Introducing Tanabe–Sugano Diagrams

Introduction

Orgel diagrams show which weak-field transitions are plausible, but they are not calibrated tools for extracting both field splitting and interelectronic repulsion. Tanabe–Sugano diagrams add a dimensionless energy scale and include states of multiple spin multiplicities. They can represent high- and low-spin regions when those exist. The key to reading one is to treat each curve as a transition energy relative to the ground state at that particular field strength .

Core explanation

The horizontal axis is Δₒ/B, often written 10Dq/B, where Δₒ is octahedral splitting and B is a Racah repulsion parameter. The vertical axis is E/B for an excited term measured from the ground term at the same horizontal position. Both coordinates are dimensionless because numerator and denominator must use the same energy units, usually cm⁻¹. A ground-state curve is therefore drawn along E/B=0 even though the absolute energy of that state may change as Δₒ grows. This is a moving zero, not a claim that the ground state has zero physical energy in every environment.

Each d count has its own diagram: a d² curve set cannot be used to analyse d³ simply because both ions are octahedral. The curves come from many-electron term energies involving a ligand field and electron–electron repulsion. At Δₒ/B=0, the left edge correlates with free-ion LS terms. As the field grows, free-ion terms split into O h states and equal-symmetry states may mix, bending their curves. Heavy or solid lines in some diagrams mark spin-allowed transitions from the ground state; dashed lines may identify spin-forbidden candidates. Line styling differs between publications, so always read the legend rather than assuming a convention.

To predict a transition, choose a horizontal position, draw an imaginary vertical line and read the ordinate where it meets an excited-state curve. Because the ground baseline is zero, that ordinate is directly the transition energy divided by B. Multiplying by the complex's B gives an energy in cm⁻¹. Comparing ratios of observed bands can locate Δₒ/B without initially knowing B. Once a matching x-coordinate is found, one observed energy gives B and then Δₒ=xB. The result depends on correct spectral assignments and on the diagram's assumptions about parameters such as C/B.

Some diagrams, notably d⁴–d⁷, show a high-spin to low-spin ground-state change. A vertical separator often marks where the identity and multiplicity of the ground state switch. Excitation energies are measured from the lowest state on each side, so plotted curves can show apparent discontinuities or change labels at the boundary. This is not a sudden physical discontinuity in every individual electronic Hamiltonian eigenvalue; it reflects re-referencing to a different ground state and the competition of states with different spin. For d¹, d², d³, d⁸ and d⁹ in the usual octahedral diagrams, the same kind of high-/low-spin switch is absent.

The diagram contains more information than a simple orbital splitting chart. A d³ spectrum can have three spin-allowed bands from ^4A₂g to different quartet targets; the second and third positions are sensitive to electron repulsion and state mixing, not just Δₒ. Spin-forbidden target curves can also explain weak peaks. Even so, the diagram is a model: covalency, low symmetry, spin–orbit coupling, charge-transfer states and broad overlapping bands can limit a literal reading.

Step-by-step reasoning

Determine the metal d count and choose its octahedral diagram. Confirm whether the complex is high or low spin, using magnetic or chemical evidence when possible. Convert observed wavelengths to wavenumbers. Tentatively assign bands to curves of the correct spin multiplicity. Find a horizontal position where ratios of plotted ordinates match observed energy ratios. Read E/B, calculate B from one band, and multiply B by the horizontal coordinate to obtain Δₒ. Check other bands and any weak spin-forbidden features.

Visual explanation

Sketch a plot with x=Δₒ/B and y=E/B. Draw the ground state as the x-axis and three excited curves rising or bending above it. At one selected x draw a vertical guideline through all curves. Mark the ordinate y₁ and write E₁=y₁B; mark x and write Δₒ=xB. Near a crossing of two equal-symmetry curves draw a gap rather than an X to represent avoided crossing.

Real-world analogy

A terrain chart may show each hill's height above the local valley rather than above sea level. The valley can rise or fall along the route while remaining the plotted zero. A Tanabe–Sugano baseline similarly sets the ground state to zero separately at each field strength, making upward distances direct excitation energies.

Real-world example

An octahedral Cr³⁺ complex is d³. Its Tanabe–Sugano diagram starts from a ^4A₂g ground state and displays several quartet excited states plus weaker-accessible doublets. Observing two broad, spin-allowed bands and a faint sharper feature can therefore be interpreted as a combination of quartet and possible doublet excitations, rather than as three independent orbital promotions of equal status.

Why?

Why divide both axes by B? The ratio scales out much of the ion-specific repulsion energy and allows one calculated curve set to be applied to different complexes of the same d count. Experimental transition energies then restore the actual B scale for a particular complex.

Common misconception

“A vertical ordinate of 20 means a transition of 20 cm⁻¹.” It means E/B=20. If B is 800 cm⁻¹, the transition energy is 16,000 cm⁻¹. Ignoring the scale factor gives an error of orders of magnitude.

Worked example

Suppose a d³ diagram matches an observed spectrum at Δₒ/B=25.0 and its first assigned excited curve has E/B=25.0. If the observed first band is 20,000 cm⁻¹, then B=20,000/25.0=800 cm⁻¹ and Δₒ=25.0×800=20,000 cm⁻¹. If the second target's ordinate is 36.0 at that x, the model predicts 28,800 cm⁻¹. A substantially different measured second band would signal a poor assignment or model mismatch.

Quick check

1. What does the horizontal zero line represent? Answer: The chosen ground state's energy has been set to zero at each Δₒ/B, so excited-state ordinates give transition energies relative to it.

Exam focus

Write the axis labels with their ratios and use consistent cm⁻¹ values. Read from the correct spin region and identify the ground state before drawing an arrow. Ratios locate a position on the diagram; absolute energies then give B and Δₒ.

Advanced insight

The plotted state energies come from diagonalising a Hamiltonian containing ligand-field and interelectronic-repulsion terms. Equal-symmetry basis states can mix, giving curved avoided crossings. A chosen C/B value affects parts of the diagram, especially singlet or spin-forbidden states, so a published diagram is a calculated approximation rather than a universal lookup table.

Summary

Tanabe–Sugano diagrams plot E/B against Δₒ/B for one d count, taking the ground state as a local zero. They support semi-quantitative extraction of B and Δₒ and include spin-forbidden or low-spin states that qualitative Orgel sketches omit. Correct band assignment remains essential.

Practice questions

1. At x=20, a target has E/B=30 and B=700 cm⁻¹. Find its transition energy and Δₒ. Answer: E=30×700=21,000 cm⁻¹ and Δₒ=20×700=14,000 cm⁻¹; the two quantities differ because the target is a many-electron term. 2. Why can plotted excitation curves change abruptly near a spin-crossover separator? Answer: The state chosen as ground changes identity and multiplicity, so energies on either side are referenced to different baselines even though underlying state energies vary with field strength. 3. Name two checks before interpreting a faint peak as a spin-forbidden d–d band. Answer: Confirm that its energy matches a differently spin-labelled target at the inferred Δₒ/B, and ensure its weak intensity and sample chemistry are consistent rather than an impurity or charge-transfer shoulder. 4. Can a d² Tanabe–Sugano diagram be reused for a d⁸ ion without modification? Answer: No. Electron–hole relationships aid qualitative comparisons, but each diagram's state ordering and energy trajectories must match the actual d count and geometry.