Icosahedron symmetry factoring

Figure 13 Symmetry factoring of a regular icosahedron using a C2 x x C2 x C3 scheme...

Oi, with 48 operations rather than Ih with 120 operations. The identical symmetry factoring scheme used above for the icosahedron, namely C2 x C2 x C2 x C3, can also be used for the cuboctahedron but the pattern of degeneracies is different. The spectrum of the cuboctahedron is found to have four distinct eigenvalues like that of the icosahedron, namely -t-4, -1-2, 0, and —2 for the cuboctahedron of multiplicities 1, 3, 3, and... 

From the above discussion, it is evident that mixed cluster ions of the type Ar M exhibit strong magic numbers at values of (n + m) = 13, 19, 55, 71, and 147 in a variety of different studies. These values correspond to the completion of the first, second, and third icosahedral shells occurring at 13, 55, and 147 whereas 19 and 71 correspond to especially stable subshells formed by interpenetrating double icosahedron structures. The size and symmetry of the dopant moiety appear to be the most important factors in observing magic numbers that can be rationalized on the basis of icosahedral-like structures. The inability to observe magic numbers has been attributed to the distortion of the icosahedral structure due to size and steric factors associated with the dopant ion which destroys the delicate balance between the monomer interactions. One of the issues that has been interpreted differently involves the location of the dopant atomic/molecular... 

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

In their formulation of the superposition method Hoare and Mclnnes omitted the effect of symmetry [132]. This omission explains why they predicted that for LJ the cluster would reside in the icosahedral global minimum up to a reduced temperature of 0.6. In fact, LJ13 melts at r 0.29. Without the inclusion of symmetry, the density of states of the icosahedron is overestimated by a factor of 120. 

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