# Irreducible tensor (second or higher rank)

1. Oct 31, 2008

### ala

When one want to find selection rules for matrix element of (for example) electric quadrupole moment tensor Qij, irreducible components of Qij are needed to apply Wigner-Eckart theorem. When symmetry group is SO(3) irreducible component can be found using what we know from addition of angular momentum. But how to do same problem if symmetry group is O (the octahedron group) or some other point group?

For example in one book I found (without explanation) following results:
1. for O group: Qxy, Qzz, Qyz are transformed by the representation F2; Qxx+kQyy+k^2Qzz, Qxx+k^2Qyy+kQzz are transformed by the representation E, where k=exp(i2Pi/3)
2. for D3d group: Qzz by A1q; Qxx-Qyy,Qxy by Eg; Qxz,Qyz by Eg.
Standard notation for irreducible representations for group O and D3d are being used.