Crystal field theory and colour
Ligands split the five d orbitals into t₂g and eg in an octahedral field. If Δₒ is larger than the pairing energy, electrons pair up; d-d transitions give colour.
How to draw and explain the octahedral splitting diagram
Crystal field theory treats the metal-ligand bond as **purely ionic**, caused by electrostatic attraction. Ligands are taken as **point charges** (anions) or **dipoles** (neutral molecules). In a free metal ion, the five d orbitals have the **same energy** (they are degenerate).
- 1Six ligands approach the metal along the x, y and z axes.
- 2dx²−y² and dz² point along the axes, straight at the ligands. They are repelled most and rise in energy: the eg set, +0.6Δₒ.
- 3dxy, dyz and dxz point between the axes. They are repelled less and fall in energy: the t₂g set, −0.4Δₒ.
- 4★The energy gap between eg and t₂g is the crystal field splitting energy, Δₒ (o for octahedral).
- 5Drawing: on the left, one level of five short lines (free ion). On the right, three lower lines labelled t₂g and two upper lines labelled eg. Draw a dotted average line between them and mark 0.6Δₒ above it, 0.4Δₒ below it, and the full gap Δₒ.
★ key step
Strong and weak field ligands
Ligands differ in how strongly they split the d orbitals. Arranged in increasing order of field strength they form the spectrochemical series: I⁻ < Br⁻ < SCN⁻ < Cl⁻ < S²⁻ < F⁻ < OH⁻ < C₂O₄²⁻ < H₂O < NCS⁻ < edta⁴⁻ < NH₃ < en < CN⁻ < CO. For d¹ to d³ ions, the electrons simply enter t₂g. For d⁴, the fourth electron has a choice. If Δₒ < P (a weak field ligand), it enters eg: t₂g³ eg¹, high spin. If Δₒ > P (a strong field ligand), it pairs in t₂g: t₂g⁴ eg⁰, low spin. The same choice arises for d⁵, d⁶ and d⁷ ions.
Electron counts in octahedral complexes
| Complex | Metal ion, d electrons | Field | Configuration | Unpaired electrons |
|---|---|---|---|---|
| [Ti(H₂O)₆]³⁺ | Ti³⁺, d¹ | Only one electron | t₂g¹ eg⁰ | 1 |
| [Cr(NH₃)₆]³⁺ | Cr³⁺, d³ | No choice needed | t₂g³ eg⁰ | 3 |
| [FeF₆]³⁻ | Fe³⁺, d⁵ | Weak (Δₒ < P) | t₂g³ eg² | 5 |
| [Fe(CN)₆]³⁻ | Fe³⁺, d⁵ | Strong (Δₒ > P) | t₂g⁵ eg⁰ | 1 |
| [CoF₆]³⁻ | Co³⁺, d⁶ | Weak | t₂g⁴ eg² | 4 |
| [Co(NH₃)₆]³⁺ | Co³⁺, d⁶ | Strong | t₂g⁶ eg⁰ | 0 |
| [Fe(H₂O)₆]²⁺ | Fe²⁺, d⁶ | Weak | t₂g⁴ eg² | 4 |
| [Ni(H₂O)₆]²⁺ | Ni²⁺, d⁸ | No choice needed | t₂g⁶ eg² | 2 |
Configuration and spin-only moment from crystal field theory
Write the d-electron configuration of the metal and calculate the spin-only magnetic moment of (a) [Mn(CN)₆]³⁻ (b) [MnCl₆]³⁻.
- Oxidation state: in both, x + 6(−1) = −3, so x = +3.
- Mn: Z = 25, [Ar] 3d⁵ 4s². So both complexes contain Mn³⁺, 3d⁴.
- (a) CN⁻ is a strong field ligand: Δₒ > P, so the fourth electron pairs up in t₂g: t₂g⁴ eg⁰, 2 unpaired electrons.
- μ = √[2(2 + 2)] = √8 ≈ 2.83 BM.
- (b) Cl⁻ is a weak field ligand: Δₒ < P, so the fourth electron enters eg: t₂g³ eg¹, 4 unpaired electrons.
- μ = √[4(4 + 2)] = √24 ≈ 4.90 BM.
(a) t₂g⁴ eg⁰, 2.83 BM, low spin (b) t₂g³ eg¹, 4.90 BM, high spin.
Colour of coordination compounds
An electron in t₂g can absorb light of energy Δₒ and jump to eg: a d-d transition. The colour we see is the complementary colour of the light absorbed. [Ti(H₂O)₆]³⁺ absorbs blue-green light (about 498 nm) and looks violet. With no ligands there is no splitting and no colour: anhydrous CuSO₄ is white, but CuSO₄·5H₂O is blue. Changing the ligand changes Δₒ and so the colour: green [Ni(H₂O)₆]²⁺ turns violet as en replaces the water, giving [Ni(en)₃]²⁺. Zn²⁺ (3d¹⁰) complexes are colourless, since no d-d transition is possible. Gemstones show the same effect: ruby is Al₂O₃ containing some Cr³⁺ ions, and emerald is the mineral beryl containing Cr³⁺ ions.
Tetrahedral splitting and limitations of crystal field theory
- In a tetrahedral field the splitting is reversed: the two e orbitals are lower and the three t₂ orbitals are higher.
- Δₜ = (4/9)Δₒ. This is usually smaller than the pairing energy, so tetrahedral complexes are almost always high spin.
- Limitation 1: ligands are treated as point charges, so anionic ligands should cause the largest splitting. In fact they lie at the low end of the spectrochemical series.
- Limitation 2: the theory ignores the covalent character of the metal-ligand bond.
The red of ruby and the green of emerald both come from Cr³⁺ ions. The crystal around Cr³⁺ is different in the two stones, so Δₒ, and therefore the colour absorbed, is different.
3 marks: “Draw the crystal field splitting of d orbitals in an octahedral field. Write the configuration of a d⁴ ion when Δₒ > P and when Δₒ < P.” eg raised by 0.6Δₒ, t₂g lowered by 0.4Δₒ. Δₒ > P: t₂g⁴ eg⁰ (low spin, 2 unpaired). Δₒ < P: t₂g³ eg¹ (high spin, 4 unpaired).
2 marks: “Why is [Ti(H₂O)₆]³⁺ coloured while [Zn(H₂O)₆]²⁺ is colourless?” Ti³⁺ is d¹: it absorbs blue-green light for a t₂g to eg jump and looks violet. Zn²⁺ is d¹⁰: no d-d transition is possible, so it is colourless.
2 marks: “Anhydrous CuSO₄ is white but CuSO₄·5H₂O is blue. Explain.” Without water ligands there is no crystal field splitting and no d-d transition. Water ligands split the d orbitals; light is absorbed for a d-d transition and the complementary colour, blue, is seen.
Tea goes down (t₂g, −0.4Δₒ), eggs go up (eg, +0.6Δₒ). Strong ligand: pair before you climb. Weak ligand: climb before you pair.