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"The average diameters given in this table for the icosahedral fullerenes are calculated from Eq. (3.5) and ac_c is taken to be 1.44 A. Other calculations, using different values for ac_c and different definitions for dn yield slightly different values for d, (see Table 3.1).

"The average diameters given in this table for the icosahedral fullerenes are calculated from Eq. (3.5) and ac_c is taken to be 1.44 A. Other calculations, using different values for ac_c and different definitions for dn yield slightly different values for d, (see Table 3.1).

(n,m) = (2,1), (3,1), (1,3), and (4,4) corresponding to C140, C260, C260, and C960. Fullerenes with n = m or with either n = 0 or m — 0 have full Ih symmetry (see Chapter 4) and were recognized at an early stage of fullerene research [3.59]. The icosahedral fullerenes for which n m, and neither n nor m vanishes, have / symmetry but lack inversion symmetry. In this case, fullerenes with indices such as (1,3) and (3,1) are mirror images of each other, as shown in Fig. 3.6 [3.58].

For a given number of carbon atoms (such as nc = 80), there may be more than one way to arrange the nc carbon atoms into a fullerene. Each of these possibilities is called an isomer. In fact, for C80 there are seven

Fig. 3.6. Icosahedral fullerenes with (n, m) = (2,1), (3,1), (1,3), and (4,4) corresponding to C140, C260, CMI, and C960. The (3,1) fullerene is below the (1,3) fullerene in the diagram [3.58].

distinct arrangements (two of which are shown in Fig. 3.4) of the 80 carbon atoms which conform to the isolated pentagon rule and have only pentagons and hexagons. In addition to the two isomers for C80 with Ih and DSd point group symmetries that were discussed above (see §4.4), there are five other isomers for C80 which obey the isolated pentagon rule, each having a different symmetry group: D2, C2v, D3, C2v, Dsd [3.60]. The distinct bond lengths for each of these isomers and their symmetries have been calculated [3.49]. Listed in Table 3.5 is the number of isomers n, for a given C„c obeying the isolated pentagon rule (out to nc =■ 88). This table also lists the symmetries for the isomers following a notation used in the literature [3.60], As another example of fullerene isomers, C78 has been shown to have five distinct isomers obeying the isolated pentagon rule, one with Z)3 symmetry, two with C2v symmetry, and two with D3h symmetry [3.61] (see Table 3.5). Figure 3.7 shows examples of isomers for C84 with six different point group symmetries [3.21,62]. The C84 molecule has a total of 24 isomers that obey the isolated pentagon rule.

Isomers giving rise to right-handed and left-handed optical activity are expected to occur in molecules belonging to group / (rather than Ih) and

Table 3.5

Isomers of common fullerene molecules [3.60],

Table 3.5

Isomers of common fullerene molecules [3.60],

C«c

«/'

Symmetries

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