The icosahedral harmonics

Table 3.10 Classification of the icosahedral harmonics up to angular momentum level 4 by descent in symmetry, into their irreducible components for the molecular point groups Ij, and I.

While the appropriate linear combinations of the spherical harmonics known as the kubic harmonics, which transform in cubic symmetry, have been known for many years from the seminal works of Bethe and van Vleck, we believe that the icosahedral harmonics available in the basis functions lists in the files Ih.xls and I.xls on the CDROM have not been identified, in this form, in the literature. Thus, for the record, this appendix contains the complete list of such harmonic functions, classified as basis functions for the irreducible representations of the groups I and Ih. 

Table Al.l The icosahedral harmonics, fashioned to be basis functions for all the irreducible representations of the regular orbit cage of Ih point symmetry. The polynomial functions are written in Elert s notation, as described in Chapter 1 and their irreducible properties under Ih and I are identified, in columns 1 and 2 of the table, using Mulliken symbols.

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... 

It is instructive to include here several examples involving the use of the general, kuMc and icosahedral harmonics and the theorems to establish the methodology of our group theory. These examples illustrate ... 

In this Appendix, Htickel theory calculations are used to demonstrate that the polynomials of Appendix 1 can be applied as basis functions for the irreducible subspaces of a Hamiltonian invariant under icosahedral point symmetry, while extended Htickel theory calculations on cubium cages of cubic point symmetry are used to demonstrate the same result for the kubic harmonics, since single bond-length regular orbits are not possible in all cases. 

T ux,Tiuy,Tiu transform as the x, y, z)-axes themselves or, for illustration purposes, a set iPx,Py, Pz) of /t-orbitals. Actually, as the Tiu orbitals form a basis for the spherical harmonics with angular momentum L = 1, this basis can be used whatever orientation is chosen. However, the same cannot be said for the HOMO orbitals as the L = 5 harmonics decompose as Tiu 0 72m 0 77 in icosahedral symmetry. 

Basic rule Standard basis functions for irreducible representations that occur in antisymmetrized (skew) direct products are chosen out of the spherical harmonics with I odd and the remaining ones out of those with I even. This rule is immediately applicable to 2 3 and without difficulty to all non-abelian point groups except the icosahedral group which, however, is not a simply reducible group" ). 

Griffith has presented the subduction of spherical JM) states to point-group canonical bases for the case of the octahedral group. Similar tables for subduction to the icosahedral canonical basis have been published by Qiu and Ceulemans [8]. Extensive tables of bases in terms of spherical harmonics for several branching schemes are also provided by Butler [9]. 

Figure 22 Construction of the icosahedral capsid of poliovirus using a low-order spherical harmonic representation of the viral protomers (one copy each of the proteins VPl, VP2, VP3, and VP4). " The individual boundaries of the four protein chains are texture mapped onto the surface. One pentameric assembly intermediate has been translated away from the capsid along a five-fold axis. The model was constructed, and can be manipulated interactively in real time using the symmetry server library developed in the Olson group, within the AVS data-flow visualization environment. (Image courtesy of Arthur J. Olson, The Scripps Research Institute, La Jolla, CA)...

A relationship actually exists between periodic and quasiperiodic patterns such that any quasilattice may be formed from a periodic lattice in some higher dimension (Cahn, 2001). The points that are projected to the physical three-dimensional space are usually selected by cutting out a slice from the higher-dimensional lattice. Therefore, this method of constmcting a quasiperiodic lattice is known as the cut-and-project method. In fact, the pattern for any three-dimensional quasilattice (e.g., icosahedral symmetry) can be obtained by a suitable projection of points from some six-dimensional periodic space lattice into a three-dimensional subspace. The idea is to project part of the lattice points of the higher-dimensional lattice to three-dimensional space, choosing the projection such that one preserves the rotational symmetry. The set of points so obtained are called a Meyer set after French mathematician Yves Meyer (b. 1939), who first studied cut-and-project sets systematically in harmonic analysis (Lalena, 2006). 

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... 

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