Background

The (time-dependent or time-independent) Schrödinger equation for a highly symmetric potential energy surface can be solved by expanding the angular dependence of a quantum-mechanical wavefunction in symmetry adapted spherical harmonics (SASHs). While the generation of the required SASHs is trivial for simple symmetries, special techniques have been developed for the point groups of highest symmetry. Our work gives complete sets of “seed functions” to obtain SASHs for all irreducible representations of the octahedral and icosahedral point groups recursively. Basically, the technique consists of two steps:

1. A wavepacket in three-dimensional space is described as a sum of products of time-dependent radial wavefunctions and spherical harmonic functions. This approach yields a set of coupled equations for the radial functions where both the numerical effort and the storage requirement depend quadratically on the number of spherical harmonics used in the expansion scheme. This approach is known in the literature as close coupled wavepackets (CCWP).

2. In order to reduce this number, symmetry adapted spherical harmonics (SASHs) are used which transform according to an irreducible representation of a point group. Note that the usual technique employing projection operators (based on the great orthogonality theorem of group theory) causes difficulties when applied to cubic point groups and to spherical harmonics for high azimuthal quantum numbers. Instabilities of the conventional scheme based on Wigner rotation matrices [1] encountered for high angular momentum states can be overcome by a numerical scheme based on Gaussian quadratures [2]. An elegant alternative can be found in an iterative scheme [3] where low order SASHs ("seed functions") can be used to recursively generate all higher order SASHs with very little computational effort.

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Tables of "seed functions" are available for the octahedral and icosahedral point groups. The tables given in the following are complete in the sense that all SASHs of all irreducible representations of the two point groups under consideration can be recursively generated from the seed functions.

1.  Octahedral point group Oh

2.  Icosahedral point group Ih

Acknowledgments 

 It is a great pleasure to thank F. Neugebaur (Humboldt University, Berlin) and P. Zdanska (Heyrovsky Institute, Prague) for their invaluable help in writing computer codes and generating the above tables of  SASHs. Furthermore, Ch. Salzmann's help in preparing the figures is highly appreciated.

References

1. C. J. Bradley and A. P. Cracknell
   The Mathematical Theory of Symmetry in Solids
   Clarendon, Oxford (1972)

2. G. W. Fernando, M. Weinert, R. E. Watson, and J. W. Davenport
   J. Comp. Phys. 112, 282 (1994)
   DOI:10.1006/jcph.1994.1100

3. W. Prandl, P. Schiebel, and K. Wulf
   Acta Cryst. A52, 171 (1996)
   DOI:10.1107/S0108767395012578