Tanabe-Sugano Diagrams: Reading d-Electron Term Energies vs Ligand Field Strength

What a Tanabe-Sugano diagram is and where it applies

When a transition-metal ion sits in an octahedral field, the free-ion Russell-Saunders terms (³F, ⁴F, ⁵D, etc.) split into ligand-field terms such as ⁴A₂g, ⁴T₂g and ⁴T₁g. A Tanabe-Sugano diagram tracks how the energies of all these terms move as the field grows stronger. Crucially, both axes are divided by the Racah parameter B, which quantifies inter-electron repulsion. This scaling makes the diagram universal: one d³ chart applies to Cr³⁺, V²⁺, Mn⁴⁺ and any d³ complex regardless of the specific ligand.

• Ground state on the baseline: the ground term is drawn as the horizontal zero-energy line, so every excited term's energy is measured relative to it.
• Coverage: diagrams exist for d² through d⁸ (d¹ and d⁹ have a single term and need only an Orgel-style line).

They are the working tool for assigning UV-vis d–d bands, extracting Δ₀ and B, ranking ligands in the spectrochemical series, and diagnosing high-spin vs low-spin ground states in synthesis and bioinorganic chemistry.

The derivation: how the term energies are computed

Tanabe and Sugano built the diagrams by diagonalizing the electrostatic-repulsion plus ligand-field Hamiltonian for each dⁿ configuration. The energies depend on three quantities:

• Δ₀ = 10Dq — the octahedral splitting between t₂g and eg orbitals.
• Racah B and C — parameters describing d-electron repulsion; C is fixed at a constant ratio (they used C ≈ 4B) so only B varies.

For each configuration they wrote the term energies as functions of Δ₀ and B. In the weak-field limit (Δ₀ → 0) energies converge to the free-ion terms, spaced by multiples of B and C. In the strong-field limit the terms sort by t₂g^xeg^y occupancy. The diagram interpolates between them. Dividing through by B gives dimensionless E/B versus Δ₀/B. Non-crossing terms of the same symmetry and spin repel each other (the non-crossing rule), which is why several curves bend rather than cross — the source of a diagram's characteristic curvature.

Key quantities and a worked Cr(III) example

For a d³ ion the three spin-allowed transitions are ⁴A₂g → ⁴T₂g (ν₁), ⁴A₂g → ⁴T₁g(F) (ν₂), and ⁴A₂g → ⁴T₁g(P) (ν₃). A convenient identity is that ν₁ = Δ₀ exactly for d³, so the lowest band gives 10Dq directly.

Take [Cr(H₂O)₆]³⁺: ν₁ ≈ 17,400 cm⁻¹ and ν₂ ≈ 24,700 cm⁻¹.

• Δ₀ = ν₁ = 17,400 cm⁻¹, so Dq ≈ 1,740 cm⁻¹.
• Racah B follows from B = (2ν₁² + ν₂² − 3ν₁ν₂) / (15ν₂ − 27ν₁). Plugging in gives B ≈ 640–700 cm⁻¹.
• The free-ion value is B₀(Cr³⁺) = 1,030 cm⁻¹, so the nephelauxetic ratio β = B/B₀ ≈ 0.63.

That β < 1 means the ligands have expanded the d-electron cloud ("nephelauxetic" = cloud-expanding), reducing repulsion by ~37% through covalent metal-ligand mixing. Reading the diagram at Δ₀/B ≈ 27 places Cr³⁺ firmly in the high-field region of its own chart.

How the diagram is used in practice

The workflow from spectrum to parameters is direct:

• Record the UV-vis-NIR spectrum and identify the spin-allowed (Laporte- and spin-allowed) d–d bands; these are the strong, broad features, molar absorptivities ε ≈ 5–100 L mol⁻¹ cm⁻¹ for centrosymmetric octahedral complexes.
• Take band-energy ratios (e.g. ν₂/ν₁) and slide along the horizontal axis until the calculated ratio matches the diagram — this fixes Δ₀/B.
• Read E/B for each term at that position; dividing an observed band energy by its E/B value returns B in cm⁻¹, and then Δ₀ = (Δ₀/B) × B.

Because the ratio method needs no absolute intensities, it works from a simple benchtop spectrophotometer. The extracted Δ₀ places the ligand in the spectrochemical series (I⁻ < Br⁻ < Cl⁻ < F⁻ < H₂O < NH₃ < CN⁻ < CO), while β places it in the nephelauxetic series. Ratio-fitting also flags spin-forbidden bands: sharp, very weak features (ε < 1) that the diagram predicts as nearly field-independent lines, such as the ruby ⁴A₂g → ²Eg emission at 694 nm.

How it compares to Orgel diagrams and crystal-field pictures

Tanabe-Sugano diagrams are refinements of the earlier Orgel diagrams (Leslie Orgel, 1950s). The differences matter:

• Ground state as baseline: T-S diagrams flatten the ground term onto the x-axis; Orgel diagrams let it slope, which is visually simpler but hides energies.
• Spin states: Orgel diagrams show only spin-allowed transitions and only high-spin cases. T-S diagrams include spin-forbidden terms and, for d⁴–d⁷, both high- and low-spin regions separated by a vertical discontinuity.
• Quantitative vs qualitative: Orgel diagrams are qualitative; T-S diagrams, being scaled by B with computed slopes, let you extract numbers.

Relative to raw crystal field theory, which treats ligands as point charges and ignores repulsion changes, the T-S approach is a ligand field theory tool: B and its reduction (β) explicitly encode covalency that pure electrostatics cannot. It sits below full molecular orbital theory in rigor but far above the point-charge model in predictive spectroscopy.

Exceptions, limits, and why it still matters

The diagrams carry real caveats:

• Fixed C/B: assuming C ≈ 4B is an approximation; real C/B ranges roughly 3.5–5.5, shifting spin-forbidden lines and introducing a few-percent error in fitted B.
• Octahedral only: tetrahedral dⁿ is handled by the electron-hole relationship (a tetrahedral dⁿ ≈ octahedral d¹⁰⁻ⁿ), with no g/u subscripts and much smaller Δ (Δ_tet ≈ (4/9)Δ_oct).
• d⁵ high-spin: every d–d transition is spin-forbidden, so [Mn(H₂O)₆]²⁺ is famously pale pink with ε ≈ 0.01–0.03; the diagram's near-horizontal excited-term lines explain the razor-sharp, field-independent bands.
• Band broadening & Jahn-Teller distortions blur assignments in real, non-ideal-O_h complexes.

Despite this, T-S analysis remains foundational: it underpins gemstone color (ruby vs emerald are both Cr³⁺, differing only in Δ₀), tunable Cr³⁺ and Ti³⁺ laser and NIR-phosphor design, spin-crossover materials, and the assignment of metalloprotein active-site spectra — all from two or three numbers on a spectrometer readout.

Ground terms, number of spin-allowed d–d bands, and whether a spin-crossover line appears, for octahedral d^n ions

d^n (example)	Free-ion ground term → octahedral ground state	Spin-allowed bands	Spin-crossover line?
d² — V³⁺	³F → ³T₁g	3	No
d³ — Cr³⁺	⁴F → ⁴A₂g	3	No
d⁴ — Mn³⁺, Cr²⁺	⁵D → ⁵Eg (HS) / ³T₁g (LS)	1 (HS)	Yes
d⁵ — Mn²⁺, Fe³⁺	⁶S → ⁶A₁g (HS) / ²T₂g (LS)	0 spin-allowed (HS)	Yes
d⁶ — Fe²⁺, Co³⁺	⁵D → ⁵T₂g (HS) / ¹A₁g (LS)	1 (HS)	Yes (≈ Δ₀/B ~ 20)
d⁸ — Ni²⁺	³F → ³A₂g	3	No