Crystal Field Theory

A free transition metal ion has five d-orbitals (dxy, dxz, dyz, dx²−y², dz²) that are all equal in energy — they are degenerate. When ligands approach to form a complex, their electron pairs (or negative charges) repel the metal's d-electrons, raising the overall energy of the d-orbitals above the free-ion level. The key insight of CFT is that this repulsion is not equal for all five orbitals, because they have different spatial orientations.

In an octahedral complex, six ligands approach along the ±x, ±y, and ±z axes. The two orbitals that point directly along these axes — dx²−y² and dz² — experience maximum repulsion and are raised in energy. These form the higher-energy eg set. The three orbitals that point between the axes — dxy, dxz, and dyz — experience less repulsion and are lowered in energy, forming the lower-energy t₂g set. The energy gap between the eg and t₂g sets is the octahedral crystal field splitting energy, Δₒ (also written 10Dq). By convention the t₂g orbitals are stabilized by −0.4Δₒ per electron and the eg orbitals are destabilized by +0.6Δₒ per electron, relative to the barycentre (the weighted average energy of all five orbitals).

In a tetrahedral complex, four ligands approach along directions that bisect the face diagonals of a cube. None of the d-orbitals points directly at a ligand, but the dxy, dxz, and dyz orbitals (now called the e set in tetrahedral symmetry) are closer to the ligand directions than the dx²−y² and dz² orbitals (the t₂ set). The splitting is therefore inverted compared with the octahedral case: the e set is higher in energy and the t₂ set is lower. Critically, the magnitude is smaller: Δₜ ≈ (4/9)Δₒ for the same metal-ligand combination, because there are only four ligands and none points directly at an orbital. This smaller splitting means most tetrahedral complexes are high-spin.

Different ligands produce different magnitudes of Δ. The spectrochemical series ranks ligands from weakest to strongest field: I⁻ < Br⁻ < S²⁻ < SCN⁻ < Cl⁻ < NO₃⁻ < F⁻ < OH⁻ < H₂O < NCS⁻ < NH₃ < en (ethylenediamine) < bipy < NO₂⁻ < CN⁻ < CO. Strong-field ligands (CN⁻, CO) produce large Δ and tend to give low-spin complexes; weak-field ligands (I⁻, Cl⁻) produce small Δ and tend to give high-spin complexes. The position of a ligand in the series cannot be fully explained by electrostatics alone — π-bonding interactions (captured better in Ligand Field Theory) also matter, which is why neutral CO is a stronger-field ligand than negatively charged F⁻.

When filling d-electrons into split orbitals, electrons can either pair in the lower set (low-spin, favoured when Δ > pairing energy P) or spread out to obey Hund's rule (high-spin, favoured when Δ < P). Crystal Field Stabilization Energy (CFSE) is calculated as: CFSE = (number of t₂g electrons × −0.4Δₒ) + (number of eg electrons × +0.6Δₒ). For a d³ ion like Cr³⁺ in [Cr(H₂O)₆]³⁺: 3 electrons in t₂g → CFSE = 3 × (−0.4Δₒ) = −1.2Δₒ. Larger negative CFSE means greater thermodynamic stabilization of the complex.

When white light strikes a coordination compound, photons whose energy matches Δ are absorbed, promoting a d-electron from the t₂g set to the eg set (a d–d transition). The complementary colour is transmitted and observed. For [Ti(H₂O)₆]³⁺ (a d¹ ion), the single d-electron in t₂g absorbs green light (~500 nm), and the compound appears violet-purple. Using E = hc/λ, the absorbed wavelength directly gives Δₒ. Compounds with no d–d transitions possible (d⁰ or d¹⁰) are colourless unless charge-transfer bands are present (e.g. MnO₄⁻ is intensely purple due to ligand-to-metal charge transfer, not a d–d transition).

• \text{CFSE} — Crystal Field Stabilization Energy (kJ mol⁻¹ (or expressed as a multiple of Δₒ)): The net energy stabilization gained by placing d-electrons in the split orbitals rather than in a hypothetical spherical field at the barycentre energy.
• n_{t_{2g}} — Number of t₂g electrons (dimensionless (integer 0–6)): Electrons occupying the lower-energy t₂g orbitals (dxy, dxz, dyz) in an octahedral complex.
• n_{e_g} — Number of eg electrons (dimensionless (integer 0–4)): Electrons occupying the higher-energy eg orbitals (dx²−y², dz²) in an octahedral complex.
• \Delta_o — Octahedral crystal field splitting energy (kJ mol⁻¹ or cm⁻¹): The energy gap between the t₂g and eg orbital sets in an octahedral complex, determined experimentally from UV-Vis spectroscopy as the wavelength of maximum d–d absorption.

Problem. Calculate the CFSE for [Fe(CN)₆]⁴⁻. Iron in this complex is Fe²⁺ (d⁶). CN⁻ is a strong-field ligand, so the complex is low-spin. The experimentally measured Δₒ = 394 kJ mol⁻¹. Also determine whether the complex is diamagnetic or paramagnetic.

Answer. [Fe(CN)₆]⁴⁻ has a CFSE of −2.4Δₒ = −945.6 kJ mol⁻¹ and is diamagnetic (zero unpaired electrons) due to its low-spin d⁶ configuration.

• Introduction to Coordination Compounds — CFT describes the electronic consequences of the metal-ligand bonding arrangement introduced in this foundational topic — understanding coordination number and geometry is prerequisite to applying CFT.
• Magnetic Properties of Coordination Complexes — High-spin and low-spin configurations predicted by CFT directly determine whether a complex is paramagnetic or diamagnetic, measured by magnetic moment calculations.
• Colour and Light Absorption in Transition Metal Complexes — The colour of a complex is the observable consequence of the d–d transition energy equal to Δ; the spectrochemical series determines which wavelength is absorbed.
• Ligand Field Theory — Ligand Field Theory extends CFT by incorporating covalent (molecular orbital) contributions, explaining anomalies in the spectrochemical series such as why neutral CO is a stronger-field ligand than F⁻.
• Atomic Orbitals and Quantum Numbers — CFT is built on the spatial shapes and orientations of the five d-orbitals, which are defined by their angular quantum numbers l = 2 and m_l = −2, −1, 0, +1, +2.