Icosahedric fullerenes

Hirsch and co-workers calculated NICS values for tetrahedral clusters of N, P, As, Sb, and Bi, as well as for the corresponding tetra-anions composed of Si, Ge, Sn, or Pb atoms, finding diatropic values for 2,n(n 1) jr-systems.296a b It was postulated by Hirsch, Schleyer, and their co-workers that for icosahedral fullerenes and their hetero-analogues the Hiickel rule, involving 4/2+2 //-electrons, should be replaced by the 2(/2+l)2 electron rule.296... 

M. Terrones, G. Terrones, H. Terrones, Structure, chirality, and formation of giant icosahedral fullerenes and spherical graphitic onions, Struct. Chem., vol. 13, pp. 373-384, 2002. 

The 1996 Nobel Prize in chemistry was awarded to three American scientists for their identification of a new allotrope of aromatic carbon molecules called fullerenes. These unusual carbon molecules form a closed-cage structure of joined carbon atoms. The original soccer ball-shaped carbon molecule called buckminsterfullerene contained 60 carbon atoms and was nicknamed Bucky Ball in honor of Buckminster Fuller (1859—1983), who used similar shapes in some of his architectural structures. Since then additional organic pentagon structures beyond the original icosahedral fullerene (C ) have been developed, all with an even... 

Table 14.1 Electron ground-state configurations of the fully (bold) and partially filled K shells of icosahedral fullerenes.

It was discovered, however, that the spherical aromaticity of the icosahedral fullerenes C20, Cjq and CgQ depends on the filling of the Jt-sheUs with electrons [107]. As pointed out in Section 14.3.1 no distortion of the cage structure is expected in these fullerenes if their shells are fully filled. Closed-shell situations are realized if the fullerene contains 2(N -1-1) Jt electrons. This is closely related to the stable noble-gas configuration of atoms or atomic ions [108]. In this case the electron distribution is spherical and all angular momenta are symmetrically distributed. Correlation of the aromatic character determined by the magnetic properties is shown in Table 14.3. 

N = 4), and Cgg (M = 5) exhibit very pronounced He chemical shifts in the center of the cage. Both the pentagons and hexagons are very paratropic in all cases. Therefore, icosahedral fullerenes that contain 2(N + Vf ji-electrons show the maximum degree of spherical aromaticity. Exceptions are possible if levels of the most outer shell are occupied before aU inner shells are completely filled. This is the case for Here, the tj levels of... 

One key conclusion is that the entire molecule must be taken into account to understand the aromatic properties of icosahedral fullerenes. The 2(N -1-1) rule of spherical aromaticity also sufficiently describes the magnetic behavior of non-icosahedral fullerenes [128], homoaromatic cage molecules [129], and inorganic cage molecules [130],... 

Icosahedral fullerenes that contain 2(N + 1) re-electrons are closed shell systems and are spherically aromatic. 

In Virology, the number t(k, l) = k2 + kl + l2 (used for icosahedral fullerenes) is called triangulation number. In terms of Buckminster Fuller, the number k 4-1 is called frequency, the case l = 0 is called Alternate, and the case l = k is called... 

The case b = 6 is the classical fullerene case. Theorem 2.2.2 gives that all 7(5,6)-fulleroids, i.e. fullerenes of icosahedral symmetry, are of the form GCk,i(Dodecahedron). See on Figure 19.1 the first three of the following smallest icosahedral fullerenes besides Dodecahedron ... 

We give an estimation of the carbon fraction locked in these molecules. We discuss the rotation rates and electric dipole emission of hydrogenated icosahedral fullerenes in various phases of the interstellar medium. These molecules could be the carriers of the anomalous microwave emission detected by Watson et al. (Astrophys. J. 624 L89,2005) in the Perseus molecular complex and Cassasus et al. (2006) in the dark cloud LDN 1622. Hydrogenated forms of fullerenes may account for the dust-correlated microwave emission detected in our Galaxy by Cosmic Microwave Background experiments. 

In Fig. 1.2 we plot the resulting transitions up to 8 eV for icosahedral fullerenes with total number of carbon atoms N < 1,500. In each subpanel we present the theoretical cross-sections for the bands convolved with a Lorentzian function of width ra (the parameter in Eq. 1.2 of (Iglesias-Groth et al. 2002)). Results are plotted for two values of the r parameter 0.01 eV (set of narrow transitions) and 0.3 eV (broad features). 

We will consider here the various families of fulleranes CN H that can be formed from icosahedral fullerenes with N = 60, 180, 240, 540, 960, 1,500, and 2,160 carbon atoms. Buckyonions can also be hydrogenated. We will consider here only... 

Lateral view of three kinds of carbon nanotubes with end caps (a) armchair (5,5) capped by one-half of (T.. (b) zigzag (9,0) capped by one-half of Cgo, and (c) an enantiomorphic pair of chiral SWNTs each capped by a hemisphere of icosahedral fullerene C140... 

For icosahedral symmetry, the whole net is completely specified by one lattice vector, i.e, by an ordered pair of integers a, ft). The atom count of the fullerene is proportional to the area of the planar net, and so it is easily shown that at least one icosahedral fullerene C exists for... 

Figure 1. The Goldberg/Coxeter construction of an icosahedral triangulation of the sphere. Twenty copies of the large equilateral triangle fit together to form the net of a master icosahedron, which yields an icosahedral fullerene on taking the dual. The Coxeter parameters in this particular example are a = 3 and b= i.e., to reach B from A, take 3 steps along the horizontal to the right, turn left through 60°, and take 1 step.

The possibility of a general rule connecting electronic and geometric structure for fullerenes first emerged from Hiickel calculations on the icosahedral series described earlier, i.e., where C with n = (fi ij + f). The electronic configurations of the icosahedral fullerenes follow a remarkably simple rule ... 

Thus, Ih Ceo is the first of an infinite magic-number series of neutrals and / , C20 the first of a series with closed-shell dications but open-shell neutrals. Inspection of the two series shows a general geometrical relationship in that any open-shell icosahedral fullerene C with n = 60k + 20 = 20(3k + 1) canbe converted into a larger icosahedral fullerene C3 with 3n = 60(3k + 1) by a specific transformation, and all icosahedral closed-shell neutrals are so produced. The conversion operation is the so-called... 

Figure 1. MNDO optimized structure of the icosahedral fullerene. The view along a C3 axis emphasizes the localization of curvature at the pentagons.

In this work, we have applied the renormalized monopole-dipole model (RMD) [1] to spherical and icosahedral fullerenes, with the aim to investigate in more detail local effects, i.e. the local electric field and the influence of Stone-Wales defects on the local field in case of spherical fullerenes. A continuous model for a metallic sphere is developed for comparison. 

The static polarizabiliy of icosahedral fullerenes from Cgo to C2160 has been characterized using density functional theory in combination with both unscreened (uncoupled) and screened (coupled) approaches. In particular, a simple expression has been employed to estimate screened RPA values from the unscreened quantities. This expression uses an effective radius, which is determined from comparing the polarizability... 

In 2000, Hirsch s 2( +1) rule of aromaticity for spherical compounds [18,46,47] was introduced as the spherical analog of Hiickers 4n + 2 mle. Hirsch s mle is based on the fact that the Ji-electron system of an icosahedral fullerene can be, in a first approximation, considered as a spherical electron gas surrounding the surface of a sphere. The corresponding wave functions of this electron gas are characterized by file angular momentum quantum number 1(1 = 0,1,2,...), with each energy level 2 Z + 1 times degenerated, and thus all jt-sheUs are completely filled when we have 2 ( + 1) electrons. For such reason, spherical species with 2(n +1) Ji-electrons are aromatic, like icosahedral 20, or Cgo ". ... 

Hirsch A, Chen Z, Jiao H (2000) Spherical aromaticity in icosahedral fullerenes the 2(N + 1) rule. Angew Chem Int Ed 39 3915-3917... 

Dunlap, B. L, Zope, R. R. (2006). Efficient quantum-chemical geometry optimization and the structure of large icosahedral fullerenes. Chemical Physics Letters, 422, 451. 

Dunlap, B. L, Brenner, D. W., Mintmire, J. W., Mowrey, R. C., White, C. T. (1991). Local density functional electronic structures of three stable icosahedral fullerenes. Journal of Physical Chemistry, 95, 8737. 

Xu, C. H., Scuseria, G. E. (1996). An 0(N) tight-binding study of carbon clusters up to C 40-The geometrical shape of the giant icosahedral fullerenes. Chemical Physics Letters, 262, 219. 

Zope, R. R., Baruah, T., Pederson, M. R., 8c Dunlap, B. I. (2008). Static dielectric response of icosahedral fullerenes from Geo to C2160 characterized by an all-electron density functional theory. Physical Review B, 77,115452. 

As in the Hiickel model, only the connection between the atoms is taken into account, the 4n and 4n + 2 rules remain valid in the case of rings where s electrons are itinerant. For example, Lij has a perfect symmetry and two itinerant s electrons. Hence it is expected to be aromatic. The Hiickel model can also be applied to three-dimensional systems, such as fullerenes [4], where electron count rules can also be formulated. In the case of icosahedral fullerenes, Hirsch et al. have formulated such a rule [5], but other electron counting rules also exist [6]. As in fullerenes the itinerant electrons are delocalized over a nearly spherical surface, these rules account for spherical aromaticity [7]. 

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