Splitting of Terms in an Octahedral Field

Every time a chemist assigns a UV-vis absorption band of a first-row transition-metal complex, they are implicitly using term-splitting tables like the ones derived here. The colors of gemstones — ruby and emerald both containing Cr³⁺ but in different host lattices with different Δₒ, giving red versus green — are textbook illustrations used in mineralogy and gemology courses worldwide, and the underlying ⁴A₂g/⁴T₂g/⁴T₁g splitting pattern of the ⁴F term is exactly what is computed above. In materials science, the same term-splitting analysis (extended to tetragonal, square-planar, and other lower-symmetry fields by further descending in symmetry from Oʰ) underlies the design of phosphors and laser gain media — the ruby laser itself operates on the sharp ²Eg → ⁴A₂g emission of Cr³⁺ discussed in Example 2, and modern solid-state lasers (Nd:YAG, Ti:sapphire) are engineered by tuning exactly this kind of term-splitting via host lattice choice. In bioinorganic chemistry, the term-splitting diagrams for high-spin d⁵ Fe³⁺ and Mn²⁺ explain why many metalloprotein active sites (met-hemoglobin, certain Mn-dependent enzymes) are pale rather than intensely colored, because the ground ⁶S term forces all d–d transitions to be spin-forbidden. Diagnostic UV-vis and EPR spectroscopy of metalloenzymes routinely uses these term-splitting assignments to determine oxidation state, spin state, and coordination geometry directly from spectral band positions and intensities, without needing crystallography. In solid-state and quantum-information research, transition-metal- and rare-earth-doped crystals (Cr³⁺:Al₂O₃, Er³⁺:Y₂SiO₅) are chosen and engineered as qubit hosts precisely because the term-splitting pattern determines which optical transitions are sharp, long-lived, and addressable — turning a 1929 group-theory exercise into modern photonic quantum technology. Finally, this same machinery is the launching point for the qualitative Orgel Diagrams and the quantitative Tanabe-Sugano Diagrams: Concept that every inorganic chemist uses daily to interpret d-block electronic spectra.

Ask the tutor: How does the Oʰ term-splitting pattern change once you lower the symmetry to tetragonal (as in a Jahn-Teller-distorted or trans-substituted complex)?

Start with the one-electron story, because it is the seed of everything that follows. A single d electron has five degenerate orbitals in a free ion. Put that ion at the center of an octahedron of six point-charge ligands along ±x, ±y, ±z. The orbitals that point directly at the ligands (dz², dx²−y²) are pushed up in energy; the orbitals that point between the ligands (dxy, dxz, dyz) are pushed down. Group theory formalizes this: the set of five d orbitals transforms as the reducible representation Γ_d = Eg + T₂g of Oʰ, and this is the origin of the familiar t₂g/eg splitting with gap Δₒ (also written 10Dq). Now the harder and more important question: what happens to a term, not a single orbital? A many-electron free-ion term like ³F or ¹G is a state with a definite total orbital angular momentum L and total spin S. Under rotations, a term with quantum number L transforms exactly like a set of (2L+1) spherical harmonics — mathematically identical to a single electron with orbital angular momentum L. This means the machinery used to split a single d-orbital set (L = 2) can be reused verbatim for any L: an F term (L = 3) splits the way a set of seven "atomic f-orbital-like" functions would split under Oʰ; a P term (L = 1) splits the way three p-orbitals would (which, since Oʰ is centrosymmetric and the p-set is already the basis of T₁u — but for even-parity terms arising from d^n configurations, gerade labels apply, giving T₁g). The spin S is a spectator: it does not couple to the spatial ligand field (in this non-relativistic treatment), so a term's spin multiplicity (2S+1) is simply carried along unchanged into every sublevel it produces. The reduction is done with characters. For a rotation by angle θ, the character of the (2L+1)-dimensional representation generated by angular momentum L is

Evaluate this at the specific rotation angles present in the Oʰ point group (E at θ=0, C₃ at θ=120°, C₂ at θ=180°, C₄ at θ=90°, and the class of C₂′ at θ=180°, plus their improper partners obtained via i and the parity of L), and you get the reducible representation's character set. Reduce it using the standard reduction formula against the Oʰ character table, and out pops the list of irreducible representations — the symmetry labels of the ligand-field states that the term splits into.

There is a beautiful shortcut hiding in this machinery: because a P term (L=1) transforms exactly like a set of three p-like functions, and a p-orbital set never splits under Oʰ (it stays triply degenerate, transforming as T₁u for odd-parity or T₁g for even-parity term functions), P terms never split — they simply shift as a whole. Likewise, S terms (L=0) are totally symmetric and never split — they transform as A₁g. This gives the two "boring but important" baseline cases against which the splitting D, F, G, H terms can be compared.

The whole machine rests on three linked equations. First, the character of the representation generated by a term of angular momentum L, evaluated at a rotation of angle θ:

where:

• $L$ — the total orbital angular momentum quantum number of the free-ion term (0,1,2,3,4,… for S,P,D,F,G,…)
• $\theta$ — the rotation angle of the symmetry operation (0° for E, 120° for C₃, 90° for C₄, 180° for C₂ and C₂′)
• $\chi_L(\theta)$ — the character (trace) contributed by that operation in the (2L+1)-dimensional representation

Reading: this formula is identical to the one used to find how a set of atomic orbitals with angular momentum l transforms under rotation — because a term's angular part behaves exactly like a single "super-orbital" with quantum number L. It converts a chemistry question (how does this term split?) into a pure group-theory computation. Second, the reduction formula that extracts the number of times each irreducible representation Γᵢ appears in the reducible representation Γ_L:

where:

• $n_i$ — number of times irreducible representation $\Gamma_i$ occurs in the term's representation
• $h$ — order of the point group (h = 48 for Oʰ)
• $g_R$ — number of operations in class $R$
• $\chi_L(R)$ — character of the term's representation for class $R$ (from the formula above)
• $\chi_i(R)$ — character of irreducible representation $\Gamma_i$ for class $R$, read off the Oʰ character table

Reading: this is the same reduction formula used everywhere in group theory (e.g., to reduce reducible representations of ligand σ-orbital sets). Applied here it tells you, term by term, exactly which ligand-field states appear and how many times. Third, a bookkeeping identity that is the fastest sanity check available and should always be run:

where:

• $2L+1$ — total degeneracy of the original free-ion term
• $d_i$ — dimensionality of irreducible representation $\Gamma_i$ (1 for A, 2 for E, 3 for T)
• $n_i$ — multiplicity of that representation in the decomposition

Reading: degeneracy must be conserved — the field mixes and separates states, but it cannot create or destroy them. Any decomposition that fails this identity is wrong by construction.

Example 1 — Foundational: the d¹ ion, Ti³⁺, term ²D. The free ion Ti³⑦ (3d¹) has a single term, ²D (L=2, S=½). Using the same procedure as above for L=2 gives characters (5, −1, 1, −1, 1) for (E, 8C₃, 6C₂, 6C₄, 3C₂′). Reducing against the Oʰ table:

That second sum looks wrong at first glance — but recomputing with correct signs for T₂g's C₂ character (+1, not the T₁g value of −1) gives instead $\frac{15+0+6-6-3}{24}=\frac{12}{24}=0.5$, which signals an arithmetic slip; redo directly: for T₂g, characters are (3,0,1,−1,−1), so

So Γ_D = Eg + T₂g, with degeneracy check 2+3 = 5 = 2(2)+1 ✓. Physically: the single d electron occupies the lower t₂g set (which is the one-electron shadow of the ²T₂g ligand-field term) in the ground state, giving the textbook single d–d absorption band of [Ti(H₂O)₆]³⁺ near 20,300 cm⁻¹ assigned as ²T₂g → ²Eg, with Δₒ read directly off that one band.

Example 2 — Applied: Cr³⁺ (d³) in ruby, ⁴F ground term. Cr³⁺ has ground term ⁴F (L=3, S=3/2) plus excited ⁴P, and several doublet terms. From the F-term derivation above, ⁴F splits as

For a d³ configuration in a reasonably strong field like Cr³⁺–O in Al₂O₃ (ruby), the ⁴A₂g component (all three electrons in t₂g, maximally spread among the three orbitals to preserve high spin) turns out to be the ground state, with ⁴T₂g and ⁴T₁g(F) lying higher. Meanwhile the free-ion ⁴P term does not split (as shown above) but transforms as ⁴T₁g and — crucially — has the same symmetry label as one of the components of ⁴F. States of identical symmetry and spin are allowed to mix ("configuration interaction" within the ligand-field problem), which is exactly why the upper ⁴T₁g branch in the Orgel/Tanabe-Sugano diagram for d³ curves rather than rising as a straight line with Δₒ. The two spin-allowed bands seen in ruby's absorption spectrum, ⁴A₂g → ⁴T₂g near 18,000 cm⁻¹ and ⁴A₂g → ⁴T₁g(F) near 25,000 cm⁻¹, together with a sharp, spin-forbidden ⁴A₂g → ²Eg emission line at 14,400 cm⁻¹ (the R-line, from a doublet term derived from ²G), produce ruby's characteristic red color and its famous sharp fluorescence used in the first laser.

Example 3 — Exam/competition level: verify the G term (L = 4) decomposition and use it to count d⁵ terms. Take L = 4 (a G term, e.g. one of the terms arising in d⁵ or d³ configurations) and derive its Oʰ decomposition from scratch, then check the degeneracy. Using $\chi_4(\theta) = \sin(4.5\theta)/\sin(0.5\theta)$: At E: χ = 9. At C₃ (120°): $\chi_4(120^\circ) = \sin(540^\circ)/\sin(60^\circ) = \sin(180^\circ)/\sin(60^\circ) = 0$. At C₂ (180°): $\chi_4(180^\circ) = \sin(810^\circ)/\sin(90^\circ) = \sin(90^\circ)/1 = 1$. At C₄ (90°): $\chi_4(90^\circ) = \sin(405^\circ)/\sin(45^\circ) = \sin(45^\circ)/\sin(45^\circ) = 1$. At C₂′ (180°): same as C₂, χ = 1. So Γ_G has characters (9, 0, 1, 1, 1). Reduce against each irrep:

So Γ_G = A₁g + Eg + T₁g + T₂g, degeneracy 1+2+3+3 = 9 = 2(4)+1 ✓ — this matches the table shown earlier. This decomposition matters directly for d⁵ high-spin ions like Mn²⁺/Fe³⁺: the ground term is ⁶S (which never splits, remaining ⁶A₁g — this is why high-spin Mn²⁺ complexes are so pale, since all d–d transitions from a sextet ground state to any quartet excited state are both spin-forbidden and orbitally forbidden), while the many excited quartet terms include a ⁴G term whose four-way split (⁴A₁g, ⁴Eg, ⁴T₁g, ⁴T₂g) generates the cluster of weak, sharp bands actually observed in the spectra of [Mn(H₂O)₆]²⁺.

The systematic use of group theory to classify how atomic terms split in crystal fields was built by physicists studying rare-earth and transition-metal ions in crystals in the 1920s–30s, most centrally by Hans Bethe, whose 1929 paper "Termaufspaltung in Kristallen" ("Splitting of Terms in Crystals") is the direct ancestor of everything in this article — indeed the title of this piece is Bethe's own. Bethe worked out, using exactly the character-table methods shown above, how free-ion terms decompose in cubic, octahedral, and other crystal symmetries, laying down the correlation tables still used unchanged today. His approach was purely group-theoretical and electrostatic (crystal-field theory), treating the ligands as point charges; it was later married to molecular-orbital and ligand-field ideas by chemists such as Leslie Orgel and Yukito Tanabe/Satoru Sugano, who turned Bethe's abstract term-splitting tables into the practical correlation diagrams that chemists use to read UV-vis spectra of coordination complexes.