Questions: Electronic Spectra and Tanabe-Sugano Diagrams

Question 1

In a Tanabe-Sugano diagram for a d² ion, the x-axis plots Δ/B and the y-axis plots E/B. Why are energies normalized to the Racah parameter B rather than plotted as absolute energies?

• A. To ensure all transition metals produce identical diagrams regardless of their identity
• B. To create a universal, dimensionless diagram where the ratio Δ/B captures the competition between crystal field splitting and interelectron repulsion — making the diagram applicable to any d² ion by adjusting B
• C. To eliminate the need for experimental measurements
• D. Because absolute energies cannot be calculated from quantum mechanics

Answer: (B)

The Racah parameter B quantifies the magnitude of electron-electron repulsion within the d-shell. By normalizing both Δ and E to B, the diagram becomes dimensionless and universal for a given d^n configuration. Every d² octahedral complex, regardless of the specific metal and ligands, falls somewhere on the same d² Tanabe-Sugano diagram — the position is determined by the ratio Δ/B. To extract actual transition energies for a specific complex, you read E/B from the diagram and multiply by the experimentally determined B value. This normalization is what makes the diagrams so powerful: one diagram serves all d² systems.

Question 2

A d⁵ Tanabe-Sugano diagram shows a discontinuity in the ground state at a specific Δ/B value. This discontinuity corresponds to the spin-crossover point between high-spin and low-spin configurations.

• T. True
• F. False

Answer: True

For configurations where both high-spin and low-spin ground states are possible (d⁴ through d⁷), the Tanabe-Sugano diagram shows two regimes. At low Δ/B (weak field), the high-spin term is the ground state; at high Δ/B (strong field), the low-spin term becomes the ground state. The transition between regimes appears as a discontinuity because the ground-state line changes from one term symbol to another. At the crossover point, both states have the same energy. For d⁵, this is the transition from ⁶A₁g (high-spin, all electrons unpaired) to ²T₂g (low-spin), and it occurs at a relatively high Δ/B value because the pairing energy for five electrons is large.

Question 3

From a Tanabe-Sugano diagram, you observe that a d³ octahedral complex has three spin-allowed absorption bands. This is consistent with the Tanabe-Sugano diagram, which shows three spin-allowed excited states above the ⁴A₂g ground state.

• T. True
• F. False

Answer: True

For d³ in an octahedral field, the ground state is ⁴A₂g (derived from the ⁴F free-ion term). The spin selection rule (ΔS = 0) allows transitions only to other quartet states. The Tanabe-Sugano diagram shows three such states: ⁴T₂g, ⁴T₁g(F), and ⁴T₁g(P). These correspond to three spin-allowed d-d transitions, each producing an absorption band. The classic example is [Cr(H₂O)₆]³⁺, which shows exactly three bands in its UV-Vis spectrum, assignable using the d³ Tanabe-Sugano diagram to determine Δ and B.

Question 4

A chemist measures two absorption band energies for a d³ octahedral complex and wants to determine both Δ and B. Explain how the Tanabe-Sugano diagram enables this from just two experimental values.

Model answer: The ratio of two transition energies (E₁/E₂) is a function of Δ/B only — it does not depend on the absolute value of B. On the Tanabe-Sugano diagram, the chemist calculates the experimental ratio E₁/E₂ and finds the Δ/B value where the diagram predicts the same ratio between the corresponding transitions. This gives Δ/B. Then, reading E₁/B from the diagram at that Δ/B value and dividing the experimental E₁ by this number gives B. Once B is known, Δ = (Δ/B) × B. Two equations, two unknowns — and the Tanabe-Sugano diagram provides the functional relationship between them.

This method works because the Tanabe-Sugano diagram encodes the full energy-level structure as a function of the single variable Δ/B. The ratio method eliminates B from the initial step, making it a self-consistent determination. A third band, if available, serves as an internal check on the assignments.