A Crystal field when inversion is absent

[ftft]

I find in many textbooks that when expanding the crystal field in spherical harmonics, those terms of odd ranks do not appear in the expansion even if the crystal lacks an inversion centre such as the tetrahedral crystal field. Why is that? and when one should include spherical harmonics of odd ranks?

 

[DrDu]

I would expect that in a tetrahedral field, there are no odd terms because of the high remaining symmetry of the tetrahedron. In mathematical language: inversion symmetry is sufficient but not necessary for the absence of odd terms.
In general the crystal field should transform as a totally symmetric representation of the symmetry group. As the symmetry group of the crystal is a subgroup of the total rotational symmetry group SO(3) of which the spherical harmonics span irreducible representations, you can look up how these irreducible representations split up when going to the crystal field subgroup, e.g. using the character table.

 

[ftft]

Does it mean that when the crystal lacks any symmetry at all, i.e. when it is classified as C1, all terms in the expansion are important?

 

[DrDu]

Yes

 

[ftft]

Thank you DrDu