Chemistry learning job: learn Tanabe–Sugano diagrams as quantitative maps of how d-electron term energies change with ligand-field strength, then use them to interpret transition-metal spectra without confusing crystal-field splitting, interelectronic repulsion, selection rules or observed band intensity.
Wait, What? A Coloured Transition-Metal Complex Is Not Showing You “One d Electron Jump”
In a one-electron orbital picture, octahedral d orbitals split into t2g and eg sets separated by Δo = 10Dq. That picture is useful, but multi-electron transition-metal ions contain electron–electron repulsion too. Their real electronic states are organised into spectroscopic terms such as 4A2g, 4T2g and 4T1g, not merely “t2g to eg”.
Tanabe–Sugano diagrams show how those many-electron states move as ligand-field strength grows relative to interelectronic repulsion.
horizontal axis: Δ/B (often written 10Dq/B)
vertical axis: E/B
B is a Racah parameter describing part of the electron–electron repulsion within the d shell. Scaling both axes by B lets one diagram describe a whole dn family rather than one particular complex.
The Direct Answer
Learn a Tanabe–Sugano diagram in four steps: determine the metal oxidation state and d count; choose the correct dn diagram; locate the weak-field or strong-field region consistent with the spin state; then compare observed absorption-band energy ratios with the spin-allowed excited-state lines. From that match you can estimate Δo and B and test assignments of d–d transitions. The diagram improves on a simple crystal-field picture because it includes many-electron term energies and high-spin/low-spin changes. It still does not predict every spectral feature: low symmetry, spin–orbit coupling, Jahn–Teller distortion, charge-transfer bands, vibronic coupling and covalency can move or intensify bands beyond the idealised octahedral model.
Beginner → Secondary → JC → University → Professional
- Beginner: ligands change the energies of metal d orbitals, so transition-metal complexes can absorb visible light.
- Secondary Chemistry: connect colour to electronic structure rather than treating colour as a memorised property.
- JC Chemistry: connect oxidation state, d-electron count, ligand strength, high-spin/low-spin ideas and absorption of light.
- Undergraduate Chemistry: move from orbital occupancy to free-ion terms, Racah parameters, selection rules and quantitative spectral assignment.
- Professional / Research: integrate Tanabe–Sugano analysis with magnetic data, EPR, Mössbauer/XAS where relevant, low-symmetry ligand-field models and computational spectroscopy.
Stage 1 — Get the d Count Right Before Opening a Diagram
A Tanabe–Sugano diagram is chosen by the metal’s d-electron configuration, not simply by element name. For a first-row transition metal, find the oxidation state, then subtract that oxidation state from the neutral-atom d/s valence count in the usual ionic bookkeeping model.
Examples:
- Cr3+ → d3
- Fe2+ → d6
- Co2+ → d7
- Ni2+ → d8
If the d count is wrong, the entire spectral assignment is wrong before any graph reading begins.
Stage 2 — Why d¹ and d⁹ Usually Do Not Need a Full Tanabe–Sugano Diagram
With one d electron, there is no d–d interelectronic repulsion to resolve. An octahedral d1 ion has a simple 2T2g ground state and 2Eg excited state. The d9 case is the corresponding one-hole analogue, though Jahn–Teller distortion often complicates the spectrum. A d10 ion has no ordinary d–d excitation at all.
The full Tanabe–Sugano machinery becomes most valuable for d2 through d8, where several electrons create multiple terms and substantial electron–electron repulsion.
Stage 3 — From Orbitals to Terms
A free transition-metal ion has many possible microstates. Russell–Saunders coupling groups these into terms labelled by total spin and orbital angular momentum, such as 3F or 4F. When an octahedral ligand field is applied, those free-ion terms split into states carrying A, E or T symmetry labels.
The superscript gives the spin multiplicity, 2S + 1. A quartet therefore has multiplicity 4, a triplet 3, a doublet 2 and a singlet 1.
Stage 4 — What the Racah Parameter B Means
B measures a major component of interelectronic repulsion among d electrons. Free ions generally have larger B than complexes because metal–ligand covalency spreads d-electron density onto ligands and reduces effective d–d repulsion. This reduction is part of the nephelauxetic effect.
That is why B is not just a graph-scaling trick. It contains chemical information about electron repulsion and covalency.
Stage 5 — Why the Axes Are Ratios
The x-axis compares ligand-field splitting with interelectronic repulsion: Δ/B. The y-axis expresses electronic-state energy in the same scaled units: E/B.
Because the axes are dimensionless ratios, one d3 diagram can be applied to many different d3 complexes. Once B is determined, the scaled result can be converted back into energy or wavenumber units.
Stage 6 — Weak Field and Strong Field Are Competing Energies
At low Δ/B, keeping electrons unpaired is often energetically favourable. At high Δ/B, occupying lower-energy t2g orbitals can justify additional pairing. For octahedral d4–d7 configurations, the ground state can therefore change from high spin to low spin as ligand-field strength rises.
Do not reduce this to “Δ bigger than pairing energy” without recognising that the real many-electron energies involve Racah parameters and term mixing.
Stage 7 — Reading Spin-Allowed Transitions
The approximate spin-selection rule is:
ΔS = 0
Transitions between states of the same spin multiplicity are spin allowed. Those changing multiplicity are spin forbidden and are usually much weaker, although spin–orbit coupling can lend them intensity.
In centrosymmetric octahedral complexes, d–d transitions are also Laporte forbidden because they are g → g. Vibronic coupling and symmetry breaking relax that rule, which is why octahedral d–d bands are weak rather than literally absent.
Stage 8 — The d³ Case Shows the Method Cleanly
For a typical octahedral d3 complex such as Cr3+, the ground state is 4A2g. Several quartet excited states are spin allowed. If two or three absorption bands can be assigned confidently, their energy ratios locate a position on the d3 Tanabe–Sugano diagram.
From that position, Δo/B is read. Matching one measured transition energy then gives B, and Δo follows.
Stage 9 — Why Energy Ratios Matter More Than a Single Peak
A single absorption band rarely fixes both Δo and B. Ratios between two assigned transitions remove the absolute energy scale and can be compared directly with E/B ratios from the diagram.
This is a general analytical idea: use a dimensionless relationship to locate the model state first, then restore absolute units.
Stage 10 — Why Lines Bend and Avoid Crossing
Electronic states with the same symmetry can mix. When their energies approach, they repel rather than crossing freely. This creates characteristic curvature in Tanabe–Sugano diagrams. The bending is physical information about state interaction, not poor drawing.
Stage 11 — Colour Is Not the Same as “the Peak You See”
The observed colour is determined by the wavelengths transmitted or reflected after all absorptions are considered. A broad visible band may contain overlapping electronic transitions. Other bands can lie in the near-infrared or ultraviolet. Charge-transfer transitions can be much more intense than d–d transitions and can dominate the colour entirely.
Stage 12 — Tanabe–Sugano vs Orgel
Orgel diagrams are useful qualitative maps for weak-field cases and show which terms move up or down. Tanabe–Sugano diagrams are more quantitative and include both weak- and strong-field regimes. That makes them much more useful for estimating ligand-field parameters and for systems where spin-state changes occur.
Observation vs Inference
Observation: a complex shows absorption maxima at measured wavenumbers with particular intensities.
Inference: a set of those bands may correspond to transitions predicted for a chosen dn term scheme.
Stronger inference: if the same Δo and B reproduce several transition energies and agree with magnetic/spin-state evidence, the assignment becomes much more persuasive.
How We Know
- UV–visible–NIR spectroscopy provides transition energies and band intensities.
- Magnetic susceptibility constrains the number of unpaired electrons and therefore likely spin state.
- EPR can identify paramagnetic electronic states in suitable systems.
- Structural data test whether octahedral symmetry is a reasonable starting approximation.
- Racah-parameter fitting checks whether several bands can be represented by one internally consistent ligand-field model.
- Complementary spectroscopy can distinguish oxidation state, covalency or charge-transfer character.
Competing Explanations
A new absorption band could be a d–d transition, ligand-to-metal charge transfer, metal-to-ligand charge transfer, ligand-centred transition, vibronic sideband or impurity. A Tanabe–Sugano assignment is only credible if energy, intensity, spin state and chemical composition all agree.
Misconceptions Worth Hunting
- “The y-axis is wavelength.” It is scaled energy, E/B.
- “The x-axis is simply ligand identity.” It is ligand-field strength relative to B.
- “Every visible band is a d–d transition.” Charge-transfer bands can dominate.
- “Spin-forbidden means impossible.” Such transitions can gain weak intensity through spin–orbit coupling.
- “Laporte-forbidden means invisible.” Vibronic coupling relaxes the rule.
- “B is the same for every ion.” It varies with ion and covalency.
- “A Tanabe–Sugano diagram predicts exact spectra for any geometry.” Classical diagrams assume idealised symmetry, usually octahedral.
- “d¹ and d⁹ require the full diagram.” Their electronic-state structure is much simpler.
Counterexample Test
A Cu(II) d9 complex may show a broad asymmetric band. Trying to force that spectrum through a d2–d8-style Tanabe–Sugano fitting exercise is the wrong model. Strong Jahn–Teller distortion and low symmetry split and mix the simple orbital picture in ways better treated explicitly.
Model Limits
Standard Tanabe–Sugano diagrams assume an idealised ligand field and a chosen C/B ratio. Real complexes can have low symmetry, strong spin–orbit coupling, covalent metal–ligand mixing, configuration interaction and multiple structural forms. Broad bands make peak positions uncertain. A fitted Δo is therefore a model parameter inferred from spectroscopy, not a directly photographed energy gap.
Transfer Checks
- A complex is Fe(II). Which base diagram is relevant? d6.
- Two bands have the correct energy ratio for one d3 position but the sample is diamagnetic. Is the assignment automatically secure? No.
- A band is extremely intense in an octahedral complex. Should you immediately call it d–d? No; test charge-transfer explanations.
- A measured B is lower than the corresponding free-ion value. Can increased covalency contribute? Yes.
- A d6 complex moves from weak to strong field. Can the ground-state spin multiplicity change? Yes.
Independent Reasoning Check
Before fitting a spectrum, predict the oxidation state, d count, geometry and likely spin state from chemistry independent of the spectrum. Then ask whether the spectral assignment agrees. If you choose the spin state only because it makes the graph fit, the reasoning has become circular.
Practical Interpretation
- establish oxidation state and d count;
- check whether octahedral symmetry is a reasonable approximation;
- use magnetic or other independent data to constrain spin state;
- identify plausible spin-allowed bands;
- compare energy ratios with the correct dn diagram;
- fit Δo and B to more than one band;
- inspect residual bands for charge transfer, forbidden transitions or low-symmetry effects;
- report uncertainty rather than pretending broad peaks are exact numbers.
Connect This Chemistry
- The Jahn–Teller Effect — understand why electronically degenerate complexes can distort away from the ideal symmetry assumed by simple ligand-field diagrams.
- Hard and Soft Acids and Bases — a separate chemical model for preference and bonding that should not be confused with ligand-field spectral splitting.
Research Foundations
Tanabe and Sugano’s 1954 treatment combined crystal-field theory with Racah’s description of interelectronic repulsion to organise absorption spectra of complex ions. Modern inorganic spectroscopy retains the diagrams as a powerful teaching and fitting framework, while contemporary interpretation supplements them with low-symmetry ligand-field calculations, magnetic measurements, spin–orbit coupling and computational spectroscopy.
The Quiet Ending
The beginner asks, “Why is this complex coloured?”
The developing chemist asks, “Which d orbitals and spin state are involved?”
The advanced chemist asks, “Which many-electron terms explain these absorption bands?”
And the professional asks: does one physically consistent ligand-field model explain the spectrum, spin state and structure together — and where does that model stop being adequate?
