Splitting of Free-Ion Terms in Octahedral Fields

Introduction

An isolated metal ion is approximately spherical, so L labels such as D and F are appropriate. Six ligands on octahedral axes reduce that symmetry to O h. The free-ion terms do not vanish; their orbital degeneracy is reorganised into states transforming as irreducible representations of O h. These correlations provide the alphabet for Orgel and Tanabe–Sugano diagrams and explain why one free-ion term can yield several absorption-relevant complex states.

Core explanation

For a given orbital L, the free-ion term has 2L+1 orbital components before spin is counted. Restricting the spherical rotation representation to octahedral operations decomposes those components into O h irreducible representations. The essential correlations for d-electron states are S→A₁g; P→T₁g; D→E g+T₂g; F→A₂g+T₁g+T₂g; and G→A₁g+E g+T₁g+T₂g. Their dimensions check: 1; 3; 2+3=5; 1+3+3=7; and 1+2+3+3=9. The g suffix follows because states derived solely from metal d orbitals are even under inversion in an ideal centrosymmetric octahedral complex.

Write spin multiplicity before each new state: ^3F gives ^3A₂g, ^3T₁g and ^3T₂g, not unlabeled A and T states. A spin-independent ligand field splits orbital components but does not by itself change S. This statement is approximate once spin–orbit coupling mixes states, yet it is the correct foundation for first-pass band assignment. Uppercase A, E and T denote many-electron states; lowercase a, e and t generally denote one-electron orbitals. Thus ^2T₂g is not synonymous with a single electron sitting in t₂g, although a d¹ ^2T₂g state can be described that way in a simple picture.

Correlation tells which symmetries occur, not their energy order . For a d¹ ^2D term, one electron enters lower t₂g before higher e g in an octahedral field, so ^2T₂g is ground and ^2E g is excited. But with several interacting electrons, ordering depends on occupancy, electron–electron repulsion and ligand-field strength. A ^3F parent in d² produces the same three symmetry types as a ^3F parent in d⁸, yet their energy order is not identical. A term-correlation table alone cannot replace an Orgel or Tanabe–Sugano diagram.

Two states can share the same O h label while coming from different parent terms. For d², ^3F yields a ^3T₁g state and ^3P yields another ^3T₁g state. Because they have identical symmetry and spin multiplicity, they can mix when their energies approach as field strength changes. Their energy curves avoid crossing in the simple model rather than pass independently through one another. If a diagram writes ^3T₁g(F) and ^3T₁g(P), the parenthetical letters identify the weak-field ancestry, not distinct point-group representations.

The splitting also preserves the total number of microstates when spin is included. For ^3F, the original count is 3×7=21. Its three O h components carry 3×1=3, 3×3=9 and 3×3=9 spin-orbital states, totaling 21. This dimension accounting prevents the frequent mistake of comparing a one-electron fivefold d manifold with a many-electron F term without their different spin and symmetry factors.

Centrosymmetry matters to selection rules. A pure dⁿ-to-dⁿ transition connects gerade states and is electric-dipole Laporte-forbidden in exact O h symmetry. A vibronic distortion or an odd-parity component can borrow intensity. The g labels therefore carry physical information about absorption strength, not merely decoration. In tetrahedral symmetry there is no inversion centre, so the corresponding labels omit g/u and many d–d bands can be stronger.

Step-by-step reasoning

Identify the free-ion parent ^(2S+1)L. Use its L letter to choose the O h correlation. Attach the same spin multiplicity to every component and, for a d-derived term, g parity. Check that component dimensions add to 2L+1; multiplying by 2S+1 then checks the term's complete microstate count. Use actual electron configuration or a term-energy diagram to establish order, rather than guessing it from the correlation alone.

Visual explanation

Draw five horizontal free-ion lines labelled S, P, D, F and G on the left, each branching to O h labels on the right. Give each branch a width proportional to degeneracy: A one unit, E two and T three. The D branch splits 2+3, F splits 1+3+3, and G splits 1+2+3+3. Put a note below: “Branch widths sum to 2L+1; vertical placement requires an energy model.”

Real-world analogy

A group of nine musicians can be reorganised into an individual, a pair and two trios without losing anyone. A G term similarly reorganises its nine orbital components into A₁g, E g, T₁g and T₂g under octahedral symmetry. The grouping describes symmetry; it does not tell which group plays first.

Real-world example

Ti³⁺ is d¹. The free-ion ^2D term contains ten states including spin. In an ideal octahedral aqua ion it splits into ^2T₂g with 2×3=6 states and ^2E g with 2×2=4. The lower-to-upper transition is the simplest ligand-field model for its visible absorption. Its weak intensity relative to a typical charge-transfer transition is consistent with the g-to-g parity restriction.

Why?

Why does P become T₁g rather than three unrelated singlets? The three orbital components of L=1 transform into one connected three-dimensional representation under O h rotations. Octahedral symmetry allows them to stay triply degenerate, so they form T₁g instead of independent A states.

Common misconception

“F means an f electron is present.” F is the total angular-momentum letter L=3. A d² or d⁸ ion can have a ^3F term although every active electron occupies a d subshell. The analogous one-electron f-orbital symmetry is a useful mathematical guide, not a claim about occupancy.

Worked example

Split a free-ion ^1G term in O h. G gives L=4 and nine orbital components. The correlation is A₁g+E g+T₁g+T₂g, with dimensions 1+2+3+3=9. Because the parent is a singlet, each component carries superscript 1: ^1A₁g, ^1E g, ^1T₁g and ^1T₂g. Their total state count is nine. The correlation does not specify which of the four is lowest for a particular metal complex.

Quick check

1. What is the O h splitting of a ^4P term? Answer: ^4T₁g; P has L=1 and three orbital components, all transforming as T₁g while quartet spin multiplicity remains four.

Exam focus

Memorise the short S-to-G correlation or reconstruct it with dimensions. Keep uppercase state labels distinct from lowercase orbital labels, preserve multiplicity, and check the total degeneracy. Never infer an energy order from an irreducible-representation list alone.

Advanced insight

The formal operation is restriction of an SO(3) angular-momentum representation to the finite octahedral point group, followed by decomposition with group characters. The same point-group irrep can occur in multiple parent terms, enabling configuration interaction where other quantum numbers also permit it. This is the symmetry basis for curved, avoided-crossing term trajectories.

Summary

An octahedral ligand field reorganises free-ion orbital terms: S→A₁g, P→T₁g, D→E g+T₂g, F→A₂g+T₁g+T₂g and G→A₁g+E g+T₁g+T₂g. The components preserve multiplicity and total degeneracy; their ordering needs a ligand-field energy analysis.

Practice questions

1. Split ^3D in O h and verify its state count. Answer: ^3E g and ^3T₂g have 3×2=6 and 3×3=9 states, totaling 15, equal to (2S+1)(2L+1)=3×5. 2. Why are ideal octahedral d–d transitions often weak even when energy-allowed? Answer: Both d-derived states are gerade, so electric-dipole g→g transitions violate the Laporte parity rule; vibrations or symmetry lowering can give them borrowed intensity. 3. Can the F→A₂g+T₁g+T₂g pattern by itself tell which state is ground for d⁸? Answer: No. It gives allowed symmetry components and dimensions only. Electron occupancy and interaction energies, represented in a proper term-energy diagram, determine the order. 4. Why can two ^3T₁g labels appear in a d² correlation diagram? Answer: One descends from ^3F and another from ^3P. They have the same complex symmetry but different free-ion parents and may mix.