Crystal Field Theory: Octahedral and Tetrahedral Splitting

Why is [Cu(H2O)6]2+ blue while [Zn(H2O)6]2+ is colourless? Why do some iron complexes attract magnets strongly and others barely at all? Crystal field theory answers these questions with a beautifully simple picture: treat the ligands as negative point charges approaching the central metal ion, and watch what happens to the five d orbitals. Orbitals pointing straight at the ligands are repelled more and rise in energy; orbitals pointing between the ligands rise less. This energy gap is called the crystal field splitting, and it controls colour, magnetism and stability.

Splitting matters because it is the reason transition metal compounds are coloured at all. The exact size of the gap decides which wavelength of light is absorbed, whether electrons pair up or stay unpaired, and how much extra stability a complex gains. Nearly every question about the colour or magnetism of a complex is really a question about splitting.

Key Concepts

• Octahedral splitting: Six ligands sit on the x, y and z axes. The two eg orbitals (dz2, dx2-y2) point directly at ligands and rise by +0.6 delta_o; the three t2g orbitals (dxy, dxz, dyz) point between ligands and fall by -0.4 delta_o. The gap between the sets is delta_o (delta octahedral).
• Tetrahedral splitting: Four ligands sit at alternate corners of a cube, so no orbital points directly at a ligand. The order reverses: the two e orbitals sit lower and the three t2 orbitals sit higher, with a smaller gap delta_t = 4/9 delta_o.
• Why tetrahedral is smaller: Fewer ligands (4 vs 6) and none pointing head-on at any orbital, so the repulsion difference is much weaker. Tetrahedral complexes are therefore almost always high spin.
• High spin vs low spin: Electrons choose between paying the pairing energy P (pairing two electrons in one orbital) or jumping the gap delta_o. If delta_o is greater than P, electrons pair up in t2g first (low spin); if P wins, they spread across all five orbitals (high spin).
• Spectrochemical series: Ligands ordered by splitting strength: I- less than Br- less than Cl- less than F- less than OH- less than H2O less than NH3 less than en less than CN- approximately CO. Weak-field ligands (halides) give high spin; strong-field ligands (CN-, CO) give low spin.
• d-electron configurations: Octahedral d4 high spin is t2g3 eg1 while low spin is t2g4 eg0; d6 high spin is t2g4 eg2 while low spin is t2g6 eg0. The d3, d8 and other counts are the same in both cases.
• Square planar note: Four ligands in one plane split the d orbitals into four levels, strongly favouring low spin d8 configurations such as [Ni(CN)4]2- and Pt(II) complexes.

Formulas

• Octahedral splitting pattern: E(eg) - E(t2g) = delta_o, with eg at +0.6 delta_o and t2g at -0.4 delta_o relative to the average (barycentre) energy.
• Tetrahedral relation: delta_t = 4/9 delta_o, approximately 0.44 delta_o. Because this is always smaller than typical pairing energies, tetrahedral complexes are high spin.
• Spin-only magnetic moment: mu = sqrt(n(n+2)) Bohr magnetons, where n is the number of unpaired electrons. This connects the splitting picture to measurable magnetism: d5 high spin (n = 5) gives 5.92 BM while low spin (n = 1) gives 1.73 BM.

Worked Example

Question: The complex [Mn(H2O)6]2+ has a magnetic moment of 5.92 BM. Deduce its d-electron configuration and its crystal field stabilization energy.

1. Find the d count: Mn2+ has lost the 4s electrons from [Ar] 4s2 3d5, leaving d5.
2. Find unpaired electrons: mu = sqrt(n(n+2)) = 5.92 gives n = 5 (since sqrt(5 x 7) = sqrt(35) = 5.92). All five electrons are unpaired.
3. Assign high or low spin: Five unpaired electrons in an octahedral field means t2g3 eg2, which is the high-spin arrangement. This fits because H2O is a weak-field ligand with small delta_o.
4. Compute CFSE: CFSE = 3 x (-0.4 delta_o) + 2 x (+0.6 delta_o) = -1.2 delta_o + 1.2 delta_o = 0. The high-spin d5 complex gains no net stabilization, which is why Mn2+ complexes are pale and labile.

Common Misconceptions

• The t2g set is always lower in energy: Only in octahedral (and square planar-derived) fields. In tetrahedral geometry the e set is lower and t2 higher, the reverse order, because the ligands approach between the axes.
• Strong-field ligands always force low spin: Only when it matters. For d1, d2, d3, d8, d9 and d10 the high-spin and low-spin configurations are identical, so the ligand strength changes delta_o but not the electron arrangement. Only d4, d5, d6 and d7 can switch.
• Splitting explains everything about bonding: Crystal field theory is electrostatic and ignores covalency. It predicts colours and magnetism brilliantly but cannot explain why CO is a strong-field ligand; that needs molecular orbital (ligand field) theory, which adds pi-backbonding.

Real-World Applications

• Gemstone colours: Ruby is red because Cr3+ in octahedral Al2O3 sites absorbs yellow-green light across its delta_o gap; emerald is green for the same ion in a different crystal field.
• Paints and pigments: The intense colours of Prussian blue, chrome yellow and cobalt blue are tuned by choosing metals and ligands with the right splitting.
• Oxygen transport: Haemoglobin's iron sits in a field that switches spin state when O2 binds, and the small size change in low-spin Fe2+ pulls the protein into its oxygen-holding shape.
• Magnetic materials: Designing single-molecule magnets and MRI contrast agents starts from controlling spin state through ligand choice and geometry.

Crystal field splitting divides the d orbitals into t2g and eg sets in octahedral fields and the reversed e and t2 sets in tetrahedral fields, with delta_t about four-ninths of delta_o. Comparing the gap with the pairing energy tells you high spin from low spin, and the spin-only formula turns that into a predicted magnetic moment. Colour, magnetism and stability all flow from this one diagram.