Q.Draw figure to show the splitting of d orbitals in an octahedral crystal field.

Concept: Crystal Field Splitting – In an octahedral field, the five degenerate $d$ orbitals split into two sets: the higher-energy $e_g$ ($d_{x^2-y^2}, d_{z^2}$) and the lower-energy $t_{2g}$ ($d_{xy}, d_{yz}, d_{zx}$).

Reasoning:

1. In an octahedral complex, six ligands approach along the $x$, $y$, and $z$ axes.
2. Orbitals with lobes pointing directly along these axes ($e_g$) experience strong repulsion, raising their energy.
3. Orbitals with lobes directed between the axes ($t_{2g}$) experience less repulsion, so they remain lower in energy.
4. The energy gap between the two sets is denoted $\Delta_o$ (or $10\,Dq$).

Figure: …

The core idea: why does splitting happen?

Imagine placing a central metal ion inside an octahedron of six negative charges (ligands). The d‑orbitals are not all shaped the same — some lobes point directly toward the ligands, others point between them. An electron in an orbital that points straight at a ligand feels a strong electrostatic repulsion, raising its energy. An electron in an orbital that points between ligands feels less repulsion, so its energy stays lower.

This difference in repulsion is the entire origin of crystal field splitting. The five originally equal‑energy d‑orbitals break into two groups.

Step‑by‑step reasoning

1. Identify the orbital shapes relative to the axes

   In an octahedral complex, the ligands lie along the $+x$, $-x$, $+y$, $-y$, $+z$, and $-z$ axes.

   • $d_{z^2}$ has a lobe along the $z$‑axis (plus a doughnut in the $xy$‑plane).
   • $d_{x^2-y^2}$ has lobes along the $x$ and $y$ axes.
   • $d_{xy}$, $d_{yz}$, $d_{zx}$ have lobes that lie between the axes — in the planes at $45^\circ$ to the axes.

2. Which orbitals point directly at the ligands?

   The $d_{z^2}$ and $d_{x^2-y^2}$ orbitals have their lobes aimed straight at the ligands on the axes. These are the $e_g$ set (two orbitals).

   The $d_{xy}$, $d_{yz}$, $d_{zx}$ orbitals point between the axes, so they avoid the ligands. These are the $t_{2g}$ set (three orbitals).

3. Energy consequence

   Because the $e_g$ orbitals experience direct repulsion from the ligand charges, their energy rises. The $t_{2g}$ orbitals, pointing into empty space between ligands, stay at a lower energy. The energy gap between the two sets is called $\Delta_o$ (or $10\,Dq$).

4. The splitting diagram

   On the left, the five d-orbitals of the free ion sit together at one (degenerate) energy level. On the right, in the octahedral field, they split into two groups: the $e_g$ pair ($d_{z^2}, d_{x^2-y^2}$) raised above that level, and the $t_{2g}$ trio ($d_{xy}, d_{yz}, d_{zx}$) lowered below it, separated by the gap $\Delta_o$.

   In its simplest form:

        e_g (higher energy)
        ↑
        |  Δ_o
        ↓
        t_{2g} (lower energy)
   ``` …
   

Crystal Field Splitting in an Octahedral Field

Method: Crystal Field Theory (CFT) — Electrostatic Repulsion Model

Core Concept

In an octahedral complex, the five $d$ orbitals are not degenerate (equal in energy) because they experience different electrostatic repulsion from the six ligands placed along the $x$, $y$, and $z$ axes.

Steps to Draw the Splitting Diagram

Step 1: Draw the energy axis

• Draw a vertical line (energy increasing upward).
• Mark a horizontal line labeled "Barycenter" or "Average energy of $d$ orbitals in spherical field" — this is the energy the $d$ orbitals would have if the field were spherically symmetric.

Step 2: Place the two groups of orbitals

• Above the barycenter, draw two degenerate (same energy) orbitals: $d_{x^2-y^2}$ and $d_{z^2}$ — these are the $e_g$ set.
• Below the barycenter, draw three degenerate orbitals: $d_{xy}$, $d_{xz}$, $d_{yz}$ — these are the $t_{2g}$ set.

Step 3: Label the energy gap

• Draw a double-headed arrow between the $t_{2g}$ and $e_g$ sets.
• Label this gap as $\Delta_o$ (or $10 Dq$).

Step 4: Add orbital shapes (optional but helpful)

• Beside each orbital label, sketch the shape:
  • $d_{x^2-y^2}$: four lobes along $x$ and $y$ axes
  • $d_{z^2}$: two lobes along $z$ + a donut in $xy$ plane
  • $d_{xy}$, $d_{xz}$, $d_{yz}$: four lobes between the axes

Why This Splitting Occurs

Common Mistakes in Drawing Octahedral Crystal Field Splitting

Mistake 1: Wrong Orbital Labels or Orientation

The error: Students often mislabel the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals, or draw them with incorrect spatial orientation.

How to avoid:

• Remember: $d_{x^2-y^2}$ has lobes along the x and y axes (pointing directly at ligands in octahedral geometry)
• $d_{z^2}$ has a unique "doughnut" shape — a lobe along z-axis with a ring in xy-plane
• The three $t_{2g}$ orbitals ($d_{xy}, d_{xz}, d_{yz}$) point between the axes

Key insight: In an octahedral field, ligands approach along x, y, and z axes. Orbitals pointing at axes ($e_g$) experience more repulsion than those pointing between axes ($t_{2g}$).

Mistake 2: Incorrect Energy Order

The error: Drawing $t_{2g}$ above $e_g$, or placing them at the same energy level.

How to avoid:

• $e_g$ orbitals ($d_{x^2-y^2}, d_{z^2}$) are higher in energy — they point directly at ligands
• $t_{2g}$ orbitals ($d_{xy}, d_{xz}, d_{yz}$) are lower in energy — they point between ligands
• The energy gap is denoted as $\Delta_o$ (or $10Dq$)

The $e_g$ set ($d_{x^2-y^2}, d_{z^2}$) sits at the higher energy level, separated by the gap $\Delta_o$ from the $t_{2g}$ set ($d_{xy}, d_{xz}, d_{yz}$) at the lower energy level.

Mistake 3: Forgetting the Barycenter

The error: Drawing the split levels without showing the average energy (barycenter) or misplacing it.

How to avoid:

• The barycenter (average energy of all five d-orbitals) must remain constant
• In octahedral field: $e_g$ orbitals go up by $+0.6\Delta_o$, $t_{2g}$ go down by $-0.4\Delta_o$
• Check: $2(+0.6) + 3(-0.4) = 1.2 - 1.2 = 0$ ✓

Mistake 4: Unequal Spacing or Wrong Gap Size

The error: Drawing the $e_g$ and $t_{2g}$ levels with unequal spacing from the barycenter.

How to avoid:

• The $e_g$ level is farther from barycenter than $t_{2g}$ level
• Ratio: $e_g$ shift : $t_{2g}$ shift = $3:2$ (because 2 orbitals × 3 parts = 3 orbitals × 2 parts)
• Draw the gap from $t_{2g}$ to barycenter as 2 units, and from barycenter to $e_g$ as 3 units

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