Electronic Spectra and Tanabe-Sugano Diagrams

Core Idea

Tanabe-Sugano diagrams plot the energies of all electronic states of a dⁿ ion as a function of the crystal field splitting parameter Δ/B, where B is the Racah interelectronic repulsion parameter. They provide a complete picture of the allowed electronic transitions for any d-electron configuration in an octahedral field, enabling quantitative analysis of absorption spectra — including the prediction and assignment of multiple absorption bands, the determination of Δ and B from experimental data, and the identification of spin-crossover points.

Explainer

The absorption spectrum of a transition metal complex typically shows multiple bands, each corresponding to a different electronic transition. Crystal field theory and the spectrochemical series tell you that the primary transition occurs across the Δ gap, but they do not explain the full set of observed bands or their relative energies. Tanabe-Sugano diagrams fill this gap by providing a complete energy-level picture for each dⁿ configuration as a function of crystal field strength.

A Tanabe-Sugano diagram is constructed by calculating the energies of all electronic terms (from Russell-Saunders coupling) of a dⁿ configuration as the octahedral crystal field is turned on from zero (free-ion limit) to large values. The x-axis is Δ/B (crystal field strength normalized to the Racah parameter B, which measures electron-electron repulsion), and the y-axis is E/B (state energy normalized to B). The ground state is always plotted along the x-axis (E/B = 0). Excited states curve upward, and their slopes and curvatures encode how each state responds to the crystal field. Lines that run roughly parallel to the x-axis correspond to states insensitive to Δ; steeply rising lines correspond to states strongly destabilized by the crystal field.

The power of the diagram lies in its direct connection to experiment. For a d³ complex like [Cr(H₂O)₆]³⁺, you measure the UV-Vis spectrum and find three absorption bands. The d³ Tanabe-Sugano diagram shows three spin-allowed excited states above the ⁴A₂g ground state: ⁴T₂g, ⁴T₁g(F), and ⁴T₁g(P). By taking the ratio of two band energies and matching it to the diagram, you determine the Δ/B value for that complex. From there, you extract both Δ and B individually. The value of B for the complex is always less than the free-ion B₀ — this reduction (measured as the nephelauxetic ratio β = B/B₀) reflects the covalency of the metal-ligand bond, a topic explored further in the nephelauxetic effect.

For d⁴ through d⁷ configurations, Tanabe-Sugano diagrams also reveal the spin-crossover boundary. At low Δ/B, the ground state is a high-spin term; at high Δ/B, it switches to a low-spin term, marked by a vertical discontinuity in the diagram. Near this crossover, both spin states are close in energy, and external perturbations (temperature, pressure) can switch the complex between them — the basis for spin-crossover materials used in molecular switches and sensors. The Tanabe-Sugano diagram thus connects spectroscopy, magnetism, and materials science through a single, elegant graphical tool.