Electronic structure of fullerenes

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

Electronic Structure of Fullerenes. As discussed above, the molecular structures of fullerenes show some very conspicuous features which will be certainly determinant in the electronic structure of this form of carbon. The principal of such structural features is certainly their spheroidal geometry. 

Complex formation also produces some alterations in the electronic structure of fullerenes. In general, the reduction potentials of the metal derivatives are... 

Thus, the electronic structures of fullerene anionic species can be obtained by filling the 3-fold degenerate LUMOs of the cluster. Accordingly, the products of the intercalation of alkali metals in fullerenes are expected to show, depending on the intercalation degree, an enhanced conductivity. In some cases superconductivity has been also observed (vide infra). 

The structure-property relations of fullerenes, fullerene-derived solids, and carbon nanotubes are reviewed in the context of advanced technologies for carbon-hased materials. The synthesis, structure and electronic properties of fullerene solids are then considered, and modifications to their structure and properties through doping with various charge transfer agents are reviewed. Brief comments are included on potential applications of this unique family of new materials. 

In contrast to carbon, which forms structures derived from both sp2 and sp3 bonds, silicon is unable to form sp2 related structures. Since one out of four sp3 bonds of a given atom is pointing out of the cage, the most stable fullerene-like structure in this case is a network of connected cages. This kind of network is realized in alkali metal doped silicon clathrate (19), which were identified to have a connected fullerene-like structure (20). In these compounds, Si polyhe-dra of 12 five-fold rings and 2 or 4 more six-fold rings share faces, and form a network of hollow cage structures, which can accommodate endohedral metal atoms. Recently, the clathrate compound (Na,Ba), has been synthesized and demonstrated a transition into a superconductor at 4 K (21). The electronic structure of these compounds is drastically different from that of sp3 Si solid (22). 

Fig. 3 Structure of fullerenes Cso, C70, Cso and single-wall carbon nanotube. The figures were taken with permission of Prof. C. Dekker from the image gallery found at http //online.itp.ucsb.edu/online/qhalLc98/dekker/. Transmission electron microscopy image of multi-wall carbon nanotube (MWCNT) treated with iodinated and platinate DNA. The figure was taken from [24] with kind permission from Prof. P. Sadler...

Several effects can influence the electronic structure of Cjq upon metal complex formation. One is the removal of one double bond from the remaining 29 fullerene double bonds. As in any polyene system, this decreased conjugation is expected to raise the energy of the LUMO and therefore decreases the electron affinity of the system. Conversely, the d-orbital backbonding transfers electron density from the metal into n orbitals of the remaining double bonds, which also decreases the electron affinity. 

Under some simplifications associated with the symmetry of fullerenes, it has been possible to perform calculations of type Hartree-Fock in which the interelec-tronic correlation has been included up to second order Mpller-Plesset (Moller et al. 1934 Purcell 1979 Cioslowski 1995), and calculations based on the density functional (Pople et al. 1976). However, given the difficulties faced by ab initio computations when all the electrons of these large molecules are taken into account, other semiempirical methods of the Hiickel type or tight-binding (Haddon 1992) models have been developed to determine the electronic structure of C60 (Cioslowski 1995 Lin and Nori 1996) and associated properties like polarizabilities (Bonin and Kresin 1997 Rubio et al. 1993) hyperpolarizabilities (Fanti et al. 1995) plasmon excitations (Bertsch et al. 1991) etc. These semiempirical models reproduce the order of monoelectronic levels close to the Fermi level. Other more sophisticated semiempirical models, like the PPP (Pariser-Parr-Pople) (Pariser and Parr 1953 Pople 1953) obtain better quantitative results when compared with photoemission experiments (Savage 1975). 

In comparison with pristine C6o and C70, the fullerene derivatives (Fig. 2) show partly different photophysical properties due to the pertubation of the fullerene s TT-electron system. The degree of variation is dependent on (1) the electronic structure of the functionalizing group, (2) the number of addends, and (3) in the case of multiple adducts on the addition pattern at the fullerene core [59-112],... 

In this chapter we have described the photophysics and photochemistry of C6o/C70 and of fullerene derivatives. On the one hand, C6o and C70 show quite similar photophysical properties. On the other hand, fullerene derivatives show partly different photophysical properties compared to pristine C6o and C70 caused by pertuba-tion of the fullerene s TT-electron system. These properties are influenced by (1) the electronic structure of the functionalizing group, (2) the number of addends, and (3) in case of multiple adducts by the addition pattern. As shown in the last part of this chapter, photochemical reactions of C60/C70 are very useful to obtain fullerene derivatives. In general, the photoinduced functionalization methods of C60/C70 are based on electron transfer activation leading to radical ions or energy transfer processes either by direct excitation of the fullerenes or the reaction partner. In the latter case, both singlet and triplet species are involved whereas most of the reactions of electronically excited fullerenes proceed via the triplet states due to their efficient intersystem crossing. 

The electronic structure of the hypothetical fulleroids also parallels that of the fullerenes in several significant respects. Properly closed n shells are found within the Hiickel approximation only for isolated p-gon isomers, and then only when the structure is either (a) leapfrogged from a smaller fulleroid of the same symmetry, or (b) a p-fold analogue of one of the fullerene carbon cylinder series (Fowler 1990). D1(l C84 is the smallest fulleroid with a properly closed shell (figure 3). 

Quite apart from their singular topology, the fullerenes are distinguished from other conjugated hydrocarbons by their non-planarity. The geometrical aspects of fullerene formation as it relates to pyramidalization of the constituent carbon atoms has been recognized for some time (Haddon et al. 1986 Haddon 1988). Here we consider the effect of non-planarity on the electronic structure of the carbon atoms as it arises in the fullerenes (Haddon et al. 1986 Haddon 1992). 

C60 shows an extremely facile reduction profile and there is evidence for the addition of up to 12 electrons to the molecule. The prediction that C60 will exhibit an exceptionally high electron affinity and that the molecule will add up to 12 electrons under suitable conditions (Haddon et al. 1986a) seems to be borne out by the experimental results. Rehybridization plays an important role in determining the electronic structure of the fullerenes and it is the combination of topology and rehybridization which together account for the extraordinary ability of C60 to accept electrons. 

Haddon, R. C. Raghavachari, K. 1992 Electronic structure of the fullerenes carbon allotropes of intermediate hybridization, buckminsterfullerenes. VCH. (In the press.)... 

In the case of metal clusters, for example, valence electrons show the shell structure which is characteristic of the system consisting of a finite number of fermions confined in a spherical potential well [2]. This electronic shell structure, in turn, motivated some theorists to study clusters as atomlike building blocks of materials [3]. The electronic structure of the metallofullerenes La C60 [4] and K C60 [5] was investigated from this viewpoint. This theorists dream of using clusters as atomlike building blocks was first realized by the macroscopic production of C60 and simultaneous discovery of crystalline solid C60, where C60 fullerenes form a close-packed crystalline lattice [6]. 

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