Basis function cubic harmonic

In most cases of the use of standard basis functions (cubic harmonic Cartesians), the basis functions on any atom will have much higher symmetry (typically Td) than the molecular point group therefore it will be the rule rather than the exception for symmetry operations to induce such linear combinations. 

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. 

We shall examine the behaviour of these functions under various symmetry operations. We need only consider the behaviour of functions centred at the origin, as functions centred elsewhere display the same behaviour together with a possible translation of the centre that is easily determined. Cubic groups are excluded as the spherical harmonics are less well suited as basis functions for these cases. 

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. 

The use of the Cartesian Monomials for the 6 d and 10 / basis functions is not always appropriate it is often preferable to work explicitly in terms of the cubic harmonic (5) d functions or (7) / functions. 

Let us assume then that we have a given raw basis and wish to transform it into a working basis which is in the form of an orthogonalised, symmetry-adapted linear combination of cubic harmonic functions. The steps are clear ... 

The parity of D > may be deduced from the fact that the d-orbitals, which are basis functions for D > contain the spherical harmonics which do not change sign upon reflection in the origin. Therefore the representation D ) is of even parity. This property is carried over into the cubic group so that we may now say that D > has been reduced to the representations eg and t2g in the group Oh, where the suffix g indicates even parity. 

It was demonstrated by von der Lage and Bethe [2] that the irreducible representation for the p states (Z - 1) in the cubic field is the three-dimensional representation 6, which corresponds to the cubic harmonics of the x, y, and z type. If Z = 2, the irreducible representations are a two-dimensional representation y with the basis functions z - r and 3i - y, and a three-dimensional representation e whose basis functions are of the type yz, and xz. 

---