Kubic harmonics

For sites of cubic symmetry the point-group symmetry elements mix the spherical harmonic basis functions. As a result, linear combinations of spherical harmonic functions, referred to as Kubic harmonics (Von der Lage and Bethe 1947), must be used. 

All of these hexafluorides are dimorphic, with a high-temperature, cubic form and an orthorhombic form, stable below the transition temperature (92). The cubic form corresponds to a body-centered arrangement of the spherical units, with very high thermal disorder of the molecules in the lattice, leading to a better approximation to a sphere. Recently, the structures of the cubic forms of molybdenum (93) and tungsten (94) hexafluorides have been studied using neutron powder data, with the profile-refinement method and Kubic Harmonic analysis. In both compounds the fluorine density is nonuniformly distributed in a spherical shell of radius equal to the M—F distance. Thus, rotation is not completely free, and there is some preferential orientation of fluorine atoms along the axial directions. The M—F distances are the same as in the gas phase and in the orthorhombic form. 

To generate an irreducible G subspace, for particular cases, f needs to be chosen with care. In the case of the kubic harmonics, first defined by Bethe in 1929 suitable functions are the mononomials x y"zP, which we identify in Elert s notation as (mnp). The kubic harmonics up to level 4 and their maps onto the irreducible representations of the cubic groups are listed in Table 3.9. 

Table 3.9 Classification of the kubic spherical harmonics np to angnlar momentnm level 4 by descent in symmetry, into their irredncible components for the molecniar point gronps Oh, O, Tj, Tjj and T.

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. 

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. 

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