Symmetries of central functions with arbitrarily high angular momentum

4 Symmetries of Central Functions with Arbitrarily High Angular Momentum 

Spherical harmonic functions are important in many problems in Chemistry and Physics. Spherical harmonic functions are central in discussions of rotation, motion in a central potential, multipole expansion, cluster bonding, spherical wave expansions and many more topics. The calculation of symmetrized powers of representations give a way of obtaining the 

In view of the importance of spherical harmonics, an alternative calculator facility is provided, allowing explicit calculation up to T (60) and, indirectly, calculation to any value of J for all point groups of practical interest. The alternative method is based on the multiplication property 

Extension to arbitrary J follows from the fact that, for any finite point group, there is an integer K such that r(7 - - A ) — r(7) is a constant (reducible) character depending (at most) on the value of J mod K. The constant term is either the regular character of the group, or, for centrosymmetric groups, is alternately the g and u halves of the regular character. 

For example, in Civ, r(7) obeys a pattern that repeats every K = 3 steps, i.e. 

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