What it does
A Tanabe–Sugano diagram shows how the electronic states of a transition-metal ion move apart as the ligand field grows. The x axis is the field strength Δ (also called 10Dq) divided by the Racah parameter B; the y axis is the energy of each state above the ground state, also divided by B. Chemists use these diagrams to assign the bands of a d–d absorption spectrum, to work out Δ and B for a complex, and to see where an ion switches from high spin to low spin.
Printed diagrams are drawn for one C/B ratio and are hard to read accurately. This page calculates the whole diagram for you:
- From first principles. Every way of placing n electrons in the five d orbitals (a Slater determinant) is a starting state. The repulsion between the electrons is set by B and C, the octahedral field puts the t2g orbitals at −4Dq and the eg orbitals at +6Dq, and the energy matrix is solved for each spin. The states are named by their octahedral symmetry (A₁g, A₂g, Eg, T₁g, T₂g), their spin (the raised number, 2S + 1) and the free-ion term they come from, such as ⁴T₁g(F).
- Any C/B. The textbook diagrams use about 4.5; you can see how the picture changes for your own value.
- High spin to low spin. For d⁴ to d⁷ the ground state changes at a critical Δ/B; the page finds it exactly and marks it. At C/B = 4.5 it is about 18.9 for d⁶, 21.2 for d⁷, 26.8 for d⁴ and 28.4 for d⁵.
- Read any point. Tap or click the diagram, or type a Δ/B, to list every state with its energy in units of B, and, given B, in cm⁻¹ and nm.
- Δ and B from a spectrum for d³ (Cr³⁺) and d⁸ (Ni²⁺) complexes, with the third band predicted and the nephelauxetic ratio β if you know the free-ion B.
- Free-ion terms (Δ = 0) with their energies, so you can see where each line starts.
How to use it
- Choose the number of d electrons. The examples (Cr³⁺ for d³ and so on) are common ions with that count.
- Leave C/B at 4.5 unless you have a better value for your ion.
- Look at the diagram. Coloured lines have the same spin as the ground state: these are the spin-allowed transitions that give the strong bands. Grey dashed lines are spin-forbidden and give weak, sharp bands. Choose “spin-allowed only” to hide the grey ones.
- Tap the diagram where your complex sits, or type Δ/B. The table lists the states at that point; a tick marks the spin-allowed ones. Type your B to get wavenumbers and wavelengths.
- For a d³ or d⁸ complex, type the two lowest band positions under “Δ and B from a spectrum”, then press “Read the diagram at this fit”.
Worked example
Choose d³ with C/B = 4.5 and read at Δ/B = 20. The ground state is ⁴A₂g(F). The table shows ⁴T₂g(F) at 20.00 B (this band is always exactly Δ for d³), then the spin-forbidden ²Eg(G) at 20.71 B and ²T₁g(G) at 21.63 B, and the next spin-allowed state ⁴T₁g(F) at 29.36 B. With the example B of 700 cm⁻¹ these are 14000, 14500, 15141 and 20552 cm⁻¹.
Now the fit: made-up bands at 17000 and 24000 cm⁻¹ give Δ = 17000 cm⁻¹ (203.4 kJ/mol), B = 707.1 cm⁻¹ and Δ/B = 24.04, with the third band, ⁴A₂g → ⁴T₁g(P), predicted at 37606 cm⁻¹ (266 nm). Typing a free-ion B (here a made-up 900 cm⁻¹) gives the nephelauxetic ratio β = 0.786.
Tips
- Only the spin-allowed bands are strong enough to fit reliably. For d³ and d⁸ the lowest band is Δ itself.
- For d⁴ to d⁷ the diagram breaks where the ground state changes: to the right of the break, energies are measured from the new low-spin ground state, so every line jumps. That is how the printed diagrams are drawn too.
- d⁴ and d⁶ are not mirror images in this diagram (their high-spin ground states are ⁵Eg and ⁵T₂g), and neither are d³ and d⁷, even though they share free-ion terms.
- C barely matters for the spin-allowed bands of d³ and d⁸ (they depend only on B) but moves the spin-forbidden states and the spin crossover a lot.
FAQ
Why are the numbers slightly different from my textbook’s diagram? Printed diagrams use slightly different C/B values (often 4.5 to 4.8) and are read by eye. The values here are exact for the C/B you set.
Does it include spin–orbit coupling or distortion? No. It is the pure octahedral model with electron repulsion, as in the original Tanabe–Sugano diagrams. Jahn–Teller distortion and spin–orbit coupling split some states further in real spectra.
What about d¹ and d⁹? They have a single excited state at exactly Δ, so a diagram adds nothing: the band position is Δ.
Why is the third band often missing? It sits in the ultraviolet, where much stronger charge-transfer bands hide it.
Where the numbers come from
- Method: Tanabe and Sugano (1954, J. Phys. Soc. Japan 9, 753 and 766); Slater–Condon rules and Racah parameters B = F2 − 5F4, C = 35F4 (Condon and Shortley, The Theory of Atomic Spectra; Griffith, The Theory of Transition-Metal Ions). fibub's code is its own and builds the diagram from the Slater determinants; it is checked in the tests against an independent NumPy calculation and the textbook free-ion term energies and d³/d⁸ band formulas.
- The ion list and C/B = 4.5 are general textbook facts. The example bands (17000 and 24000 cm⁻¹) and B = 700 cm⁻¹ are made up. Nothing you type is sent or stored.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.