Lesson 7 · 35 min

Electronic spectra: Δo and B from the bands

[Cr(H₂O)₆]³⁺ has one Δo but two bands. Repulsion between the electrons pulls them apart; a Tanabe–Sugano diagram yields Δo and the Racah parameter B together, and six chromium(III) complexes give the nephelauxetic series.

In this lesson

Explain why a complex with a single Δo shows two or three d–d bands
Obtain Δo and the Racah parameter B from the bands of d³ and d⁸ ions with a Tanabe–Sugano diagram or its equations
Rank ligands in the nephelauxetic series from β = B(complex)/B(free ion), and say what covalency it reflects

A violet solution of $\ce{[Cr(H2O)6]^3+}$ has two absorption bands in the visible, at 17 400 and 24 600 cm⁻¹. The last lesson said the band position is Δo — but the complex has only one Δo, so why two bands? $\ce{[Ni(H2O)6]^2+}$ has three: 8500, 13 800 and 25 300 cm⁻¹. The extra bands come from something crystal-field theory left out: the repulsion between the d electrons.

Predict first

$\ce{Cr^3+}$ is d³, with a $t_{2g}^3$ ground state. Promoting one or two electrons to $e_g$ without changing the total spin, how many spin-allowed bands can there be?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

1 2 3 4

Free-ion terms and the Racah parameter B

Even without ligands, the three d electrons of a d³ ion can be arranged in several ways that electron repulsion gives different energies: the terms. The two terms of highest spin, ⁴F (the ground term) and ⁴P, lie exactly 15B apart; B is the Racah parameter, the measure of how strongly the d electrons repel each other. The ³F and ³P terms of d⁸ are also 15B apart. Take the atomic-spectra data of the US National Institute of Standards and Technology (NIST), find each term’s centre of gravity with weights 2J + 1, and the free-ion B follows: 917 cm⁻¹ for $\ce{Cr^3+}$ and 1056 cm⁻¹ for $\ce{Ni^2+}$ .

E(\mathrm{T_{2g}}) = \Delta_\text{o}, \qquad E(\mathrm{T_{1g}}) = 7.5B + 1.5\Delta_\text{o} \mp \tfrac{1}{2}\sqrt{225B^2 - 18B\Delta_\text{o} + \Delta_\text{o}^2}

Energies of the spin-allowed excited states of octahedral d³ and d⁸ complexes above the A₂g ground state. At Δo = 0 the two T₁g states return to the free-ion F and P terms (0 and 15B).

$\Delta_\text{o}$: octahedral splitting
$B$: Racah parameter in the complex
$E(\mathrm{T_{1g}})$: the two T₁g states: minus for T₁g(F), plus for T₁g(P)

The Tanabe–Sugano diagram

Divide energies and Δo by B and the curves serve every ion of the same dⁿ: the Tanabe–Sugano diagram, with Δo/B across, E/B up and the ground state drawn along the axis. For d³ and d⁸ the first band is Δo itself; the ratio of the second band to the first depends only on Δo/B, so two bands fix Δo and B together. The third band then becomes a test: for nickel(II) complexes one pair of Δo and B reproduces all three bands to within about 2 %.

Worked example

Δo and B for [Cr(H₂O)₆]³⁺

$\ce{[Cr(H2O)6]^3+}$ absorbs at 17 400 and 24 600 cm⁻¹. Find Δo, B and β, and predict the third band.

d³ and d⁸ · spin-allowed transitions

Tanabe–Sugano: Δ and B from the bands

Transition	Measured / cm⁻¹	Model / cm⁻¹
³A₂g → ³T₂g	8500	8500
³A₂g → ³T₁g(F)	13800	14107
³A₂g → ³T₁g(P)	25300	24993

Δo = 8500 cm⁻¹ · B = 907 cm⁻¹

β = B(complex) / B(free ion) = 907 / 1056 = 0.86

Δo is the first band; B comes from the sum rule ν₂ + ν₃ = 15B + 3Δo, and then each of ν₂ and ν₃ is checked on its own.

Free Ni²⁺ has B = 1056 cm⁻¹, from the 15B separation of its ³P and ³F terms in the NIST atomic spectra.

Nephelauxetic series: the smaller β, the more covalent the metal–ligand bond

$\ce{[Ni(en)3]^2+}$ β = 0.87
$\ce{[Ni(H2O)6]^2+}$ β = 0.86
$\ce{[Ni(NH3)6]^2+}$ β = 0.85

Each of the three nickel(II) complexes has three bands: once Δo and B are fixed by the sum rule, see how far the model’s open circles sit from the filled dots. Then switch to “drag them yourself” and try to line up all three with the sliders alone.

Try the calculation

$\ce{[Cr(NH3)6]^3+}$ absorbs at 21 550 and 28 500 cm⁻¹. What is its Racah parameter B?

The nephelauxetic effect

B in a complex is always smaller than in the free ion: the d electrons share some electron cloud with the ligands, the orbitals expand and repel each other less. This is the nephelauxetic effect (“cloud-expanding”), and the smaller β = B(complex)/B(free ion), the more covalent the metal–ligand bond. Feeding the bands of six chromium(III) complexes through the equations gives F⁻ (β = 0.98) > H₂O (0.79) > NH₃ (0.72) > en (0.68) > ox (0.67) > CN⁻ (0.53) — exactly the order of the nephelauxetic series in textbooks. It is not the spectrochemical series: NH₃ has both a larger Δo and a smaller β than H₂O, while F⁻ has a small Δo and the largest β.

Concept check

Of these six chromium(III) complexes, which has the most covalent metal–ligand bonds?

\ce{[CrF6]^3-}

\ce{[Cr(H2O)6]^3+}

\ce{[Cr(en)3]^3+}

\ce{[Cr(CN)6]^3-}

Key takeaways

Electron repulsion splits one configuration into several terms, so a complex with one Δo can have two or three d–d bands; d³ and d⁸ each have three spin-allowed transitions.
d³ and d⁸: ν₁ = Δo and E(T₁g) = 7.5B + 1.5Δo ∓ ½√(225B² − 18BΔo + Δo²); two bands fix Δo and B, and a third tests them.
Free-ion B values come from atomic spectra: 917 cm⁻¹ for Cr³⁺ and 1056 cm⁻¹ for Ni²⁺; B shrinks in complexes, and the smaller β, the more covalent the bonding.
The nephelauxetic series computed from measured chromium(III) bands: F⁻ > H₂O > NH₃ > en > ox > CN⁻.

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Data and references

Y. Tanabe and S. Sugano, “On the absorption spectra of complex ions I”, J. Phys. Soc. Jpn. 9, 753–766 (1954).
A. B. P. Lever, Inorganic Electronic Spectroscopy, 2nd ed., Elsevier, 1984 (band maxima and the d³/d⁸ energy expressions).
A. Kramida, Yu. Ralchenko, J. Reader and NIST ASD Team, NIST Atomic Spectra Database, version 5.12 (2024): Cr IV and Ni III levels.
C. E. Housecroft and A. G. Sharpe, Inorganic Chemistry, 5th ed., Pearson, 2018, chapter 20 (electronic spectra; the nephelauxetic series).