Irreducible tensor (second or higher rank)

[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.

Thanks in advance!

 

[azntoon]

To determine selection rules for matrix elements of electric quadrupole moment tensor, Qij, for a particular point group, such as O or D3d, one needs to know the irreducible components of Qij. The irreducible components are found by using the group theory techniques associated with the particular point group. For example, if the point group is O, then the irreducible components of Qij can be determined by using the addition of angular momentum principle, which states that any irreducible representation of a given point group is the direct sum of all possible combinations of the irreducible representations of the subgroups that make up the point group. For O, this requires considering the subgroups C2v and C3v.In the case of D3d, the irreducible components can be determined by using the character table of the group and the Wigner-Eckart theorem. The Wigner-Eckart theorem states that the matrix element of an operator between two states belonging to different irreducible representations of a symmetry group is proportional to the product of the Clebsch-Gordan coefficient and the reduced matrix element of the operator in the same irreducible representation. This can be used to find the selection rules for the matrix elements of Qij for the D3d point group.In summary, the selection rules for matrix elements of electric quadrupole moment tensor, Qij, for a particular point group, such as O or D3d, can be found using the techniques of group theory specific to that point group.