Octahedral d-Orbital Splitting

Introduction

Six ligands in an ideal octahedral complex approach a metal along three Cartesian axes. This particular geometry separates five d orbitals into a lower group of three and an upper group of two. Knowing which orbitals belong to each group turns a rough “ligands change energies” idea into a working electron diagram.

Core explanation

The d x²−y² and d z² orbitals point strongly along coordinate axes. Octahedral ligands approach along ±x, ±y and ±z, so these orbitals interact more strongly with ligand electron density in the CFT electrostatic picture. They form the upper e g set. The d xy, d xz and d yz orbitals point between axes and form the lower t₂g set. The separation between the groups is Δ oct. The subscript g comes from symmetry terminology and need not be derived to use the diagram correctly.

The three lower orbitals are equal in energy in an ideal octahedron, as are the two upper ones. A lower t₂g electron is assigned −0.4Δ oct relative to the five-orbital average, and an upper e g electron +0.6Δ oct. The weighted average is unchanged because 3(−0.4) + 2(+0.6) = 0. These numbers let an advanced problem calculate crystal-field stabilization, but the central introductory task is correct ordering and occupancy.

Electrons occupy available orbitals subject to the Pauli principle and Hund's rule. For d¹, place one electron in t₂g; for d², place two singly in separate t₂g orbitals; for d³, place one in each lower orbital. Beyond d³, the choice between pairing in t₂g and entering e g depends on the pairing energy P relative to Δ oct. For d⁶, a weak field can give high-spin t₂g⁴e g² with four unpaired electrons, while a strong field can give low-spin t₂g⁶e g⁰ with none. Both have six d electrons.

These diagrams assume an idealized octahedral field. Actual complexes may distort, further separating levels within t₂g or e g. Copper(II) d⁹ octahedral environments frequently show distortion, so a measured spectrum may not fit a single simple gap perfectly. A useful model identifies the dominant ordering while acknowledging smaller changes induced by lower symmetry.

Oxidation-state arithmetic must precede the d count. Fe²⁺ has atomic number 26, and its common cation configuration is [Ar]3d⁶ after outer 4s electrons are removed. Fe³⁺ is d⁵. Placing six electrons for Fe³⁺ or five for Fe²⁺ creates an incorrect spin prediction even if the t₂g/e g labels are otherwise right. The complex charge may not equal the metal oxidation state because ligands have charges of their own.

The splitting also enables visible-light absorption if an allowed electronic transition has an energy within the visible range, although selection rules and charge-transfer transitions affect intensity. Therefore a Δ oct diagram suggests a mechanism but does not by itself guarantee a strong colour or give the perceived colour without spectral information.

Step-by-step reasoning

1. Confirm six-coordinate octahedral geometry. 2. Derive metal oxidation state and d count. 3. Draw three lower t₂g boxes and two upper e g boxes. 4. Fill electrons with Hund's rule and Pauli exclusion. 5. For relevant d counts, compare Δ oct with pairing energy to choose spin state. 6. Count unpaired arrows for qualitative magnetism.

Visual explanation

Draw three boxes on a lower line labelled d xy, d xz and d yz, and two boxes on an upper line labelled d x²−y² and d z². Show six ligand arrows arriving on axes and label the vertical energy gap Δ oct.

Real-world analogy

An octahedral room has doors along the x, y and z directions. Objects pointing directly at the doors encounter incoming traffic more strongly than objects pointing between doors. This helps remember upper e g versus lower t₂g, though orbitals are wavefunctions, not physical rods.

Real-world example

Magnetic measurements of Fe²⁺ complexes can distinguish high-spin and low-spin arrangements. A d⁶ count alone cannot do so; ligand-field strength determines whether four or zero unpaired electrons are predicted in the ideal octahedral cases.

Why?

Why is e g higher than t₂g in octahedral CFT? The e g orbital lobes point toward six ligand approach directions and experience greater electrostatic repulsion than t₂g lobes between axes.

Common misconception

“Six-coordinate means six d electrons.” Coordination number counts donor atoms, while d count follows metal identity and oxidation state. An octahedral Co(III) complex may be d⁶; an octahedral Fe(III) complex is d⁵.

Worked example

Fill a strong-field octahedral Fe²⁺ d⁶ diagram. Put three electrons singly in t₂g, then pair the next three in those same lower orbitals because Δ oct is taken larger than pairing cost. The arrangement is t₂g⁶e g⁰, with zero unpaired electrons and a diamagnetic spin-only prediction. Under weak-field conditions, t₂g⁴e g² would instead have four unpaired electrons.

Quick check

1. Which octahedral d orbitals form the upper e g set? Answer: d x²−y² and d z².

Exam focus

Keep the lower three and upper two boxes clear. State the spin assumption whenever more than one filling is possible. Do not infer d count from coordination number.

Advanced insight

Crystal-field stabilization energy is found by summing −0.4Δ oct for each t₂g electron and +0.6Δ oct for each e g electron, then considering pairing separately. This energy bookkeeping helps compare configurations but is not a full metal–ligand bond energy.

Summary

Octahedral ligands raise axis-directed e g orbitals above the between-axis t₂g set. The gap is Δ oct. Filling the levels with the correct d count and pairing assumption predicts a qualitative spin state and helps interpret spectra.

Practice questions

1. How many orbitals are in the octahedral t₂g set? Answer: Three. 2. How many orbitals are in the e g set? Answer: Two. 3. What is Fe²⁺'s d count in a standard ionic calculation? Answer: d⁶. 4. How many unpaired electrons does ideal low-spin octahedral d⁶ have? Answer: Zero, with t₂g⁶e g⁰ occupancy.