# What is the meaning of these d states names?

I have read in the article at the reference below that :

the Mn substitution leads to localized states (impurity bands) within the band gap. The Mn 3d orbitals split into a single a$$_1$$ (d$$_{z^2}$$) state and two twofold degenerate e$$_1$$ (d$$_{xz/x^2-y^2}$$) and e$$_2$$ (d$$_{xz/yz}$$) since the Mn atom under the crystal field has a C3v symmetry.

My question is what is the meaning of a$$_1$$, e$$_1$$ and e$$_2$$ states? What do they refer to?

Reference paper : 10.1039/c4cp00247d

The following screencast is from the cited paper (Seems there is a typo in the description):

The $$d$$-orbitals of Mn atoms are split into three groups due to the $$C_{3v}$$ point group symmetry, which is a general conclusion from the crystal field theory based on group theory.

The $$e_1$$, $$e_2$$ and $$a_1$$ are the symbols of the corresponding irreducible representation from group theory.

• What about the t2g and eg states ? what do they mean ? – Chi Kou Apr 10 at 10:52
• @ChiKou You may take a look at this post: mattermodeling.stackexchange.com/questions/2210/… – Jack Apr 10 at 12:54
• @ChiKou The $t_{2g}$ and $e_g$ may correspond to another partition of the five $d$-orbitals with another point group symmetry. – Jack Apr 10 at 12:57
• Well explained ! Thanks a lot. – Chi Kou Apr 17 at 17:31

To add to Jack's answer, I'll point out that the $$a$$, $$e$$, $$t$$ etc. labels are symmetry labels of the orbitals in an irreducible representation (irrep). They are lower-case versions of their corresponding Mulliken symbols found in character tables. There are four letters that tend to show up in $$d$$-orbital systems:

• $$e$$, indicating two-fold degeneracy
• $$t$$, indicating three-fold degeneracy
• $$a$$ and $$b$$, indicating non-degeneracy (or single degeneracy). $$a$$ ($$b$$) is picked if the irrep is symmetric (anti-symmetric) with respect to rotation about the principal axis. In addition, $$a$$ is picked if there's no well-defined $$C_n$$ or $$S_n$$ operation.

In the case of $$C_\mathrm{3v}$$ you can take the principal axis to be $$\hat{z}$$, in which case it's clear that $$d_{z^2}$$ must transform symmetrically, and should be labeled $$a$$.

For non-degenerate irreps ($$a$$ and $$b$$), the subscripts $$_1$$ and $$_2$$ indicate symmetry and anti-symmetry, respectively, under rotation about a non-principal axis (perpendicular to the principal axis). In case of multidimensional representations ($$e$$, $$t$$), the numerical subscript is used to distinguish between inequivalent irreps that aren't separated under other rules.

Finally, you also asked about the $$t_{2g}$$ and $$e_g$$ orbitals that show up in e.g. an octahedral environment. Here, the subscript $$g$$ indicates the wave function is symmetric around an inversion center. $$g$$ comes from the German word "gerade", meaning even. This can be contrasted with the subscript $$u$$ for "ungerade", indicating antisymmetry around the inversion center.

• Well explained ! Thanks a lot. – Chi Kou Apr 17 at 17:31