What is the meaning of these d states names?

Question

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

Answer 1

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.

Answer 2

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.