Structure and electronic properties of fullerenes C(52)q+: is C(52)2+ an exception to the pentagon adjacency penalty rule?

A structure-property interplay between the width and height of cages and the static third order nonlinear optical responses for fullerenes: applying gamma density analysis

To reveal a new structure-property relationship regarding the nonlinear optical (NLO) properties of fullerenes that are associated with gamma (γ) density, fullerenes I (C40, C50, C60 and C70), whose heights range from 4.83 to 7.96 Å, and II (C24, C36, C48 and C72), whose widths range from 4.45 to 8.22 Å, have been the research objects. Calculation of their geometric and electronic structures, absorption spectra, and the second hyperpolarizability (γ) and the γ density analysis have been performed. It is found that the electronic spatial extent and the polarizability (α) value increase linearly as the fullerenes increase by every 12 carbon atoms. Similarly, the γ values are also proportional to the fullerene size. It is worth noting that the relative magnitude of γ_xxxx and γ_zzzz was exactly consistent with that of the width and height of fullerenes. The analysis of γ density provides the essential reason for this result, that is, the magnitude of the contribution to γ values associated with γ densities is proportional to the density amplitudes multiplied by the distance between them. Larger fullerenes possess larger density amplitudes and longer distances, resulting in larger γ values with respect to smaller fullerenes. This work presents a new structure-property interplay between the width and height of the fullerenes and their second hyperpolarizability γ. Moreover, the γ density analysis provides a new insight to explore the nature of the relationship between the structure and the NLO properties.

From C(58) to C(62) 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.

Comprehensive theoretical study of all 1812 C60 isomers

All 1812 C₆₀ isomers are investigated with high-level quantum chemical methods to benchmark semiempirical approaches and find appropriate stability criteria.

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.

Doping the Buckminsterfullerene by Substitution: Density Functional Theory Studies of C59X (X = B, N, Al, Si, P, Ga, Ge, and As)

The heterofullerenes C59X (X = B, N, Al, Si, P, Ga, Ge, and As) were investigated by quantum chemistry calculations based on density functional theory. These hybrid cages can be seen as doping the buckminsterfullerene by heteroatom substitution. The geometrical structures, relative stabilities, electronic properties, vibrational frequencies, dielectric constants, and aromaticities of the doped cages were studied systemically and compared with those of the pristine C60cage. It is found that the doped cages with different heteroatoms exhibit various electronic, vibrational, and aromatic properties. These results imply the possibility to modulate the physical properties of these fullerene-based materials by tuning substitution elements.

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 (B(55) bonds), and many of them are even more stable than some classical isomers with fewer B(55) bonds. The nonclassical isomers with the lowest energy are higher in energy than the classical ones with the lowest energy, because they have more B(55) 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.

A global search for the lowest energy isomer of C(26)

The complete set of 2333 isomers of C(26) fullerene composed of square, pentagonal, hexagonal, and heptagonal faces together with some noncage structures is investigated at the Hartree-Fock and density functional theory (DFT) levels. For the singlet states, a nonclassical isomer C(26)-10-01 with a square embedded is predicted by the DFT method as the lowest energy isomer, followed by the sole classical isomer C(26)-00-01. Further explorations reveal that the electronic ground state of C(26)-10-01 is triplet state in C(s) symmetry, while that of C(26)-00-01 corresponds to its quintet in D(3h) symmetry. Both the total energies and nucleus independent chemical shift values at DFT level favor the classical isomer. It is found that both C(26)-00-01 and C(26)-10-01 possess high vertical electron affinity. The addition of electron(s) to C(26)-10-01 increases its aromatic character and encapsulation of Li atom into this cage is highly exothermic, indicating that it may be captured in the form of derivatives. To clarify the relative stabilities at elevated temperatures, the entropy contributions are taken into account based on the Gibbs free energy at the B3LYP/6-311+G( *) level. C(26)-10-01 behaves thermodynamically more stable than the classical isomer over a wide range of temperatures related to fullerene formation. The IR spectra of these two lowest energy isomers are simulated to facilitate their experimental identification.

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.

Ab initio calculation of carbon clusters. II. Relative stabilities of fullerene and nonfullerene C24

Chemical stabilities of six low-energy isomers of C24 derived from global-minimum search are investigated. The six isomers include one classical fullerene (isomer 1) whose cage is composed of only five- and six-membered rings (56-MRs), three nonclassical fullerene structures whose cages contain at least one four-membered ring (4-MR), one plate, and one monocyclic ring. Chemical and electronic properties of the six C24 isomers are calculated based on a density-functional theory method (hybrid PBE1PBE functional and cc-pVTZ basis set). The properties include the nucleus-independent chemical shifts (NICS), singlet-triplet splitting, electron affinity, ionization potential, and gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital (HOMO-LUMO) gap. The calculation suggests that the neutral isomer 2, a nonclassical fullerene with two 4-MRs, may be more chemically stable than the classical fullerene (isomer 1). Analyses of molecular orbital NICS show that the incorporations of 4-MRs into the cage considerably reduce paratropic contributions from HOMO, HOMO-1, and HOMO-2, which are mainly responsible for the sign change in NICS from positive for isomer 1 (42) to negative (-19) for isomer 2, although C24 clusters satisfy neither 4N+2 nor 2(N+1)2 aromaticity rule. Anion photoelectron spectra of four cage isomers, one plate, one monocyclic ring, and one tadpole isomer, as well as three bicyclic ring isomers are calculated. The simulated photoelectron spectra of mono- and bicyclic rings (with C1 symmetry) appear to match the measured HOMO-LUMO gap (between the first and second band in the experimental spectra) [S. Yang et al., Chem. Phys. Lett. 144, 431 (1988)]. Nevertheless, the nonclassical fullerene isomers 3 and 4 apparently also match the measured vertical detachment energy (2.90 eV) reasonably well. These results suggest possible coexistence of nonclassical fullerene isomers with the mono- and bicyclic ring isomers of C24(-) under the experimental conditions.

Structures, stabilities, and electronic and optical properties of C52 fullerene, ions, and metallofullerenes

The 437 classical isomers of fullerene C52 have been studied by PM3, HCTH/3-21G, and B3LYP6-31G(d). C(2):029 with the least number of adjacent pentagons is predicted to be the most stable isomer. The investigations show that both the number of adjacent pentagons and the degree of aromaticity play important roles in the relative stabilities of fullerene isomers. To clarify the relative stabilities of the C52 isomers in a wide range of temperatures, the entropy contributions are taken into account on the basis of the Gibbs energy at the B3LYP6-31G(d) level. C(2):029 prevails in a wide temperature range. In addition, the electronic spectra and second-order hyperpolarizabilities are determined by means of ZINDO and sum-over-states model. The static second-order hyperpolarizability of C(2):029 is 51% larger than that of C60. Furthermore, intensity-dependent refractive index gamma (-omega;omega,omega,-omega) (omega=1.1653 eV) of C(2):029 is 13 times larger than that of C60. The encapsulation of Ca atom in C52 fullerene is exothermic and the metallofullerene Ca-C52 is described as Ca2+-C52(2-).

Computational Studies on the Cyclization of Polycyclic Aromatic Hydrocarbons in the Synthesis of Curved Aromatic Derivatives

Computational studies on the cyclization reactions of some polycyclic aromatic hydrocarbons (PAHs) were performed at the DFT level. Compounds C26H14 and C24H14, which show the connectivity of C60 fullerene fragments, were chosen as suitable models to study the formation of curved derivatives by six- or five-membered ring formation, upon oxidation to their radical cations. Four possible pathways for the cyclization process were considered: a) initial C-C bond formation to afford a curved derivative, followed by dehydrogenation; b) homolytic C-H cleavage prior to cyclization; c) initial concerted H2 elimination and subsequent cyclization; and d) deprotonation of the radical cations prior to cyclization. Computed reaction and activation energies for these reactions show that direct cyclization from radical cations (pathway a) is the lowest-energy mechanism. The formation of five-membered rings is somewhat more favourable than benzannulation. After new cycle formation, homolytic C-H dissociation to afford the corresponding cations is the most favourable process. These cations react with H* without barrier to give H2* Intermediate deprotonations are strongly disfavoured. The relatively low activation energies compared with carbon cage rearrangements suggest that ionization of PAHs can be used for the tailored preparation of nonplanar derivatives from suitable precursors.

Structure and electronic properties of highly charged C60 and C58 fullerenes

We present a theoretical study of the structure and electronic properties of positively charged C60(q+) and C58(q+) fullerenes (q = 0-14). Electronic energies and optimum geometries have been obtained using density-functional theory with the B3LYP functional for exchange and correlation. We have found that closed- and semiclosed-shell C60(q+) ions (q = 0, 5, and 10) preserve the original icosahedral symmetry of neutral C60. For other charges, significant distortions have been obtained. The C58(q+) fullerenes are, in general, less symmetric, being C58(8+) the closest to the spherical shape. Most C60(q+) fullerenes follow Hund's rule for spin multiplicity, while most C58(q+) fullerenes are more stable with the lowest spin multiplicity. The calculated ionization potentials for both kinds of fullerenes increase almost linearly with charge, except in the vicinity of C60(10+) and C58(8+). We have also explored the region of the potential-energy surface of C60(q+) that leads to asymmetric fission. Minima and transition states corresponding to the last steps of the fission process have been obtained. This has led us to conclude that, for 3 < or = q < or = 8, C2(+) emission is the preferred fragmentation channel, whereas, for higher q values, emission of two charged atomic fragments is more favorable. The corresponding fission barrier vanishes for q > 14.

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.

Comparative Investigation on Non-IPR C68 and IPR C78 Fullerenes Encaging Sc3N Molecules

A computational study on the experimentally detected Sc(3)N@C(68) cluster is reported, involving quantum chemical analysis at the B3LYP/6-31G level. Extensive computations were carried out on the pure C(68) cage which does not conform with the isolated pentagon rule (IPR). The two maximally stable C(68) isomers were selected as initial Sc(3)N@C(68) cage structures. Full geometry optimization leads to a confirmation of an earlier assessment of the Sc(3)N@C(68) equilibrium geometry (Nature 2000, 408, 427), namely an eclipsed arrangement of Sc(3)N in the C(68) 6140 frame, where each Sc atom interacts with one pentagon pair. From a variety of theoretical procedures, a D(3h) structure is proposed for the free Sc(3)N molecule. Encapsulated into the C(68) enclosure, this unit is strongly stabilized with respect to rotation within the cage. The complexation energy of Sc(3)N@C(68) cage is found to be in the order of that determined for Sc(3)N@C(80) and exceeding the complexation energy of Sc(3)N@C(78). The cage-core interaction is investigated in terms of electron transfer from the encapsulated trimetallic cluster to the fullerene as well as hybridization between these two subsystems. The stabilization mechanism of Sc(3)N@C(68) is seen to be analogous to that operative in Sc(3)N@C(78). For both cages, C(68) and C(78), inclusion of Sc(3)N induces aromaticity of the cluster as a whole.