New structural parameters of fullerenes for principal component analysis

The Kekulé structure count and the permanent of the adjacency matrix of fullerenes are related to structural parameters involving the presence of contiguous pentagons p, q, r, q/p and r/p, where p is the number of edges common to two pentagons, q is the number of vertices common to three pentagons and r is the number of pairs of nonadjacent pentagons adjacent to another common pentagon. The cluster analysis of the structural parameters allows classification these parameters. Principal component analysis (PCA) of the structural parameters and the cluster analyses of the fullerenes permit their classification. PCA clearly distinguishes five classes of fullerenes. The cluster analysis of fullerenes is in agreement with PCA classification. Cluster analysis shows greatest similarity for the q–q/p and r–r/p pairs. PCA provides five orthogonal factors F ₁–F ₅. The use of F ₁ gives an error of 28%. The inclusion of F ₂ decreases the error to 2%.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresión Latina (QUITEL 2002)     

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.     

Fullerenes as tilings of surfaces

If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Klein-bottle fullerenes, none. Klein-bottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Klein-bottle, and toroidal cases, respectively.     

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.     

Sizes of pentagonal clusters in fullerenes

Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes.     

Strain energies due to nonplanar distortion of fullerenes and their dependence on structural motifs

Accurate strain energies due to nonplanar distortion of 114 isolated pentagon rule (IPR) fullerenes with 60-102 carbon atoms have been calculated based on B3LYP/6-31G(d) optimized structures. The calculated values of strain energy due to nonplanar distortion (Enp) are reproduced by three simple schemes based upon counts of 8, 16, and 30 distinct structural motifs composed of hexagons and pentagons. Using C180 (Ih) and CN (Ih) (N is very large) as test molecules, the intrinsic limitations of the motif model based on six-membered rings (6-MRs) as the central unit have been discussed. On the basis of the relationship between the contributions of motifs to Enp and the number of five-membered rings (5-MRs) in motifs, we found that IPR fullerenes with dispersed 5-MRs present smaller nonplanar distortions.     

Non-IPR fullerenes with properly closed shells

Fullerenes with properly closed shells (having exactly half their adjacency eigenvalues strictly positive) are rare. All reported examples obey the isolated-pentagon rule (IPR), usually considered a necessary condition of overall stability, and fall into three series (leapfrogs, carbon cylinders and sporadic closed shells). It is shown here that there also exist fullerenes with properly closed shells that violate the IPR ('super-sporadic' fullerenes). All have negative LUMO eigenvalues of small magnitude. Exhaustive search finds four examples with 160 or fewer vertices: one isomer of C(120), two of C(156) and one of C(160). The first three contain single pentagon pairs and the fourth, a linear triple of fused pentagons. Larger examples can be found. A capping construction gives a series of properly closed shell fullerenes of C(3)/C(3v) symmetry, each with a single fully fused triple of pentagons and ≥632 vertices. Tubular extension of the C(120) example leads a series of C(1)/C(s) isomer pairs with ≥168 vertices, retaining the single pentagon adjacency and approaching isospectrality with increasing size. Both constructions are conjectured to lead to an infinite number of super-sporadic fullerenes.     

A catalogue of isomerization transformations of fullerene polyhedra

Representation of isomerization and carbon insertion or extrusion mechanisms as patch replacements on a fullerene surface allows construction of a catalogue of topologically distinct local transformations of fullerenes, classified by patch boundary and pentagon content. All isomerization patches and isomerization pairs containing up to five pentagons and with an upper limit for the boundary length depending on the number of pentagons are listed. Several infinite series of transformations are identified.     

Pentaheptite modifications of the graphite sheet

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-packed to hexagonal-close-packed packings in 3D. Symmetries realizable for these pentaheptite layers are three strip groups (periodic in one dimension), p1a1, p112, and p111, and five Fedorov groups (periodic in two dimensions), cmm, pgg, pg, p2, and p1. All can be constructed by simultaneous rotation of the central bonds of pyrene tilings of the graphite sheet. The unique lattice of cmm symmetry corresponds to the previously proposed pentaheptite carbon metal. Analogous pentagon-heptagon tilings on other surfaces including the torus, Klein bottle, and cylinder, face-regular tilings of pentagons and b-gons, and a full characterization of tilings involving isolated pairs and/or triples of pentagons are presented. The Kelvin paradigm of a continuum of structures arising from propagation of two original motifs has many potential applications in 2D and 3D.     

Structure and Electronic Properties of Fullerenes C52q+: Is C522+ an Exception to the Pentagon Adjacency Penalty Rule?

The structure, vibrational spectra and electronic properties of the neutral, singly and doubly charged C52 fullerenes were studied by means of the Hartree-Fock method and density functional theory. Different isomers were considered, in particular those with the lowest possible number (five or six) of adjacent pentagons, and an isomer with a four-atom ring. For neutral and singly charged species, the most stable isomer is that with the lowest number of adjacent pentagons, namely five. However, for C(52)2+, the most stable structure has six adjacent pentagons. This finding, which contradicts the pentagon adjacency penalty rule, is a consequence of complete filling of the HOMO pi shell and the near-perfect sphericity of the most stable isomer. The simulated vibrational spectra show important differences in the positions and intensities of the vibrations for the different isomers.     

A catalogue of growth transformations of fullerene polyhedra

Carbon insertion or extrusion mechanisms transforming one fullerene to another are presented as patch replacements on the fullerene surface. A systematic catalogue is constructed for the topologically distinct local insertion/extrusion transformations of fullerenes, classified by patch boundary and pentagon content. All pairs of patches with the same boundary but different numbers of atoms, i.e., growth patches, containing up to five pentagons, with an upper limit for the boundary length that depends on the number of pentagons, are listed. New transformations and infinite series of transformations are identified.     

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     

Zigzags, railroads, and knots in fullerenes

Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways: from zigzags, i.e., circuits (possibly self-intersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly self-intersecting) of edge-sharing hexagonal faces, such that the shared edges occur in opposite pairs. A z-knot fullerene has only a single zigzag, doubly covering all edges: in the range investigated (n /= 38, all chiral, belonging to groups C(1), C(2), C(3), D(3), or D(5). An r-knot fullerene has a railroad corresponding to the projection of a nontrivial knot: examples are found for C(52) (trefoil), C(54) (figure-of-eight or Flemish knot), and, with isolated pentagons, at C(96), C(104), C(108), C(112), C(114). Statistics on the occurrence of z-knots and of z-vectors of various kinds, z-uniform, z-transitive, and z-balanced, are presented for trivalent polyhedra, general fullerenes, and isolated-pentagon fullerenes, along with examples with self-intersecting railroads and r-knots. In a subset of z-knot fullerenes, so-called minimal knots, the unique zigzag defines a specific Kekulé structure in which double bonds lie on lines of longitude and single bonds on lines of latitude of the approximate sphere defined by the polyhedron vertices.     

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.     

富勒烯合成化学研究进展

富勒烯是一类由12个五元环和若干六元环组成的笼状分子, 自20世纪80年代中期被发现以来就以其独特的结构和新奇的性质而成为科学界研究的热点, 25年来, 无论在基础研究还是在实际应用领域都有了长足的进步, 人们在发展富勒烯合成新方法和寻找富勒烯新结构方面做了大量的工作。本文对富勒烯的各种宏量合成方法进行了回顾, 并概述了迄今已发表的60余种富勒烯新结构,包括各种富勒烯空笼、内嵌富勒烯、富勒烯笼外修饰衍生物及氮杂富勒烯等结构。     

C64H4: production, isolation, and structural characterizations of a stable unconventional fulleride

Unconventional fullerenes are those smaller than C(60) or those intermediate between C(60) and C(70), which are not stable in structure as none of the unconventional fullerene isomers satisfying the "isolated-pentagon-rule" (IPR). Below we report the synthesis of a stable unconventional fullerene derivative C(64)H(4) by introducing methane in the fullerene productions with the normal Kr?tschmer-Huffman method. We also applied various spectroscopic measurements such as mass spectrometry, (13)C NMR, IR, UV-vis absorption spectrometry, etc. to characterize the structural and electronic properties of this molecule, revealing an unprecedented fullerene cage with a triplet of directly fused pentagons in the framework of C(64)H(4). Four hydrogen atoms are added to the carbons at vertexes of fused pentagons to allow the bond angles at these sites close to the sp(3) tetrahedral angle, which essentially release the sp(2) bond strains on the abutting-pentagon sites of C(64). Ab initio calculations were performed to explore the electronic property and simulate the (13)C NMR and IR spectra of this fulleride, which reproduced well the experimental results and confirmed the structural assignment of the C(64)H(4).     

Three-stage transformation pathway from nanodiamonds to fullerenes

The dynamics of structure evolution of nanodiamonds ranging from 22 to 318 atoms of various shapes is studied by density functional tight-binding molecular dynamics. The spherical and cubic nanodiamonds can be transformed into fullerene-like structures upon heating. A number of the transformed fullerenes consist of pentagons and hexagons only. Others contain squares, heptagons, and octagons. One simulated fullerene is an isomer of C(60). The temperature of the transformation depends on the size, shape, and orientation of initial cluster. To be transformed into onion-like fullerenes, the spherical nanodiamonds should have 200 atoms or more, while the cubic ones require 302 atoms or more. The time-resolved energy profiles of all the transformations clearly reveal three-stage transformation character. During the first stage, the energy reduces quickly due to converting sp(3) carbon with dangling bond at the surface into sp(2) one, and the formation of partial sp(2) envelope wrapping the cluster. For the second stage, energy decreases slowly. The remaining interior carbon atoms come to the surface through the hole in the sp(2) envelope, and similar amount of sp(3) and sp(2) atoms coexist. The third stage involves the closure of holes, accompanied by the detachment of C(2) molecules and carbon chains from the edges. The energy decreases relatively fast in this stage. The proposed three-stage transformation pathway holds for all the simulations performed in this work, including those with the instant heating.     

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.