Fullerenes represent a fascinating class of carbon allotropes characterized by their closed-cage structures composed entirely of sp²-hybridized carbon atoms. These molecules exhibit unique geometric and electronic properties due to their curvature and symmetry, distinguishing them from other carbon nanomaterials like graphene or nanotubes. The most well-known fullerene, C60, commonly called buckminsterfullerene, consists of 60 carbon atoms arranged in a truncated icosahedron, resembling a soccer ball. Other prominent members include C70, which adopts a more elongated structure, and larger fullerenes such as C76, C78, and C84, each with distinct geometric configurations.

The cage-like structure of fullerenes arises from the incorporation of pentagonal and hexagonal rings, following Euler’s theorem for polyhedra, which states that any closed, convex polyhedron must contain exactly 12 pentagons, while the number of hexagons can vary. In C60, this results in 12 pentagons and 20 hexagons, forming a highly symmetric structure with icosahedral (Ih) symmetry. The curvature induced by the pentagons causes the carbon atoms to deviate from planarity, leading to bond angles that differ from the ideal 120° found in flat graphene. In C60, the bond angles between two hexagons (6:6 bonds) are approximately 138°, while those between a hexagon and a pentagon (6:5 bonds) are around 142°. This strain contributes to the molecule's reactivity and stability.

C70, the next most stable fullerene, contains 12 pentagons and 25 hexagons, resulting in a more oblong shape with D5h symmetry. The additional hexagons elongate the structure, creating distinct equatorial and polar regions. The bond angles in C70 vary depending on their position: near the poles, the angles resemble those of C60, while the equatorial region exhibits angles closer to those of a carbon nanotube. This structural difference influences the electronic properties, making C70 less symmetric and slightly less stable than C60. The strain energy per carbon atom in C70 is marginally higher due to the increased deviation from ideal sp² hybridization.

The stability of fullerenes is governed by the isolated pentagon rule (IPR), which states that the most stable fullerenes are those in which no two pentagons share an edge. This minimizes the strain caused by adjacent pentagons, which would otherwise introduce excessive curvature and destabilize the structure. C60 and C70 both satisfy the IPR, contributing to their abundance and ease of synthesis. Larger fullerenes, such as C76 and C84, also adhere to the IPR but exhibit multiple isomeric forms due to the different arrangements of hexagons. For example, C76 has two IPR-satisfying isomers with D2 and Td symmetry, while C84 has at least four major isomers with varying symmetries (D2, D2d, D6h, and Td).

The symmetry of fullerenes plays a crucial role in their physical properties. C60’s icosahedral symmetry results in a highly degenerate electronic structure, with energy levels that are often multiply degenerate. This symmetry leads to unique optical and electronic behaviors, such as a relatively large bandgap (approximately 1.7 eV for C60) and characteristic absorption peaks in the UV-visible spectrum. The curvature of the molecule also affects its vibrational modes, giving rise to distinct infrared and Raman spectra. For instance, C60 exhibits four active infrared modes and ten Raman-active modes due to its high symmetry.

In contrast, lower-symmetry fullerenes like C70 display more complex spectra due to the reduced degeneracy of their electronic states. The elongation of C70 breaks the spherical symmetry, splitting energy levels and introducing additional optical transitions. This makes C70’s absorption spectrum broader and more feature-rich compared to C60. The vibrational modes of C70 are also more numerous, with 53 distinct frequencies compared to C60’s 46, reflecting its lower symmetry.

The curvature of fullerenes also influences their mechanical properties. The strain from bending sp²-hybridized carbon into a closed cage results in a slight weakening of the carbon-carbon bonds compared to graphene. However, the closed structure provides exceptional resilience to external pressure. Studies have shown that C60 can withstand hydrostatic pressures up to several gigapascals before undergoing structural changes. The stiffness of the fullerene cage is also reflected in its high bulk modulus, which is comparable to that of diamond for C60.

Thermodynamically, fullerenes are metastable relative to graphite but are stabilized by entropy at high temperatures. The formation of fullerenes from vaporized carbon involves complex kinetic pathways, with the IPR-satisfying structures emerging as the most probable products due to their lower strain energy. The stability trend generally follows the order C60 > C70 > larger IPR fullerenes, though specific isomers of larger fullerenes can exhibit comparable or even higher stability depending on their symmetry and strain distribution.

The geometric differences between fullerene isomers also manifest in their packing behavior in solid-state forms. C60 crystallizes in a face-centered cubic (fcc) lattice at room temperature, with weak van der Waals interactions between molecules. At lower temperatures, a phase transition occurs to a simple cubic structure due to orientational ordering of the molecules. C70, with its lower symmetry, packs into a hexagonal close-packed (hcp) or fcc lattice depending on the conditions, and its phase behavior is more complex due to the anisotropic shape of the molecule.

In summary, the structural properties of fullerenes are defined by their cage-like geometry, bond angle strain, and symmetry. The interplay of these factors determines their stability, electronic structure, and physical behavior. C60 and C70 serve as archetypal examples, demonstrating how subtle variations in structure—such as the number of hexagons or the arrangement of pentagons—can lead to significant differences in properties. Larger fullerenes further illustrate the diversity of possible configurations, each with unique characteristics arising from their distinct geometries. Understanding these relationships is essential for harnessing the potential of fullerenes in applications ranging from electronics to materials science.