Point groups icosahedral

At higher frequencies (above 200 cm ) the vibrational spectra for fullerenes and their cry.stalline solids are dominated by the intramolecular modes. Because of the high symmetry of the Cgo molecule (icosahedral point group Ih), there are only 46 distinct molecular mode frequencies corresponding to the 180 6 = 174 degrees of freedom for the isolated Cgo molecule, and of these only 4 are infrared-active (all with Ti symmetry) and 10 are Raman-active (2 with Ag symmetry and 8 with Hg symmetry). The remaining 32 eigcnfrequencies correspond to silent modes, i.e., they are not optically active in first order. 

In a famous paper by Shechtman et al. (1984) electron diffraction patterns were shown of rapidly quenched and solidified aluminium-manganese alloys. Sharp diffraction peaks, suggesting long-range translational order, were observed with the presence however of five-fold symmetry (that is of a non-crystallographic symmetry see 3.6.1.1). By different orientation of the specimen five-fold axes (in 6 directions), three-fold axes (in 10 directions) and two-fold axes (in 15 directions) were identified with the subsequent observation of the existence also of an inver-sion centre the assignment of this phase to the icosahedral point group, m36, was defined. 

The large fourfold and fivefold degenerate shells of the icosahedral point group embody internal symmetries that are immensely richer than the ones we became acquainted with while studying the Chromium doublets. 

Molecules with icosahedral symmetry are not new but the discovery of the newest of them, C6o or buckminsterfullerene, has had such a profound effect on chemistry in recent years that I thought it useful to include a discussion of the icosahedral point group to which C60 belongs. 

Objects which have C or S axes with n> 1 have degenerate vibrations, i.e., two (or more, for cubic or icosahedral point groups or linear molecules) vibrations have the same frequency. Carrying out a symmetry operation leads to a linear combination of degenerate vibrations. 

Vibrations of totally symmetric species (defined by the first row of the character tables) emit Raman lines which are polarized, the depolarization ratio pk can assume, according to Eqs. 2.4-11. .. 13 values of 0 < p < 6/7. All other Raman-active vibrations are emitting lines which are depolarized, they have a depolarization ratio of 6/7. The value 6/7 is appropriate for an arrangement, where the Raman radiation s investigated without an analyzer. If an analyzer is used 3/4 has to be taken instead. Cubic and icosahedral point groups are a special case the depolarization ratio for totally symmetric vibrations is 0. 

Extra considerations are required to construct suitable sets of polynomials, which provide basis functions for the irreducible subspaces of the cubic and icosahedral point groups. Clearly, such a set of central functions is invariant under the point group G. Eor such a function, f, then (fgi, fg2,. .., fgn> is a subspace of the central functions invariant under G. But it is not, in general, an irreducible subspace, i.e. it may contain further subspaces that transform according to different irreducible representations. 

In the case of the icosahedral point groups, Ih and I, Table 3.10, the analysis is more complicated and there is a need to identify the combinations of the spherical harmonics, which will generate higher dimensional irreducibile subspaces. For example, at level 3, there are 7 harmonics, but the irreducible subspaces in icosahedral symmetry are four-fold [Gu] and three-fold [T2u]. It is found that three of the original functions can be carried over to provide basis functions in icosahedral, symmetry but that four distinct linear combinations of... 

The fullerene, Cso, exhibits a permutation character over its 80 vertices, which, as set out in Table 3.18, is the sum of the reducible permutation characters [Table 3.4] on the vertices of the O20 and Oeo orbits of the icosahedral point group. 

Table 3.18 The direct sum components of the permutation character over the vertices of the Cgo fullerene, Table 3.19, formed by summation of the direct sums for the O20 and Oso orhits of the icosahedral point group Ij, Table 3.4.

A case in point is the pentagonal subgroup Dsd of the icosahedral point group. This subgroup is a maximal subgroup, and the six pentagonal directions are equidistant , in the sense that any pair of them can be mapped onto any other pair. Induction then yields the five-fold degenerate H representation ... 

Icosahedral and decagonal quasicrystals of intermetallic compounds are multiple twins of cubic or orthorhombic crystals composed of very large atomic complexes with icosahedral point-group symmetry in cubic close packing Structure of decagonal AlePd. Proc. Natl. Acad. Sci. 86 (1989) 9637-9641. 

Fig. 3.5 Genealogical tree for the cubic and icosahedral point groups...

Fig. 3.9c) and [Fe(CN)6]. There is no centre of symmetry in a tetrahedron but there is one in an octahedron, and this distinction has consequences with regard to the observed electronic spectra of tetrahedral and octahedral metal complexes (see Section 20.7). Members of the icosahedral point group are uncommon, e.g. [B]2Hj2] (Fig. 3.9d). 

Molecules in the tetrahedral, octahedral, and icosahedral point groups behave like spherical electron distributions in this respect the induced dipole moment is independent of the molecule s orientation in the electric field. Most molecules, however, are more easily polarized along one axis than another. The polarizability in this case is actually represented by a matrix called the polarizability tensor, with elements that describe the polarizability along the molecule s principal inertial axes. 

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