many reported metal‐free carbon materials. These results show the influence of intrinsic pentagon defects for developing carbon‐based nanomaterials toward energy conversion and storage devices.     

The saturated hydrides of C<Subscript>60</Subscript>:F<Subscript>5</Subscript>F<Subscript>6</Subscript> PK F<Subscript>5</Subscript>F<Subscript>7</Subscript> isomers

The carbon cages composed of pentagons and heptagons (F₅F₇ isomers) are the analogs of fullerenes composed of pentagons and hexagons (F₅F₆ isomers). To provide insight into the structures and stability of the hydrides of F₅F₆ and F₅F₇ isomers, systematical density functional theory calculations are performed on all the 1,812 F₅F₆–C₆₀H₆₀ and 56 F₅F₇–C₆₀H₆₀. The calculated results demonstrate that the isomer with lowest/highest energy has most/fewest fused pentagons for both F₅F₆ and F₅F₇ hydrides and the stability of these hydrides increase with the number of fused pentagons roughly. The lowest energy F₅F₆–C₆₀H₆₀ and F₅F₇–C₆₀H₆₀ are 237.1 and 152.5 kcal/mol lower in energy than the isolated pentagon rule (IPR) C₆₀H₆₀, respectively; however, these two parent cages are 529.6 and 660.0 kcal/mol higher in energy than the IPR C₆₀. The calculations suggest that heptagon-containing cages, not only those violating the IPR can be the candidate cages for fullerene derivatives and the possible repulsion between the added atoms may play an important role in determining the structures and stability of the hydrides of carbon cages.     

Dicyclohepta[ijkl,uvwx]rubicene with Two Pentagons and Two Heptagons as a Stable and Planar Non‐benzenoid Nanographene

Polycyclic aromatic hydrocarbons with hexagons/pentagons or hexagons/heptagons have been intensively investigated in recent years, but those with simultaneous presence of hexagons, pentagons and heptagons remain rare. In this paper, we report dicyclohepta[ijkl,uvwx]rubicene ( DHR ), a non‐benzenoid isomer of dibenzo[bc,kl]coronene with two pentagons and two heptagons. We developed an efficient and scalable synthetic method for DHR by using Scholl reaction and dehydrogenation. Crystal structure of DHR shows that the benzenoid rings, two pentagons and two heptagons are coplanar. The bond lengths analysis and the ICSS(1)zz and LOL‐π calculations indicate that the incorporation of two formal azulene moieties has an effect on the conjugated structure. The π‐electrons of benzenoid and pentagon rings are more delocalized. Cyclic voltammetry studies indicate that DHR shows multiple oxidation and reduction potentials. Interestingly, DHR exhibits unusual S₀ to S₂ absorption and abnormal anti‐Kasha S₂ to S₀ emission. Moreover, crystals of DHR exhibit semiconducting behaviour with hole mobility up to 0.082 cm² V^?1 s^?1.     

Synthesis of a Hemispherical Geodesic Phenine Framework by a Polygon Assembling Strategy

A synthetic strategy to construct large geodesic structures of phenine (1,3,5‐trisubstituted benzene) was devised. In this strategy, five pentagons were assembled on an omphalos pentagon, and bridging peripheral pentagons furnished five additional hexagons. Thirty phenine units were synthetically assembled to afford a large C₂₂₀H₁₈₀ molecule with a phenine framework isoreticular to a hemispherical, bisected segment of C₆₀. Although a hemispherical structure of the phenine framework was suggested by solution‐phase NMR spectra, crystallographic analysis revealed an oval‐like deformation of the molecular shape. In‐depth structural analyses, including theoretical calculations, showed that structural fluctuations observed as variations in the biaryl torsion angles allowed structural deformations and, at the same time, that the dynamic fluctuations resulted in the spectroscopic observation of a hemisphere as a time‐averaged structure.     

"Deconvoluted fullerene" derivatives: synthesis and characterization

We have designed the synthesis of "deconvoluted fullerene" derivatives that present an ordered pattern of hexagons and pentagons in the backbone of the molecule. We not only mimicked the fullerene structure in dihedral planes, but also preserved its electron accepting behavior and enlarged its optical absorption. Moreover, very preliminary photoluminescence (PL) quenching experiments also confirmed the potentiality of these materials as acceptors in the field of organic photovoltaics (OPV)s. A brief discussion of the surface morphology, based on AFM analysis, is also presented.     

Unusual Chlorination Patterns of Three IPR Isomers of C88 Fullerene in C88(7)Cl12/24, C88(17)Cl22, and C88(33)Cl12/14

High‐temperature chlorination of three IPR isomers of fullerene C₈₈, C₂‐C₈₈(7), C_s‐C₈₈(17), and C₂‐C₈₈(33), resulted in the isolation and X‐ray structural characterization of C₈₈(7)Cl₁₂, C₈₈(7)Cl₂₄, C₈₈(17)Cl₂₂, and C₈₈(33)Cl_12/14. Chlorination patterns of C₈₈(7) and C₈₈(33) isomers are unusual in that one or more pentagons remain free from chlorination while some other pentagons are occupied by two or three Cl atoms. The addition patterns of the isolated chlorides are discussed in terms of the distribution of twelve pentagons on the carbon cages and the formation of stabilizing isolated C=C bonds and benzenoid rings.     

Electronic structure and stability of fullerene C82 isolated-pentagon-rule isomers

All nine isolated-pentagon-rule isomers of fullerene C(82) 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 (C(3v)), 8 (C(3v)), and 9 (C(2v)) 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 (C(2)), 2 (C(s)), 4 (C(s)), 5 (C(2)), and 6 (C(s)) with closed electronic shell are supposed to be stable, resembling isomer 3 (C(2)), which has just been extracted experimentally as an empty fullerene. We assume they can be produced as empty molecules.     

Some aspects of the symmetry and topology of possible carbon allotrope structures

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C₆₀ of I _h symmetry and ellipsoidal C₇₀ of D _5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C₇₆, C₇₈, C₈₂ and C₈₄ can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n ² carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C₆₀ fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C₁₆₈ schwarzite unit cell has local O _h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C₆₀ fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

Theoretical Studies on Structures,Stabilities, NMR Spectra and Designing Methods of Dihedral Fullerenes of C3 Series

Using bowl shaped carbon intermediates to construct dihedral fullerenes is an advisable method. Assu- ming that cap shaped C21 extends the size through building pentagons and hexagons at the U and V clefts of the brims, a series of homologous carbon intermediates was generated, in which most of the members have been unknown up to now. The joins between these homologous intermediates gave the C3 dihedral series under the restriction of C3 sym- metrical axis. The investigations point out that the stabilities of these fullerenes not only relate to the shapes of cages and the co-planarities of polygons, but also associate with the equalizations of bond lengths and the pentagonal dis- tributions. The stabilities reveal that the pentagonal distribution in cages is not negligible to the Jr delocalization, be- sides the co-planarities of hexagons and pentagons. Analyzing the possible Stone-Wales（SW） rearrangements in those fullerenes with dehydrogenated pyracyclene units（DPUs） can help us to find out the highly stable isomers. Based on the geometrical optimizations, the calculations provided the theoretical chemical shifts of unknown fullerenes and the data reconfirmed the existence of members C78 and C84. The symmetry adaptation analyses for the frontier orbitals support the formative mechanism of consecutive pentagonal and hexagonal fusions, but the simulated routes are more complicated than the pentagon road（PR） mechanism, which include not only C2 but also C3 additive reactions.