§ C-23 Coordination Complexes & Crystal Field Theory (d-Orbital Splitting Δₒ) (Grade 11 · Coordination Chemistry)
Octahedral [ML₆]ⁿ⁺ vs. tetrahedral [ML₄]ⁿ⁺ complexes, t₂g / e_g splitting (Δₒ), spectrochemical series, and color. The vivid colors of ruby, emerald, and transition-metal solutions are the visible signature of five d-orbitals splitting in the electric field of ligands.
Theory Summary & Scientific Mechanism
In Crystal Field Theory (CFT), electrostatic repulsion from six octahedral ligands along the ± x, ± y, ± z axes lifts the five-fold degeneracy of metal d-orbitals into a lower t₂g triplet (dₓy, dₓz, d_yz at -0.4Δₒ) and an upper e_g doublet (d_z², dₓ²-y² at +0.6Δₒ). In a tetrahedral field [ML₄]^n+, the splitting is inverted and smaller (Δₜ ≈ (4)/(9)Δₒ).
The splitting parameter Δₒ increases across the spectrochemical series: I⁻ P) force spin pairing into low-spin states.
Absorption of a visible photon matching Δₒ = hc / λₐbs promotes an electron from t₂g to e_g (d–d transition), imparting vivid complementary colors to transition-metal complexes.
Everyday Analogy
Six octahedral ligands along the x, y, z axes act like bumper cars: d-orbitals pointing straight at the ligands (e_g) get pushed up in energy, while orbitals tucked between the axes (t₂g) sit lower in energy by Δₒ!
Interactive Question & Derivation
How do six ligands along the ± x, ± y, ± z axes split the five degenerate d-orbitals in an octahedral complex [ML₆]^n+, and when does a low-spin configuration form?
- A: The d-level splits into a lower triplet t₂g (dₓy, dₓz, d_yz) and an upper doublet e_g (d_z², dₓ²-y²) separated by Δₒ; strong-field ligands (Δₒ > P) produce a low-spin complex
- B: All five d-orbitals shift equally in energy without splitting, so transition metal complexes are always colorless
- C: In an octahedral field, the d_z² and dₓ²-y² orbitals lie lower in energy than dₓy, dₓz, d_yz
Study Hint: In an octahedral field, d-orbitals split into a lower t₂g triplet and upper e_g doublet; strong-field ligands (Δₒ > P) create a low-spin state.
Proof & Explanation: In an octahedral complex [ML₆]^n+, ligand lone pairs approach directly along the coordinate axes, strongly repelling the axial e_g orbitals (d_z², dₓ²-y²) and raising their energy by +0.6Δₒ, while the inter-axial t₂g orbitals (dₓy, dₓz, d_yz) are stabilized by -0.4Δₒ. When a strong-field ligand (such as CN⁻ or CO) makes Δₒ exceed the spin-pairing energy P, electrons pair up in the lower t₂g level (low-spin state). Photons of wavelength λₐbs = hc / Δₒ promote an electron across the gap (d–d transition), giving the solution its complementary color.
Key Equations & 3D Simulation Parameters
Δₒ = E(e_g) - E(t₂g) = (h c)/(λₐbs), CFSE = (-0.4 nₜ₂g + 0.6 nₑ_g)Δₒ
- Central metal dⁿ electron count (d¹ … d⁹): 1 … 9 e⁻ (default: 6 e⁻)
- Ligand field splitting Δₒ (spectrochemical series: Cl⁻ → H₂O → NH₃ → CN⁻): 130 … 320 kJ/mol (default: 210 kJ/mol)
- Metal–ligand coordinate bond length d(M←L): 1.75 … 2.45 Å (default: 2 Å)
- Coordination geometry (1=Octahedron [ML₆], 2=Tetrahedron [ML₄]): 1 … 2 (default: 1)