The σ-Only Octahedral MO Diagram

Introduction

The σ-only molecular-orbital diagram gives the familiar octahedral d split a bonding interpretation. Six ligand donor functions form A₁g, E g and T₁u combinations. Their symmetry-matched interactions with metal s, d(e g) and p orbitals create bonding and antibonding MOs, while metal t₂g has no radial σ partner and remains approximately nonbonding. This picture explains why occupying e g can lengthen metal–ligand bonds.

Core explanation

Begin with six ligands that each donate a filled orbital directed toward the metal. Their twelve donor electrons fill six predominantly bonding MOs in a simple σ framework. The six ligand combinations transform as A₁g (one), E g (two) and T₁u (three). Metal ns has A₁g symmetry, metal np has T₁u, and metal d(z²)/d(x²−y²) has E g. Matching pairs can mix, producing lower bonding and higher antibonding combinations in each symmetry block. How far apart they lie depends on overlap and energy matching.

The three metal d(xy), d(xz) and d(yz) orbitals form T₂g. Since the radial σ ligand set contains no T₂g SALC, these metal d-like orbitals are approximately nonbonding in the σ-only idealisation. The two d(e g) orbitals do interact with ligand E g σ combinations; their metal-rich antibonding products are commonly denoted e g . The energy separation between t₂g and e g is the ligand-field Δₒ in the simplified diagram. The star is a reminder that the upper level has explicit antibonding character, not just an electrostatic shift.

After the twelve ligand-donor electrons fill the low-energy bonding MOs, the formal metal d electrons occupy t₂g and e g according to splitting and pairing. For d¹–d³, they enter t₂g singly. For low-spin d⁶, t₂g⁶ is filled and e g empty; for high-spin d⁶, t₂g⁴e g ² has two electrons in metal–ligand σ-antibonding levels. The latter often has longer metal–ligand bonds than the former for comparable chemistry, linking orbital occupancy to structure.

This diagram refines but does not invalidate formal dⁿ counting. The bonding MOs are mixtures of ligand and metal functions, and e g may retain some ligand character. Formal oxidation state remains a bookkeeping convention for assigning n. The phrase “six ligand pairs plus d⁶ gives eighteen electrons” can be useful for some complexes, but an 18-electron tally does not by itself prove a spin state or guarantee stability. Electron counting and orbital energy are different questions.

The energetic order of metal s- and p-derived antibonding MOs relative to the d-like levels can vary, so a schematic diagram should label symmetry and predominant character rather than assert one universal ordering of every high empty MO. For many introductory ligand-field questions, the crucial frontier portion is t₂g below e g . More detailed computations include additional metal and ligand orbitals, π interactions and different ligand energies.

Adding π-donor orbitals changes the t₂g set from σ-nonbonding to π-antibonding metal-rich levels; adding π-acceptor orbitals can stabilise it through back-bonding. Thus “nonbonding t₂g” is specifically a σ-only result, not a property of every real octahedral complex. The diagram also assumes equivalent ligands and O h symmetry. Mixed ligands split exact degeneracies.

For spectroscopy, a simple d¹ excitation from t₂g to e g can approximate Δₒ, but many-electron dⁿ spectra involve terms and selection rules. The MO diagram explains the orbital energy framework; it is not a complete Tanabe–Sugano diagram.

Step-by-step reasoning

List ligand σ SALCs and matching metal symmetries. Draw bonding and antibonding combinations for A₁g, T₁u and E g blocks. Leave metal T₂g as σ-nonbonding, then fill twelve donor electrons into lower bonding MOs. Add formal d electrons to t₂g and e g according to the field/pairing competition, and infer bond-length or magnetic consequences.

Visual explanation

Draw metal orbitals on the left, six ligand SALCs on the right and mixed MOs in the centre. Connect only matching A₁g, E g and T₁u lines. Leave the T₂g metal line unconnected to σ donors and place e g above it as a metal-rich antibonding pair.

Real-world analogy

Two compatible musical notes can combine into a smooth harmony and a discordant combination. A metal and ligand orbital of matching symmetry similarly produce bonding and antibonding combinations. A metal orbital with no matching ligand pattern remains a solo line in this restricted score.

Real-world example

Low-spin Fe²⁺ d⁶ complexes occupy t₂g⁶ without e g electrons in the simple diagram, whereas high-spin Fe²⁺ d⁶ occupies e g twice. Shorter bonds often accompany the low-spin state, providing structural evidence alongside magnetic measurements.

Why?

Why are e g electrons associated with longer bonds? The orbital combination has antibonding metal–ligand σ character. Occupying it reduces the net bonding advantage and can favour greater metal–ligand separation, all else comparable.

Common misconception

“t₂g is exactly nonbonding in every octahedral complex.” It has no radial σ donor match in this ideal diagram, but ligand π orbitals can interact with T₂g symmetry and change its energy and bonding character.

Worked example

Compare two formal d⁶ octahedral configurations using the σ-only diagram. Low spin fills t₂g⁶e g ⁰, with zero unpaired electrons and no occupation of the upper σ-antibonding pair. High spin fills t₂g⁴e g ², with four unpaired electrons and two upper antibonding electrons. The orbital CFSE values remain −2.4Δₒ and −0.4Δₒ respectively, while the MO picture adds a bond-length interpretation.

Quick check

1. Which metal d set lacks a radial σ ligand partner in O h? Answer: T₂g, the d(xy), d(xz) and d(yz) set.

Exam focus

Label SALC symmetries and orbital character, fill ligand bonding MOs before formal d-like frontier levels, and qualify “t₂g nonbonding” as a σ-only statement.

Advanced insight

The magnitude of Δₒ in the MO model reflects level repulsion and bonding interactions, not merely direct electrostatic repulsion. π ligands can change t₂g while leaving the main σ origin of e g intact, making ligand trends more intelligible.

Summary

Six σ donors combine into A₁g, E g and T₁u ligand orbitals that mix with matching metal functions. Metal t₂g is σ-nonbonding, while metal-rich e g is σ-antibonding; formal d electrons fill these frontier levels.

Practice questions

1. How many electrons do six filled monodentate σ-donor orbitals contribute to the simple MO diagram? Answer: Twelve electrons, two from each donor orbital, filling six predominantly bonding combinations in the basic σ framework. 2. Why can metal t₂g shift when π ligands are included? Answer: Ligand π orbitals can form T₂g symmetry combinations that mix with metal t₂g, even though the radial σ set had no such partner. 3. Which d⁶ filling is expected to have more occupation of σ-antibonding e g ? Answer: High-spin t₂g⁴e g ² has two such electrons; low-spin t₂g⁶e g ⁰ has none. 4. What does the star in e g indicate? Answer: Antibonding character from metal e g and ligand E g σ mixing.