where C}a creates an electron with spin a on the ith site, and ni(T = C]aCia is the density of electrons with spin a on site i, and i,-■ denotes the transfer integral for the transfer of charge. U and V represent on-site and offsite electron-electron interactions, the sums in Eq. (15.24) are taken over orbital and spin states, and H.c. denotes Hermitian conjugate. The exact diagonalization of given by Eq. (15.24) has been done for small clusters with N electrons on a one-dimensional ring [15.111], torus (N = 16), cube (N = 8), and truncated tetrahedron (N = 12) [15.108], which are all smaller than the case of N = 60 for C60. The calculated results show that £pajr of Eq. (15.23) can be positive or negative, depending on the relative ratios of the parameters t, U, and V. If an electron-electron pairing mechanism were to be important, it would need to explain not only superconductivity arising from the partial filling of the ilu-derived band but also, for the case of the alkaline earth compounds, the partial filling of the tlg-derived band, including the strong hybridization of the tlg electronic states with the alkaline earth metal s and d states.

Recently, the total energy of the C^ ion has been calculated using semi-empirical self-consistent field (SCF) and configurational interaction (CI) programs from the library MOPAC [15.112]. The calculated results show that the total energy as a function of n has a positive curvature, which implies that is always negative. In this sense the effective attractive interaction between two electrons in a molecule might not be relevant as a mechanism for superconductivity.

When the electron-phonon interaction screens the Coulomb repulsion between two electrons, the effective Coulomb interaction, [/eff, can be negative. In this case we can apply an attractive Hubbard model to such a system. Starting from an attractive Hubbard model, it has been shown [15.113] that the superconducting state is more stable than the charge density wave (CDW) state in the fee structure of K3C60, similar to the case in BaBi03. The effective intramolecular interaction between tt electrons, Ue{i, can be negative for the case in which the optic phonons involving the K+ ions couple strongly to the tt electron carriers on C60 molecules of the K3C60 crystal. The absolute value of UeS caused by two optic modes that are related to octahedral-tetrahedral and octahedral-octahedral sites (estimated as t/eff ~ —0.65 and —0.27 eV, respectively) may be larger than that of the repulsive interaction caused by the screened intramolecular Coulomb repulsion Ue_e{\.2-\.l eV) or the negative values caused by a bond dimer-ization effect (~ —0.026 eV) or the Jahn-Teller interaction (~ -25 meV) [15.114], Within mean field theory, the superconducting state is more stable than the CDW state for any negative Ueff. This fact can be understood both by the calculation of Tc in the weak coupling limit of Ueff and by the perturbation expansion in t of the total energy for the two states in the strong coupling limit of \Uell/t\ » 1.

A number of proposed mechanisms for superconductivity in doped fullerenes involve the Hubbard model. Therefore criticisms of using a Hubbard model should be mentioned. The validity of the Hubbard model requires that the intra-site screening be much smaller than the intersite screening and that the bare intrasite Coulomb repulsion should be much larger than the bare intersite repulsion. Since the nearest-neighbor C-C distance between adjacent Qq molecules is ~3 Á and of the same order as the diameter of C60 (~7 Á), the bare intrasite and intersite repulsions should be of comparable magnitude. The large number of on-site electrons (240) for C60 suggests that the intrasite screening is important. Thus use of the Hubbard model for describing the electronic structure of fullerenes has been questioned. It has also been argued that the energy needed to transport an electron should be the difference Umm - £/inler rather than the intracluster repulsion t/imra, and it is expected that t/intra — i/inter £/intra [15.107,115].

Another proposed theory of superconductivity for doped C60 is based on a parity doublet [15.116] formed from an h^tfjl configuration in the doped C60 material. In this case, the t\utjg electron pair forms the parity doublet that triggers the superconducting transition. Also, the possibility of plasmons mediating an attractive interaction between electrons has been suggested as a pairing mechanism for low-carrier-density systems [15.117].

Although many attempts have been made to explain the superconducting state in doped fullerenes as special cases of a Hubbard system with U ranging from negative to positive values, none of the theories are yet conclusive, partly because of their inability to account for many of the published experimental results. Likewise, many first principles calculations elucidate certain fundamental issues but do not make systematic predictions which can be tested. Most experiments suggest that the transfer energy is comparable to the phonon energy and to the electron-electron interaction. Quantitative applications of the theory using realistic parameters are needed. Furthermore, justification is also needed of the methodology that was used to determine reliable values of the parameters that are employed in theoretical calculations. Particular insights into the dominant superconducting mech anism are expected through gaining a better understanding of the isotope effect and pressure-dependent phenomena.

While it is generally believed that superconductivity in doped fullerenes arises from some kind of pairing mechanism between electrons, the detailed nature of the pairing mechanism is not well established. Although the electron-phonon mechanism has been most widely discussed, it appears that electron-electron pairing mechanisms have not yet been ruled out. For those who believe that the electron-phonon interaction is the dominant pairing mechanism, heated discussion currently focuses on which phonons play a dominant role in the coupling. To produce pairing, many authors have invoked a dynamic Jahn-Teller mechanism induced by pertinent intramolecular modes, and further clarification is needed of the role of the Jahn-Teller effect in fullerene superconductivity. Experimental evidence in support of Jahn-Teller distortions for anions comes from both EPR studies (§16.2.2) and optical studies (§13.4.1).

A host of other fundamental theoretical issues remain to be clarified. Since the charge distribution in the superconducting fullerenes is mainly on the surface of the icosahedral anions, in contrast to the more distributed charge distribution found in conventional solids, modifications are probably necessary to the traditional interpretation of a variety of measurements, such as the plasma frequency, specific heat, magnetic susceptibility, and temperature- and magnetic-field dependent transport measurements, to mention but a few. While concern about the validity of the Migdal theorem has been raised, since the vibrational frequencies are comparable in magnitude to the Fermi energy and to the bandwidth of the LUMO levels, no clear picture has yet emerged on which aspects of the Migdal theory need modification. While many workers in the field agree that many-body effects are important, no consensus has been reached on how to handle strong correlations between electrons in a system where the diameter of the C60 molecules and the nearest-neighbor distances are of comparable magnitude and the electronic energy bandwidths are very narrow.

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