Question

I have read so far that it is about whether the d-Orbital is symmetric to a C2 element perpendicular to its main rotational axis. If all the given orbitals in a group are symmetric to that element, you assign a 1 towards it, otherwise a 2. Anyway, I still don't get a few things:

First: Is it enough for one orbital of a given group to be not symmetric concerning that C2 element, for the whole group being asymmetric towards C2?

Second: I cannot get my head around e1 and e2 in the crystal field splitting of linear complexes. Do I have to keep the C2 axis the same to check whether it is e1 or e2? Because if I change the axis I get e1 twice, if I don't change it, I get e2 twice. My textbook says it is e1 for d(xz)/d(yz) and e2 for d(x2-y2)/d(xy)...

Answer 1

In crystal field theory, if any orbital within a set is asymmetric with respect to a C2 axis, the whole set is considered asymmetric. The e1 and e2 labels for d orbitals depend on their symmetry properties and do not change with different orientations of the axis. Crystal field splitting energy affects the distribution of energies within the d orbitals without changing their total energy.

Step-by-step explanation:

Understanding Symmetry and Crystal Field Splitting in d Orbitals

When examining the symmetry of d orbitals in relation to a C2 axis in a crystal field, it is indeed the case that if even one orbital in a group is asymmetric concerning the C2 element, then the entire group of orbitals is considered asymmetric. This pertains to how the orbitals transform under the operation of the symmetry element (such as a C2 axis), which is perpendicular to the principal axis of an octahedral complex.

For the second concern regarding e1 and e2 sets in linear complexes, it's important to consider the d orbitals' orientations: d(xz) and d(yz) are labeled e1, while d(x2-y2) and d(xy) are labeled e2. This labeling is based on the symmetry properties of these orbitals with respect to the C2 axis that is preserved in the complex. It might seem confusing because changing the orientation of your C2 axis could appear to alter the symmetries, but for proper symmetry assignments, you should keep the axis consistent and not change it arbitrarily. The labels e1 and e2 reflect distinct symmetry properties relative to this axis, and thus, they do not change even when considering different C2 axes in the complex.

The difference in energy between eg and t2g sets of orbitals is termed the crystal field splitting energy (Δ), and though it redistributes the energies of the d orbitals, it does not change the total energy of all five d orbitals. This concept is essential for explaining the electronic structure and properties of coordination complexes.