Tanabe–Sugano Diagrams for d³ and d⁸

Introduction

Chromium(III) and nickel(II) offer practical examples of multi-band coordination spectra. Cr³⁺ is d³ and Ni²⁺ is d⁸; both have an octahedral A₂g ground state derived from an F free-ion term in the common model. Their diagrams share a pattern of three principal spin-allowed bands, but they are not interchangeable calculations. Different d counts, spin multiplicities and values of B and Δₒ require distinct diagrams and independent fits.

Core explanation

For octahedral Cr³⁺, d³ gives ^4A₂g(F) ground. Spin-allowed targets are ^4T₂g(F), ^4T₁g(F) and ^4T₁g(P). Quartet-to-quartet absorption preserves S=3/2. The first excitation often tracks Δₒ closely in the common d³ treatment, though an actual parameter fit should use the diagram rather than assume exact equality under every perturbation. The higher two bands involve interelectronic repulsion and F/P mixing, which is why a second peak can reveal B information that the first alone cannot.

For octahedral Ni²⁺, d⁸ gives ^3A₂g(F) ground with spin-allowed targets ^3T₂g(F), ^3T₁g(F) and ^3T₁g(P). Here S=1. The analogous target labels can tempt one to use a d³ graph after changing only the superscript, but each d count has its own calculated energy trajectories. A d⁸ spectrum may have a near-infrared first band and higher visible bands depending on ligands; a spectrometer that measures only the visible region can miss the first transition and distort a naive band numbering.

Both cases show g→g transitions in ideal O h, so d–d bands are Laporte-forbidden and gain intensity through vibronic effects. A sharp weak feature could be spin-forbidden and should not automatically be counted as one of the three spin-allowed targets. A very intense ultraviolet band may be charge transfer. The model's predicted target count is a guide to possible transitions, not a guarantee of three separately resolved maxima in the chosen measurement window.

For a real Cr³⁺ series, changing ligands changes Δₒ and can change B through covalency. The d³ Tanabe–Sugano plot lets band ratios locate Δₒ/B, then absolute wavenumbers give B and Δₒ. The same workflow applies independently to Ni²⁺ on the d⁸ diagram. Comparing β=B complex/B free for each ion requires its own free-ion B reference. It would be chemically invalid to treat the difference between a chromium B and a nickel B as a direct ligand effect.

An advantage of the A₂g ground state is that it is orbitally nondegenerate in cubic symmetry. This simplifies the first-order orbital contribution to magnetism and reduces some complications from ground-state Jahn–Teller distortion compared with E g or T ground states. Excited states can still be vibronically coupled, and real octahedra can be distorted. Combining electronic spectra with magnetic moment and structural data therefore strengthens a d³ or d⁸ assignment.

The similarity between d³ and d⁸ also has limits. d³ has three unpaired electrons in t₂g³ in the simple octahedral picture, whereas d⁸ has two unpaired electrons in t₂g⁶e g². Their spin-only magnetic moments differ, and the Ni²⁺ ion can adopt other geometries, notably square planar in sufficiently strong ligand environments. Geometry must be established before a d⁸ octahedral diagram is applied.

Step-by-step reasoning

Use charge balance to distinguish Cr³⁺ d³ from Ni²⁺ d⁸. Establish six-coordinate approximately octahedral geometry. Mark ^4A₂g(F) or ^3A₂g(F) ground and list same-spin T₂g(F), T₁g(F) and T₁g(P) targets. Convert wavelengths to cm⁻¹, check intensities and measurement range, then match at least two bands to the correct d-count diagram. Calculate B and Δₒ and compare the third predicted band with observation.

Visual explanation

Place two diagrams side by side. Both have an A₂g baseline and three upward arrows to T₂g(F), T₁g(F) and T₁g(P). Label the Cr³⁺ arrows with superscript 4 and three unpaired electrons; label Ni²⁺ with superscript 3 and two unpaired electrons. Underneath, draw separate B and Δₒ boxes to warn that equal-looking symmetry labels do not imply equal parameter values.

Real-world analogy

Two maps may use the same symbols for roads, junctions and hills, yet have different distances and elevations. The d³ and d⁸ diagrams share a readable pattern of state labels, but the plotted curves and scales belong to their own electron counts. One must use the right map before measuring a route.

Real-world example

The visible colour of octahedral Cr³⁺ complexes changes when ligands alter splitting, and octahedral Ni²⁺ salts can show multiple bands extending into the near infrared. Measuring only one visible maximum in either case can be misleading. A broad scan, known geometry and band-intensity data allow an assignment that distinguishes ligand-field bands from charge transfer.

Why?

Why are there three main spin-allowed bands from either A₂g ground state? F provides T₂g and T₁g excited components in addition to ground A₂g, while a same-spin P term supplies another T₁g. All three targets retain the ground-state spin multiplicity.

Common misconception

“Cr³⁺ and Ni²⁺ share A₂g(F), so their spectra must be identical after rescaling.” Their d counts and spin multiplicities differ, and the energy trajectories depend on electron interactions. The analogy helps with target labels but does not justify substituting one diagram for the other.

Worked example

Suppose an octahedral Cr³⁺ spectrum has assigned quartet bands at 17,000 and 24,000 cm⁻¹. Their ratio is 1.412. On the appropriate d³ plot, assume that ratio matches x=21.0, where y₁=21.0 and y₂=29.65. Then B=17,000/21.0≈810 cm⁻¹ and Δₒ=21.0×810≈17,010 cm⁻¹. If a third quartet curve has y₃=45.0 there, it predicts about 36,450 cm⁻¹. These example ordinates are hypothetical readings to show the method; measured complexes require their own plot and assignments.

Quick check

1. What are the spins of the ordinary octahedral Cr³⁺ and Ni²⁺ ground states? Answer: Cr³⁺ ^4A₂g has S=3/2; Ni²⁺ ^3A₂g has S=1. Their multiplicities are four and three, respectively.

Exam focus

Use the correct d³ or d⁸ diagram and preserve spin superscripts. Mark the measured spectral range and distinguish weak d–d from intense charge transfer. If reporting Δₒ≈ν̃₁ for d³, state it as the familiar approximation and verify with the full diagram when accuracy is required.

Advanced insight

The A₂g ground state is orbitally nondegenerate, so first-order orbital angular momentum is largely quenched. A measured moment close to the spin-only value can support the model, whereas strong deviations or unusual band intensities may signal geometry, covalency or spin–orbit effects that the ideal octahedral plot omits.

Summary

Octahedral Cr³⁺ d³ and Ni²⁺ d⁸ have A₂g(F) ground states and three principal same-spin F/P-derived target states. Their qualitative band pattern is similar, but each requires its own Tanabe–Sugano diagram, B and Δₒ values, and verification of geometry and spectral coverage.

Practice questions

1. List the three spin-allowed targets from octahedral Ni²⁺ ^3A₂g(F). Answer: ^3T₂g(F), ^3T₁g(F) and ^3T₁g(P), with triplet multiplicity conserved for each transition. 2. Why is a very intense UV band in a Cr³⁺ spectrum suspect as the third quartet d–d band? Answer: Ideal octahedral d–d bands are Laporte-forbidden and generally weak; charge-transfer absorption can be much stronger, so intensity and predicted energy must be checked together. 3. Can a square-planar Ni²⁺ complex be fitted to the ordinary octahedral d⁸ diagram? Answer: No. The orbital and term splitting follow a different, lower-symmetry field. Establish structure before choosing an energy diagram. 4. Why might a d⁸ first band be absent from a visible-only scan? Answer: It can occur at a lower energy in the near infrared, outside the instrument's wavelength range, even while higher ligand-field bands are visible.