Decoding t2g, eg, t2, and e in CFT | Chemca.in

Decoding $t_{2g}$, $e_g$, $t_2$, and $e$

In Crystal Field Theory, we don't just call orbitals "lower" or "upper". We use Mulliken Symbols from Group Theory. Understand exactly what these letters and subscripts mean.

The Mulliken Symbol Dictionary

Triply Degenerate

Represents a set of 3 orbitals that have exactly the same energy. (e.g., $d_{xy}, d_{yz}, d_{zx}$).

Doubly Degenerate

Represents a set of 2 orbitals that have exactly the same energy. (e.g., $d_{x^2-y^2}, d_{z^2}$).

Gerade (Even/Symmetric)

German for "Even". If you pass through the Center of Inversion ($i$) $(x,y,z \rightarrow -x,-y,-z)$, the orbital's sign/phase does not change.

Antisymmetric

Indicates the orbital is antisymmetric with respect to a perpendicular $C_2$ rotation axis. (While $1$ would mean symmetric).

⚠️ The Ultimate JEE Trap: The Missing 'g'

Many students blindly write $t_{2g}$ and $e_g$ for Tetrahedral complexes. This is absolutely incorrect! The subscript "g" stands for gerade (symmetry about a center of inversion). A tetrahedron has NO center of inversion. Therefore, you MUST drop the "g".

Octahedral Splitting ($O_h$)

Center of Inversion (i) is PRESENT

An octahedron is highly symmetric. It possesses a perfect Center of Inversion (also known as a Center of Symmetry). If you draw a straight line from any ligand, through the central metal atom, and out the other side, you hit an identical ligand.

Because this center of inversion exists, the gerade rule applies.

The 3 lower energy orbitals ($d_{xy}, d_{yz}, d_{zx}$) are triply degenerate and symmetric $\rightarrow$ $$t_{2g}$$
The 2 higher energy orbitals ($d_{x^2-y^2}, d_{z^2}$) are doubly degenerate and symmetric $\rightarrow$ $$e_g$$

Tetrahedral Splitting ($T_d$)

Center of Inversion (i) is ABSENT

A tetrahedron lacks a center of inversion. If you draw a line from one corner (ligand) through the center, you emerge into empty space on the opposite side between the other three ligands.

Because there is no center of inversion, the concept of gerade vs ungerade cannot be applied. The "g" subscript is strictly forbidden!

The 3 higher energy orbitals ($d_{xy}, d_{yz}, d_{zx}$) are triply degenerate $\rightarrow$ $$t_2$$
The 2 lower energy orbitals ($d_{x^2-y^2}, d_{z^2}$) are doubly degenerate $\rightarrow$ $$e$$

Property	Octahedral ($O_h$)	Tetrahedral ($T_d$)
Splitting Energy Symbol	$\Delta_o$	$\Delta_t$
Center of Inversion?	Yes (Present)	No (Absent)
Lower Energy Orbitals	$t_{2g}$ (3 orbitals)	$e$ (2 orbitals)
Higher Energy Orbitals	$e_g$ (2 orbitals)	$t_2$ (3 orbitals)
Use of 'g' subscript?	Required	Incorrect

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