## Structure of Fullerenes

Many independent experiments show that the crystalline materials formed from fullerenes are molecular solids. Therefore the structure and properties of these solids are strongly dependent on the structure and properties of the constituent fullerene molecules. In this chapter we address the internal structure of the molecules, while in Chapter 7, the structures of the related molecular solids are reviewed.

3.1. Structure of C60 and Euler's Theorem

The 60 carbon atoms in C60 are now known to be located at the vertices of a truncated icosahedron where all carbon sites are equivalent [see Fig. 1.1(b)]. This is consistent with the observation of a single sharp line in the nuclear magnetic resonance (NMR) spectrum [3.1,2] (see ยง16.1.1). Since (1) the average nearest-neighbor carbon-carbon (C-C) distance ac_c in C60 (1.44 A) is almost identical to that in graphite (1.42 A), (2) each carbon atom in graphite and in C60 is trigonally bonded to three other carbon atoms in an s/?2-derived bonding configuration, and (3) most of the faces on the regular truncated icosahedron are hexagons, we can, to a first approximation, think of the C60 molecule as a "rolled-up" graphene sheet (a single layer of crystalline graphite). A regular truncated icosahedron has 90 edges of equal length, 60 equivalent vertices, 20 hexagonal faces, and 12 additional pentagonal faces to form a closed shell, consistent with Euler's theorem discussed below. Two single C-C bonds are located along a pentagonal edge at the fusion of a hexagon and a pentagon. In C60, the bond lengths for the single bonds a5 are 1.46 A as measured by NMR [3.1-4] and 1.455 A by neutron scattering [3.5] (see Fig. 3.1). The third bond is located at the fusion between two hexagons and is a double bond with a

Fig. 3.1. The C60 molecule showing single bonds (a5) and double bonds (a6).

bond length a6 which is measured to be 1.40 A by NMR and 1.391 A by neutron diffraction (see Fig. 3.1). Since the bond lengths in CgQ are not exactly equal (i.e., a5 - a6 & 0.06 A), the vertices of the C60 molecule form a truncated icosahedron but, strictly speaking, not a regular truncated icosahedron. In many descriptions of C60, however, the small differences between the a5 and a6 bonds are neglected and C60 is often called a regular truncated icosahedron in the literature.

Since the bonding requirements of all the valence electrons in Qo are satisfied, it is expected that C60 has filled molecular levels. Because of the closed-shell properties of C60 (and also other fullerenes), the nominal sp2 bonding between adjacent carbon atoms occurs on a curved surface, in contrast to the case of graphite where the sp2 trigonal bonds are truly planar. This curvature of the trigonal bonds in C60 leads to some admixture of sp3 bonding, characteristic of tetrahedrally bonded diamond, but absent in graphite.

Inspection of the C60 molecular structure [see Fig. 3.1] shows that every pentagon of C60 is surrounded by five hexagons; the pentagon, together with its five neighboring hexagons, has the form of the corannulene molecule [see Fig. 3.2(a)], where the curvature of the molecule is shown in Fig. 3.2(b). The double bonds in corannulene are in different positions relative to C60 because the edge carbons in corannulene are bonded to hydrogen atoms. Another molecular subunit on the C60 molecule is the pyraclene (also called pyracylene) subunit [see Fig. 3.2(c)], which consists of two pentagons and two hexagons in the arrangement shown. Again, the double bonds differ between the subunit of C60 and the pyracylene molecule because of the edge hydrogens in the molecular pyracylene form. The pyracylene molecule is of

Corannulene

Corannulene b

Corannulene