1.1 Buckminsterfullerene!

In the seminal 1985 work [1], Kroto et al. correctly guessed that a particularly stable molecular cluster of carbon with a molecular weight of 60 carbon atomic masses took the form of a truncated icosahedron (Fig. 1.1). This insight, together with the explosion of research confirming the form of the molecule and exploring its properties in a wide variety of contexts culminated in the award of the 1996 Nobel Prize for Chemistry to Curl, Kroto and Smalley (e.g. see [2]). The study of C₆₀ in the solid-state, or indeed any study requiring macroscopic quantities of the material, had to wait, however, until 1990 when Krätchmer, Huffman and coworkers discovered[3] that C₆₀ could be made easily by arc vapourization of carbon electrodes in a low pressure helium environment. From the remnant soot of such a process, C₆₀ can be extracted by dissolution in an organic solvent such as toluene or decalin, and purified by liquid chromatography. To get very pure C₆₀, one can sublime previously purified C₆₀ in vacuum at high temperatures ($\sim 500 ^\circ$C).

The intriguing features of the C₆₀ molecule which are also at the heart of many of its interesting properties are: its high symmetry, the curvature of its carbon surface, its relatively large size (moment of inertia $\sim 10^{-43}$kgm²), and its hollow core.

There are $4 \times 60$ valence electrons in the C₆₀ molecule, of which $3 \times 60$ are involved in typical covalent bonding between the atoms of the molecule. The remaining 60 electrons are mainly of atomic 2p_z character. The curvature of the C₆₀ surface causes a hybridization of the atomic 2s and 2p levels into the $\pi$ and $\sigma$ orbitals which have hybridization between planar (sp²) and tetrahedral (sp³). The electrons in the $\sigma$ orbitals participate in the C-C covalent bonding, while the $\pi$ orbitals protrude from the C₆₀ surface with asymmetric lobes outside and inside the carbon framework. This hybridization causes the extremely high electronegativity of C₆₀, i.e. the electron affinity is $\sim 2.7$ eV (see the review [5]). Another consequence of this hybridization is that the tendency for chemical bonding differs for the inside and outside of the molecule. Like other aromatic hydrocarbons, such as benzene, these $\pi$ orbitals interact to form highly delocalized molecular orbitals which make up the highest energy manifold of occupied molecular electronic states. The large number of such electrons suggests that the spectrum of these molecular orbitals could be very complex; however, the high degree of symmetry greatly simplifies the situation. The interesting electronic properties of an object are mainly derived from its highest filled and lowest unfilled levels. For a molecule, these are known as the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO). Using the appropriate $\pi$ atomic orbital basis, the spectrum in the HOMO-LUMO region is well described by the Hückel theory of molecular orbitals, which is essentially a tight-binding model for the $\pi$ electrons From this theory, the HOMO is fivefold degenerate ($\times 2$ for the electron spin), the LUMO is threefold degenerate, and the HOMO-LUMO energy difference is on the order of 1 eV. The symmetries of the HOMO and LUMO may also be found in this theory. The HOMO has h_u symmetry while the LUMO has t_1u symmetry (which has 5 nodal lines on the molecule surface). For more details, see [4,5,6].

High symmetry also simplifies the molecular vibrational spectrum. Free C₆₀ has $3\times 60 - 6 = 174$ degrees of freedom, but there are only 46 distinct vibrational modes (10 Raman active, 4 IR active).

For the negatively charged C₆₀, with a partially filled LUMO, the situation above is complicated by the possibility of a Jahn-Teller distortion. In such a situation, the degeneracy of the t_1u level is lifted by a static distortion of the molecule, which amounts to a lowering of the symmetry from pure I_h. Such a distortion is driven by the gain in electronic energy which may outweigh the elastic energy cost of the distortion. For more details, see section VI.B of the review [7] and references therein.

The hollow core of C₆₀ allows the formation of endohedral fullerene complexes, in which an atom is trapped on the inside of the C₆₀ cage. Such complexes are conventionally denoted by the (unsanctioned) notation X@C₆₀, where the ``@'' symbol pictographically represents the endohedral nature of the complex. Many different kinds of atoms and clusters have been encapsulated in fullerenes (C₆₀ as well as other, less symmetric fullerenes). For example metallic atoms and clusters (Ca, U₂, etc.)[8], noble gas atoms (³He, Ne, etc.)[9], atomic nitrogen[10], and, most importantly for this thesis, atomic muonium ($\mu^+e^-$, essentially an unstable light isotope of hydrogen)[11].