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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)...

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2 Answers 2

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Labels such as $T_{2g}$ means that the irreducible representation, irreps, (the row of characters as in the point group in the answer by Buttonwood) is triply degenerate, hence the $T$. Singly degenerate irreps are labelled $A$ or $B$ and doubly degenerate $E$ (not the be confused with the identity). The subscript $g$ means gerade i.e. of even symmetry wrt to inversion at the centre of symmetry and $u$ (ungerade) means odd symmetry here. Not all point groups have irreps with this $u,g$ label, e.g. $C_{2v}$.

In singly degenerate irreps only the Mulliken label is

(a) $A$ if it is symmetric about rotation about the highest order axis, but if it is antisymmetric then the label is $B$.

(b) A subscript $1$ is given if the irrep is symmetric about a $C_2$ axis perpendicular to the principal axis or if no such axis exists to reflection in a $\sigma_v$ plane. If antisymmetric then the subscript is $2$.

(c) The superscripts are $'$ or $"$ for symmetric and antisymmetric wrt reflection in $\sigma_h$ plane.

However, the subscripts $1$ or $2$ are not straightforward to generate for $E,T$ groups and it is best just to consider them as labels and to work other things out, look at the symmetry operations and the value in the irrep in that particular column; the irreps do contain all the information.

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The denomination $t_{2g}$ in the ligand/crystal field theory of transition metals consists of one part which describes the field. It can be tetrahedral ($t$), or cubic ($c$), octahedral ($o$), square/quadratic ($q$). Second, the subsequent subscript relates to group theory and character tables. Here, entries may be considered either as gerade (German for even, hence subscript $g$), or ungerade (German for odd, hence subscript $u$). The integer is both a label of book keeping and to "denote symmetry and antisymmetry". See, for example, the character table for point group $O_h$ in Wikipedia's compilation:

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  • $\begingroup$ Thanks! Which symmetry element did you refer to when you said "denote symmetry and antisymmetry"? $\endgroup$
    – Rivinius
    Aug 31, 2023 at 13:40
  • $\begingroup$ To quote the introduction of Wikipedia's List of character tables for chemically important 3D point groups: «...g and u subscripts denote symmetry and antisymmetry, respectively, with respect to a center of inversion. Subscripts "1" and "2" denote symmetry and antisymmetry, respectively, with respect to a nonprincipal rotation axis. Higher numbers denote additional representations with such asymmetry. ...» $\endgroup$
    – Buttonwood
    Aug 31, 2023 at 13:45

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