Octahedral Symmetry

The lowest-order combinations of spherical harmonics that are compatible with the octahedral symmetry of the related surfaces in Eq. (3) correspond to $\lambda=4,6,8$ and 10. Again, within each multipolarity there is only one degree of freedom (one single parameter) that determines the allowed combinations of the allowed spherical harmonics. We denote those parameters, treated as independent, by $o_4$, $o_6$, and $o_8$. The three lowest-order solutions are:

$$\alpha_{4,0}\equiv o_4 \quad \textrm{and} \quad \alpha_{4,\pm4}\equiv -\sqrt{\frac{5}{14}}\,o_4$$ (14)

for $\lambda=4$,

$$\alpha_{6,0}\equiv o_6 \quad \textrm{and} \quad \alpha_{6,\pm4}\equiv +\sqrt{\frac{7}{2}}\,o_6$$ (15)

for $\lambda=6$, and

$$\alpha_{8,0}\equiv o_8, \quad \alpha_{8,\pm4}\equiv -\sqr... ...nd} \quad \alpha_{8,\pm8}\equiv +\sqrt{\frac{65}{198}}\,o_8$$ (16)

for $\lambda=8$.

In the following we are going to limit ourselves to the lowest-order degrees of freedom.