aRef. [12.15], To improve the model at higher I values, higher-order terms in the phenomenological Hamiltonian must be included [Eq. (12.2)]. fcRef. [12.24], aRef. [12.15], To improve the model at higher I values, higher-order terms in the phenomenological Hamiltonian must be included [Eq. (12.2)]. fcRef. [12.24],
12.3. Many-Electron States for C60 and Other Icosahedral Fullerenes
In the previous sections the molecular energy levels were considered from a one-electron point of view, whereby the energy levels of the electron are treated in terms of an average potential due to the other ions and to the other valence electrons in the molecule. This simplified treatment is appropriate to the case of filled-shell configurations or for the excitation of one electron (hole) from the filled-shell configurations. Because of the large number of valence electrons on a single fullerene molecule, it is often necessary to go beyond the one-electron Hamiltonian and to consider more explicitly the interaction between electrons through exchange and correlation effects. In enumerating the Pauli-allowed states for many-electron excitations, a many-electron approach is implicit.
The phenomenological Hamiltonian for the many-electron (N) problem can be written as
;=1 i,/=l where ^ refers to the one-electron Hamiltonian, is the interaction Hamiltonian between electrons i and and the prime on the sum avoids double counting. Without loss of generality ^ can be written as where is the spin angular momentum operator and Jtj is an exchange integral. For two-electron states based on C60, this exchange interaction gives rise to a singlet-triplet splitting which is very important in discussing excited states (§12.4.2) and optical properties (see §13.1.2) [12.17],
In this section we discuss many-electron states from the point of view of forming the ground state for molecular fullerenes. The specification of the excited states then follows. We argue that it is necessary that a ground state be nondegenerate, because if it were degenerate, the molecule would then distort, causing a level splitting of this degenerate state and the lowering of the energy of the system. Such distortions are important for considering the electronic states of charged fullerenes, as discussed below.
In Table 4.10, which we have already discussed in terms of the filling of the angular momentum states, we can see that in full rotational symmetry the filling of each angular momentum state I results in a many-body non-degenerate state with Ag symmetry, which is a good candidate for a ground state. Also listed in Table 4.10 are the symmetries of these filled levels in icosahedral symmetry. Thus the filling of the I = 5 state in full rotational symmetry corresponds to the filling of levels h10, /,6U, and /2u in the lower-symmetry icosahedral group with 10, 6, and 6 electrons, respectively. From strictly symmetry considerations, filled icosahedral levels for I = 5 could occur for various numbers of electrons, such as 56, 60, 62, or 66. However, when the energetics of the problem are also considered (see Fig. 12.1 for the Huckel calculation), only the cases of 60 electrons and perhaps 66 electrons correspond to actual ground states for icosahedral fullerenes.
Another view of the many-electron ground state problem is provided by Table 4.11, which considers the filling of states for all icosahedral fullerenes up to C980, with the full electronic configurations given for the spherical model approximation, as discussed in §12.2. Using Hund's rule, the quantum numbers for the ground state for each electronic configuration are found, and the symmetries of the resulting multiplet levels in icosahedral symmetry are given in the right-hand column. It is seen that only a few of the icosahedral fullerenes correspond to nondegenerate states, namely C60, C180, and C420, while the others listed in Table 4.11 would appear to correspond to degenerate ground states. For example, the ground state J value for icosahedral Qo is 4, which yields a ninefold degenerate multiplet in full rotational symmetry which in icosahedral symmetry splits into a Gg(4) and a HJ5) level. In the case of a degenerate ground state, a Jahn-Teller dis tortion [12.1,5] to fullerenes with lower symmetry is expected, resulting in a further splitting of the ground state level and a consequent lowering of the energy of the system. The advantage of using a spherical approximation for discussing the ground states of higher fullerenes is the simplicity of the approach, which enables qualitative arguments to be presented for complex systems, such as the large fullerenes shown in Table 4.11.
Before discussing the excited states for the neutral C60 molecule explicitly, we next consider the more general issue of the ground states and excited states for the related free molecular ions C^f.
The electronic states for C60 anions and cations are important for a variety of experiments in the gas-phase, in solutions, and in the solid state. For example, experiments in the gas-phase relating to the addition of electrons by the interaction of C60 with electron beams lead to negatively charged C60 molecules. As another example, experiments on C60 in electrolytic solutions could, for example, involve electrochemical processes such as voltammetry experiments, where charge is transferred (see §10.3.2) or optical studies, whereby electrons and holes are simultaneously introduced as a result of photon excitation (see §13.1). Since solid C60 is a highly molecular solid, the electronic states of the molecule [described by the intramolecular Hamilto-nian $?intra of Eq. (12.1)] form the basis for developing the electronic structure for both solid C60 and doped Qq, for which intermolecular interactions Winter are a'so of importance. In the limit that $fintra » ^¡nter, the molecular approach is valid, while for cases where $fintra ^ <^nter> a band approach is necessary. Thus for alkali metal M3C60 compounds, which exhibit superconductivity (see §15.1), band theory is expected to be applicable. Whether band theory with correlated electron-electron interactions or a molecular solid with weakly interacting molecules is the most convergent approach for specific situations is further discussed in §12.7.
In qualitative discussions of physical phenomena, such as the filling of states by doping, use is made of the electronic states of the molecular ions C£g. For example, referring to Fig. 12.1, where the level ordering of C60 is presented, it is often argued that the maximum conductivity is expected when the LUMO tlu (/,„) level is half filled with electrons at the stoi-chiometry MjC«, and that an insulator is again achieved when the tlu level is completely filled by six electrons at the stoichiometry M6C60, assuming that one electron is transferred to each C60 molecule per alkali metal atom uptake. These simple arguments are, in fact, substantiated experimentally. An extension of this argument would imply that 12 electrons would fill both the tlu and tlg levels, following the level ordering of Fig. 12.1. The evidence regarding the filling of the tlg levels in accordance with these simple ideas has been presented by experiments on alkaline earth doping of C6() with Ca, Ba, and Sr [12.27-30] as well as by theoretical studies [12.31]. These simple level-filling concepts have also been used in a more limited way to account for the alkali metal doping of Oj0.
12.4.1. Ground States for Free Ions C^
More quantitative information about the electronic states for the C^f ions can be found first by considering the Pauli-allowed states associated with these ions, for both their ground states and excited states. The ground state configurations for the various C^ ions can be specified by Hund's rule starting from a spherical model, and the results are summarized in Table 12.2 for CnM anions (-6 < n < 0) and in Table 12.3 for Q0+ cations (0 < n < 5). On the left-hand side of these tables, the angular momentum values for S,L, and J from Hund's rule are given, assuming spherical symmetry for the ions, — 6 < n < +5. In going from the spherical ground state approximation, specified by J in spherical symmetry, to icosa-hedral Ih symmetry, level splittings occur, as indicated on the right-hand side of Tables 12.2 and 12.3. Here the electronic orbital configuration for each ion is listed in Ih symmetry, together with levels of the ground state Pauli-allowed multiplet of this configuration in Ih symmetry, and the hyperfine structure for each level of the multiplet. Since the spin-orbit interaction in carbon is small (see Table 3.1), the L ■ S coupling scheme is valid for fullerene molecules. This has an important bearing on the optical transitions which are spin-conserving (AS — 0), as described in §13.1.3.
Referring to Table 12.2, we see that neutral C«, has a nondegenerate Ag ground state, and so does C^. However, the other ions have degenerate ground states. For example, C^ corresponding to the half-filled tlu (/lu) level (mentioned above) has a Hund's rule quartet 4AU ground state. We would therefore expect a Jahn-Teller symmetry-lowering distortion to occur for Cj?0 [12.32], as discussed further in §13.4.1. Likewise, Jahn-Teller symmetry-lowering distortions would be expected for the other degenerate C£f ground states listed in Tables 12.2 and 12.3 [12.5]. The effect of strong interactions between electrons and the intramolecular vibrations has been shown to lead to an electron pairing for the anions [12.33].
12.4.2. Excited States for Negative Molecular Ions Q0~
In general, the ground states discussed in §12.4.1 are the lowest-energy states of a ground state multiplet. In this section we discuss the excited
Various Pauli-allowed states associated with the ground state configurations for the icosahedral CJ^ anions.
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