Tanabe–Sugano Diagram

A Tanabe–Sugano diagram is a quantitative electronic correlation diagram used in coordination chemistry, Crystal Field Theory, and Ligand Field Theory to predict and assign the UV–Visible–Near-IR electronic absorption spectra and magnetic ground states of octahedral (O_h) transition metal complexes containing two to eight d electrons (d²–d⁸). Published in 1954 by the Japanese physicists Yukito Tanabe and Satoru Sugano at the University of Tokyo, these diagrams solve the complete multi-electron Hamiltonian combining interelectronic Coulomb/exchange repulsion with the octahedral ligand-field potential across the entire continuum from the weak-field limit (free gaseous ion) to the strong-field covalent limit.

Unlike earlier Orgel diagrams—which are qualitative, restricted to high-spin complexes, and plot absolute energies relative to the barycenter—Tanabe–Sugano diagrams plot the dimensionless state energy E/B on the vertical axis against the dimensionless ligand-field strength Δ_o/B (or Dq/B) on the horizontal axis, using the lowest-energy electronic ground term as the horizontal baseline (E/B = 0) at every ligand-field strength. Because the ground state coincides with the x-axis, the vertical distance from the horizontal axis to any excited state curve of the same spin multiplicity gives the transition energy (hν/B) observed in an experimental UV–Vis spectrum, while configurations capable of spin crossover (d⁴, d⁵, d⁶, and d⁷) display a vertical discontinuity at the critical pairing threshold where the ground state changes spin multiplicity.

Interactive 3D Polyhedra: Compare octahedral (O_h) and tetrahedral (T_d) ligand fields inside the VSEPR Lab 3D simulator!

Free-Ion Russell–Saunders Terms and Racah Parameters (A, B, C)

For a single d¹ or d⁹ electron/hole configuration (such as [Ti(H₂O)₆]³⁺ or [Cu(H₂O)₆]²⁺), there is no d–d interelectronic repulsion variation between states, so the single ²D free-ion term splits into ²T_2g and ²E_g separated by Δ_o (modified by the Jahn–Teller effect). However, in a multi-electron d²–d⁸ ion, electrons repel each other via the two-electron Coulomb operator ∑_i<j e²/r_ij, which splits a single dⁿ configuration into multiple Russell–Saunders (LS) spectroscopic terms ^2S+1L (S, P, D, F, G, H, I).

As Giulio Racah demonstrated in 1942, the energy separations between all Russell–Saunders terms arising from any dⁿ configuration can be expressed as linear combinations of three radial Slater–Condon integrals known as the Racah interelectronic repulsion parameters:

• A: Shifts all terms of a given dⁿ configuration equally and cancels out when calculating energy differences between states.
• B: Controls the energy separation between terms of the same spin multiplicity (for example, in a d² or d⁸ ion, the separation between the ground ³F term and the excited ³P term equals 15B). In free first-row transition-metal ions, B₀ typically ranges from ~650 to 1,150 cm⁻¹.
• C: Controls the energy separation between terms of different spin multiplicities (spin-pairing energies). Across most transition-metal ions, the ratio C/B is approximately constant at ~4.0 to 4.7, allowing Tanabe and Sugano to construct a single universal diagram for each dⁿ configuration by fixing C/B (e.g., C/B = 4.42 for d⁶ Co³⁺/Fe²⁺) and dividing both axes by B.

The Nephelauxetic ("Cloud-Expanding") Effect (β = B_complex / B_free)

When a transition-metal ion is coordinated by six ligands in a complex, the experimentally fitted Racah parameter B_complex is smaller than its value B₀ in the free gaseous metal ion (by 10% to 65%). Discovered and named by Christian Klixbüll Jørgensen, this reduction in interelectronic repulsion is called the nephelauxetic effect (from the Greek for "cloud-expanding"), quantified by the nephelauxetic ratio β:

β = B_complex / B₀ < 1.0

As explained by Ligand Field Theory, covalent σ- and π-mixing of metal 3d orbitals with ligand orbitals delocalizes the metal d-electron cloud over a larger spatial volume encompassing the ligands (and positive ligand charge transferred to the metal screens the effective nuclear charge Z_eff, radially expanding the 3d orbitals). Because the d electrons occupy a larger volume, their average interelectronic distance r_ij increases and their mutual repulsion B drops. Ordering ligands by decreasing β yields the nephelauxetic series, which reflects increasing metal–ligand orbital covalency (and decrease in ligand electronegativity):

F⁻ (β ≈ 0.90) > H₂O > NH₃ > en > ox²⁻ > SCN⁻ > Cl⁻ > CN⁻ > Br⁻ > S²⁻ ≈ RS⁻ > I⁻ (β ≈ 0.40)

Configuration Interaction and Curvature of Same-Symmetry States

In an octahedral molecular point group (O_h), the free-ion F term (7-fold orbital degeneracy, L = 3) splits into A_2g + T_1g + T_2g, while the higher-lying P term (3-fold orbital degeneracy, L = 1) transforms without splitting as T_1g. Consequently, in d², d³, d⁷, and d⁸ complexes, there are two states of identical symmetry and spin multiplicity: ^2S+1T_1g(F) and ^2S+1T_1g(P).

By the quantum-mechanical von Neumann–Wigner non-crossing rule, electronic states possessing the same symmetry label and spin multiplicity mix via off-diagonal matrix elements of the ligand field (configuration interaction) and repel each other rather than crossing:

• In a d³ (Cr³⁺) or d⁸ (Ni²⁺) Tanabe–Sugano diagram, the ⁴T_2g(F) state (or ³T_2g(F)) increases linearly with slope +1 (E/B = Δ_o/B), so the lowest-energy absorption band ν₁ equals Δ_o. Meanwhile, the two ⁴T_1g(F) and ⁴T_1g(P) curves bend away from each other as Δ_o/B increases. Using the experimentally measured transition ratio ν₂/ν₁ from a UV–Vis spectrum, a chemist locates the matching vertical line on the x-axis (Δ_o/B) and solves for both Δ_o and the nephelauxetic Racah parameter B_complex.
• In a d² (V³⁺) or high-spin d⁷ (Co²⁺) diagram, the ground state itself is ³T_1g(F) (or ⁴T_1g(F)), which mixes with the upper T_1g(P) state and curves downward in absolute energy. Because the Tanabe–Sugano diagram resets the curved ground state to E/B = 0 at every point, all excited lines appear curved slightly upward at weak fields before becoming linear at strong fields.

Vertical Spin-Crossover Discontinuities in d⁴, d⁵, d⁶, and d⁷ Diagrams

In d², d³, and d⁸ octahedral complexes, the t_2g shell holds at most 3 electrons (or already holds 6 paired electrons in d⁸), so the ground state spin multiplicity remains unchanged regardless of how large Δ_o/B becomes. In contrast, for d⁴, d⁵, d⁶, and d⁷ complexes, a low-spin state born from a higher-lying free-ion term (such as ¹I in d⁶ or ²I in d⁵) plunges steeply downward as Δ_o/B increases because placing extra electrons into the lower t_2g orbital set gains −(2/5)Δ_o per electron over the +3/5Δ_o penalty of e_g*.

At a critical value of (Δ_o/B)_crit (approximately 27 for d⁴, 28 for d⁵, 20 for d⁶, and 22 for d⁷), the descending low-spin state crosses below the high-spin ground state and becomes the new E/B = 0 horizontal axis. A vertical dividing line on the Tanabe–Sugano diagram marks this spin-crossover boundary: to the left lies the weak-field (high-spin) regime, to the right lies the strong-field (low-spin) regime, and complexes poised right on the vertical line exhibit thermally or photochemically switchable spin crossover. Notice also that in the high-spin d⁶ region (left of Δ_o/B = 20, e.g., [Fe(H₂O)₆]²⁺), only one spin-allowed d–d transition exists (⁵T_2g → ⁵E_g), whereas in the low-spin d⁶ region (right of Δ_o/B = 20, e.g., [Co(NH₃)₆]³⁺), two prominent spin-allowed transitions appear (¹A_1g → ¹T_1g and ¹A_1g → ¹T_2g).

Spin-Forbidden Transitions and the d³ Ruby Laser (Cr³⁺:Al₂O₃)

Electronic transitions in octahedral complexes are governed by two quantum-mechanical selection rules:

1. The Laporte (Parity) Selection Rule (Δl = ±1, g ↔ u): Because all five d orbitals have gerade (g, centrosymmetric) inversion symmetry, electric-dipole d–d transitions (g → g) are strictly forbidden in an ideal centrosymmetric octahedron. They become weakly allowed (ε ≈ 5–100 M⁻¹ cm⁻¹) via vibronic coupling with odd-parity (t_1u, t_2u) asymmetric stretching and bending vibrations that momentarily break the center of inversion.
2. The Spin Selection Rule (ΔS = 0): Photon absorption cannot flip an electron's spin in the absence of spin–orbit coupling, so transitions between states of different spin multiplicity are ~10² to 10³ times weaker (ε < 0.1 M⁻¹ cm⁻¹).

On a Tanabe–Sugano diagram, excited states whose curves run horizontally (parallel to the ground-state x-axis) belong to the exact same t_2g^xe_g^y strong-field configuration as the ground state (an intra-configurational spin flip where only spin pairing changes, not orbital occupancy). Because both states have the same equilibrium metal–ligand bond length (zero horizontal displacement on a Franck–Condon diagram), transitions to horizontal lines appear as razor-sharp, narrow lines rather than broad vibronic bands.

The most famous example occurs in the d³ Tanabe–Sugano diagram for ruby (Cr³⁺ doped into α-Al₂O₃, Δ_o/B ≈ 25). Pumping ruby with blue and green light excites the broad, spin-allowed bands ⁴A_2g → ⁴T_2g (550 nm) and ⁴A_2g → ⁴T_1g(F) (400 nm). The excited complex rapidly undergoes non-radiative intersystem crossing down to the flat, horizontal ²E_g state (t_2g³ with one flipped spin, lying at E/B ≈ 21). Because the radiative return ²E_g → ⁴A_2g is spin-forbidden (ΔS = −1), electrons accumulate in the metastable ²E_g level (lifetime τ ≈ 3 ms), creating a population inversion that emits coherent, monochromatic deep-red laser light at 694.3 nm (the ruby R₁ line).

Why Jahn–Teller Distortions Split Single Tanabe–Sugano Bands into Doublets

According to the O_h Tanabe–Sugano diagram for a high-spin d⁶ complex such as [Fe(H₂O)₆]²⁺ (⁵T_2g ground state) or a d¹ complex such as [Ti(H₂O)₆]³⁺ (²T_2g ground state), the UV–Vis spectrum should consist of a single symmetric absorption peak corresponding to the transition T_2g → E_g. In experimental aqueous spectra, however, both [Ti(H₂O)₆]³⁺ (peak at 20,300 cm⁻¹ with a distinct shoulder at 17,400 cm⁻¹) and [Fe(H₂O)₆]²⁺ (split doublet at 10,400 and 8,300 cm⁻¹) exhibit pronounced band splitting or an asymmetric double-humped profile.

This doublet arises from the excited-state Jahn–Teller effect: when the photon promotes an electron from the weakly non-bonding t_2g shell into the strongly σ-antibonding e_gESCAST doublet, the electronically excited state (²E_g in Ti³⁺ or ⁵E__g in Fe²⁺) possesses an asymmetric e_gESCAST occupation and immediately splits into two tetragonal (D_4h) potential energy surfaces (²B_1g and ²A__1g, separated by 4δ₁ ≈ 2,000–3,000 cm⁻¹). Vertical Franck–Condon excitation from the nearly octahedral ground state intersects both sheets of the Mexican hat potential, splitting the single O_h Tanabe–Sugano transition into two resolved absorption components.

Why is there no Tanabe–Sugano diagram for d¹, d⁹, d⁰, or d¹⁰ complexes?

Complexes with d⁰ (such as Ti⁴⁺, V⁵⁺, or MnO₄⁻) have no d electrons, and d¹⁰ complexes (such as Zn²⁺, Cu⁺, or Ag⁺) have a completely filled d subshell, making d–d transitions impossible (their colors, if any, arise purely from LMCT or MLCT charge-transfer bands). Meanwhile, d¹ (one d electron) and d⁹ (one d hole) possess only a single free-ion Russell–Saunders D term (²D) with no multi-electron d–d repulsion differences between states; in an octahedral field, ²D splits linearly into T_2g and E_g separated by Δ_o without requiring Racah parameter B scaling or configuration interaction.

Why is aqueous [Mn(H₂O)₆]²⁺ almost colorless (very pale pink) compared to other transition-metal ions?

Aqueous Mn²⁺ has a high-spin d⁵ configuration (t_2g³e_g²) lying to the left of the Δ_o/B = 28 discontinuity on the d⁵ Tanabe–Sugano diagram. Its ground state is the orbital singlet, spin sextet ⁶A_1g(S), in which all five d orbitals contain one electron with parallel spins (S = 5/2). Promoting any electron between t_2g and e_g requires pairing two electrons in the same orbital with antiparallel spins, reducing total spin to S = 3/2 (quartet excited states ⁴T_1g, ⁴T_2g, ⁴A_1g, ⁴E_g). Because there are no other sextet states anywhere on the d⁵ Tanabe–Sugano diagram, every d–d transition in high-spin Mn²⁺ (and high-spin Fe³⁺) is doubly forbidden (both Laporte-forbidden and spin-forbidden, ΔS = −1), resulting in tiny molar absorptivities (ε ≈ 0.01–0.04 M⁻¹ cm⁻¹).

How do you use a Tanabe–Sugano diagram to applicability-test tetrahedral complexes?

Because tetrahedral (T_d) splitting inverts the orbital ordering of an octahedron (e below t₂), a tetrahedral dⁿ complex has the same qualitative hole/electron term splitting as an octahedral d¹⁰⁻ⁿ complex. Thus, within the high-spin weak-field region (left of any spin-pairing discontinuity, since tetrahedral complexes have small Δ_t ≈ (4/9)Δ_o and are almost universally high-spin), the Tanabe–Sugano diagram for octahedral d¹⁰⁻ⁿ can be used directly to assign the spectrum of a tetrahedral dⁿ complex (for example, using the octahedral d³ diagram to analyze tetrahedral d⁷ [CoCl₄]²⁻).

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2. Tanabe, Y., & Sugano, S. (1954). On the absorption spectra of complex ions. II. Journal of the Physical Society of Japan, 9(5), 766–779. https://doi.org/10.1143/JPSJ.9.766
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