Tanabe-Sugano Diagrams and Electronic Spectra — How to Read Them

By Aniket Bhardwaj · 17 September 2026 · Advanced Chemistry

Crystal field theory explains why a complex is coloured. It does not explain why [Ni(H₂O)₆]²⁺ shows three absorption bands rather than one, or why high-spin Mn(II) salts are almost colourless. For that you need the interaction between the ligand field and electron–electron repulsion, and the standard tool for handling both together is the Tanabe-Sugano diagram. This is core CSIR-NET and GATE inorganic material, and the good news is that for two of the d configurations you can extract the numbers by hand, without ever reading a curve off a graph.

What the diagram plots

Since this page carries no figure, here is the layout in words. A Tanabe-Sugano diagram is a plot of energy against ligand field strength, but both axes are divided by the Racah parameter B:

Feature	What it is	Why it is done that way
x-axis	Δ_o/B (often written 10Dq/B)	Dimensionless, so one diagram serves every metal ion with that dⁿ count
y-axis	E/B, the energy of each term above the ground state	Also dimensionless; energies are read as multiples of B
The horizontal line along y = 0	The ground state	Every transition is measured from it, so it is forced flat
Each curved line	An excited term (e.g. ³T_2g, ³T_1g)	Its height above the axis is the transition energy in units of B
Vertical line partway across (d⁴–d⁷ only)	The high-spin / low-spin crossover	To the left the ground state is high spin; to the right it changes, and the whole diagram restarts
Curves that run almost flat	Terms of a different spin multiplicity from the ground state	Their energies depend mainly on electron repulsion, hardly on the ligand field — they give sharp, narrow bands

The extra ingredient compared with a plain crystal field treatment is the pair of Racah parameters, B and C, which quantify the repulsion between d electrons. B is the one you calculate from a spectrum; C rarely appears in exam numericals.

Free-ion terms and how many bands to expect

Before the ligand field acts, the dⁿ electrons give free-ion terms. Only the ground term and the next term of the same spin multiplicity matter for the strong bands:

dⁿ	Free-ion ground term	Splits in O_h into	Spin-allowed bands expected
d¹, d⁹	²D	T_2g + E_g	1
d², d⁷ (HS)	³F / ⁴F	T_1g + T_2g + A_2g	3
d³, d⁸	⁴F / ³F	A_2g + T_2g + T_1g	3
d⁴, d⁶ (HS)	⁵D	E_g + T_2g	1
d⁵ (HS)	⁶S	A_1g (does not split)	0

The last row explains a fact every student has seen in the laboratory: high-spin d⁵ ions such as Mn²⁺ have no spin-allowed d–d transition at all, because there is no other sextet term. Their pale pink colour comes only from very weak spin-forbidden bands.

Selection rules — why d–d bands are weak

Spin rule: ΔS = 0. A transition that flips a spin is forbidden and therefore extremely weak.
Laporte rule: in a centrosymmetric molecule, g → g transitions are forbidden. All octahedral d–d transitions are g → g, so all of them are Laporte-forbidden.

They are seen anyway because unsymmetrical vibrations momentarily destroy the centre of symmetry — vibronic coupling. That is why octahedral d–d bands are weak and broad. A tetrahedral complex has no centre of symmetry to begin with, so its bands are markedly more intense, which is why tetrahedral cobalt(II) species are so much more deeply coloured than their octahedral counterparts. Charge-transfer bands obey neither restriction and are stronger again by orders of magnitude — that is the origin of the intense colour of permanganate, which is d⁰ and has no d–d transition available at all.

The shortcut for d³ and d⁸

For these two configurations the ground state is A_2g, and two exact results fall out that let you skip the graph completely:

ν̄₁ = Δ_o 15B = ν̄₂ + ν̄₃ − 3ν̄₁

ν̄₁, ν̄₂ and ν̄₃ are the three band positions in wavenumbers (cm⁻¹), in increasing energy. The first relation holds because the lowest transition is A_2g → T_2g, which is exactly a t_2g → e_g promotion. The second comes from the sum of the two T_1g energies, which the underlying 2 × 2 secular determinant fixes at 15B + 3Δ.

Worked example 1 — a d⁸ spectrum

Data. An octahedral nickel(II) aqua complex shows bands at about 8500, 13 800 and 25 300 cm⁻¹. (Values quoted in standard texts vary by a few hundred wavenumbers; use one consistent set.)

Step 1 — the ligand field splitting.
Ni²⁺ is d⁸, so ν̄₁ = Δ_o = 8500 cm⁻¹.

Step 2 — the Racah parameter.
15B = ν̄₂ + ν̄₃ − 3ν̄₁
= 13 800 + 25 300 − (3 × 8500)
= 39 100 − 25 500 = 13 600 cm⁻¹
B = 13 600 ÷ 15 = 907 cm⁻¹

Step 3 — the nephelauxetic ratio. The free Ni²⁺ ion has B₀ ≈ 1041 cm⁻¹.
β = B ÷ B₀ = 907 ÷ 1041 = 0.87

β below 1 means the d electrons repel each other less in the complex than in the free ion, because their cloud has expanded onto the ligands. That expansion is the nephelauxetic ("cloud-expanding") effect, and it is direct evidence of covalency in the metal–ligand bond. β ≈ 0.87 for water is a modest expansion, as expected for a hard, mainly ionic ligand.

Step 4 — sanity check against the colour. Convert the bands to wavelength with λ = 1/ν̄:
1 ÷ 8500 cm⁻¹ = 1.176 × 10⁻⁴ cm = 1176 nm (near infrared)
1 ÷ 13 800 = 7.246 × 10⁻⁵ cm = 725 nm (red)
1 ÷ 25 300 = 3.953 × 10⁻⁵ cm = 395 nm (violet)
Red and violet absorbed, green transmitted — which is exactly the colour of nickel(II) in water.

Step 5 — Δ in familiar energy units. Using 1 cm⁻¹ = 11.96 J mol⁻¹:
Δ_o = 8500 × 11.96 = 101 660 J mol⁻¹ ≈ 102 kJ mol⁻¹, comparable to a weak chemical bond.

Worked example 2 — a d³ spectrum

Data. A chromium(III) aqua complex shows bands near 17 400, 24 600 and 37 800 cm⁻¹. The highest band is often partly buried under charge transfer, which is why the third value is the least reliable one in any such data set.

Step 1: Δ_o = ν̄₁ = 17 400 cm⁻¹. Much larger than the nickel case, as expected for a 3+ ion — higher charge pulls the ligands closer and raises Δ.

Step 2: 15B = 24 600 + 37 800 − (3 × 17 400)
= 62 400 − 52 200 = 10 200 cm⁻¹
B = 10 200 ÷ 15 = 680 cm⁻¹

Step 3: free Cr³⁺ has B₀ ≈ 918 cm⁻¹, so β = 680 ÷ 918 = 0.74. A larger cloud expansion than in the nickel complex, consistent with the more highly charged metal polarising the ligands more strongly.

Cross-check. Both worked examples give B values close to those quoted in the literature for these ions (roughly 900 cm⁻¹ and 700 cm⁻¹ respectively), which is the reassurance that the two relations were applied correctly.

Why the sharp lines matter

On a d³ diagram, the ²E_g term runs almost horizontally: its energy above the ⁴A_2g ground state barely changes as Δ changes. Any transition to a flat line gives a sharp absorption or emission band, because small distortions of the complex do not shift its energy. Steeply sloping lines give broad bands for the opposite reason. This is why ruby — Cr³⁺ substituted into Al₂O₃ — emits its famous narrow red line, and it is a favourite conceptual question in NET papers: predict whether a band will be sharp or broad from the slope of its Tanabe-Sugano line.

When you cannot avoid the diagram

The two shortcut relations work only for d³ and d⁸. For d², d⁶ and d⁷ the ground state is a T term that mixes with an excited T term of the same multiplicity, so ν̄₁ is not equal to Δ_o and the numbers must come from the diagram or from the full secular determinant. The standard graphical route is: compute the ratio ν̄₂/ν̄₁ from the experiment, find the value of Δ/B on the x-axis where the diagram gives that same ratio, then read E/B for either band and solve for B.

Common mistakes that cost marks

Using ν̄₁ = Δ for every configuration. It is true only for d³ and d⁸ (A_2g ground state). Applying it to d² or d⁷ gives a wrong Δ and then a wrong B.
Mixing units. Every relation here needs wavenumbers. Convert nm to cm⁻¹ before you start, not halfway through.
Forgetting the axes are divided by B. A y-axis reading of 15 means 15B, not 15 cm⁻¹.
Reading the wrong side of the vertical line. For d⁴–d⁷ the diagram has two regions with different ground states. Decide high spin or low spin first, from the ligand.
Counting spin-forbidden bands as the main bands. A d⁵ high-spin complex has zero spin-allowed d–d transitions; any bands seen are the weak spin-forbidden ones.
Reporting β greater than 1. B in a complex is essentially always smaller than the free-ion value. A β above 1 means an arithmetic slip or a misassigned band.
Assigning a very intense band as d–d. If the molar absorptivity runs into the thousands, it is charge transfer, not a d–d transition.

Where this appears in exams

Exam	Typical use
CSIR-NET	Calculating Δ and B from band positions; term symbols; sharp vs broad band reasoning
GATE (Chemistry)	Numerical answer type on Δ_o and the Racah parameter; band-count questions
IIT-JAM	Colour and selection rules at a qualitative level; why Mn(II) is nearly colourless
CUET-PG / M.Sc. entrance	Free-ion terms, splitting in O_h, spectrochemical series

Always confirm the depth expected from the current official syllabus and notification for your paper.

Convert between the units this topic lives in. Spectra are reported in nm, Tanabe-Sugano work is done in cm⁻¹, and bond-energy comparisons want kJ mol⁻¹. The Photon Energy / de Broglie tool takes a wavelength and returns the photon energy, which is the conversion step in the middle of every one of these problems.

Open the Photon Energy & de Broglie Calculator →

Preparing for CSIR-NET, GATE, IIT-JAM or CUET-PG? ABC Chemistry runs dedicated competitive-exam batches at the coaching centre and as live online classes for students across India — details at abcchemistry.in.