Crystal Field & Ligand Field Theory

In an octahedral field, the six ligands lie along the x, y, and z axes. The d_z² and d_x²−y² orbitals point directly at the ligands, so they experience greater electrostatic repulsion and are raised in energy (the e_g set). The d_xy, d_xz, and d_yz orbitals point between the ligand axes, experience less repulsion, and are stabilized (the t_2g set). The energy gap between these two sets is Δ_oct.

Three factors increase Δ_oct: (1) higher metal oxidation state increases the effective nuclear charge, pulling ligands closer; (2) moving down a group (first → second → third row) increases the spatial extent of d orbitals, enhancing metal–ligand overlap; (3) stronger-field ligands (e.g. CN⁻, CO) interact more strongly. Combining all three — high charge, third-row metal, strong-field ligand — gives the largest Δ_oct.

Two factors make Δ_tet smaller. First, there are only four ligands instead of six, reducing the total electrostatic interaction. Second, in a tetrahedron, no ligand points directly along the d orbital axes — the interaction is indirect. Combined, these effects give Δ_tet ≈ ⁴⁄₉ Δ_oct for the same metal–ligand combination. The splitting pattern is also inverted: the t₂ set is raised and the e set is lowered.

For d⁶ high-spin: t_2g⁴e_g². CFSE = 4(−0.4Δ) + 2(+0.6Δ) = −0.4Δ_oct. For d⁶ low-spin: t_2g⁶e_g⁰. CFSE = 6(−0.4Δ) = −2.4Δ_oct, but three extra electron pairings cost 2P (there is already one forced pairing in high-spin, giving a net cost of 2P additional). The low-spin configuration is favored when Δ_oct is large enough to offset the pairing energy penalty.

When Δ_oct < P (pairing energy), electrons prefer to occupy the higher-energy e_g orbitals rather than pair up in t_2g, giving a high-spin configuration. When Δ_oct > P, the energy cost of pairing is less than the cost of promoting an electron to e_g, so electrons fill t_2g first — a low-spin configuration. This competition applies to d⁴–d⁷ octahedral complexes.

The spectrochemical series ranks ligands by their ability to split d orbitals: I⁻ < Br⁻ < Cl⁻ < F⁻ < H₂O < NH₃ < en < NO₂⁻ < CN⁻ < CO. F⁻ is near the weak-field end, producing a small Δ_oct that cannot overcome the pairing energy P — hence high-spin (4 unpaired electrons). CN⁻ is near the strong-field end, generating a large Δ_oct > P, so all six d electrons pair in t_2g — low-spin (0 unpaired electrons).

In high-spin d⁵: all five d orbitals are singly occupied (t_2g³e_g²), giving 5 unpaired electrons and maximum spin multiplicity. In low-spin d⁵: electrons fill t_2g first, giving t_2g⁵e_g⁰. Two of the three t_2g orbitals hold paired electrons and one holds a single electron, yielding 1 unpaired electron. This dramatic difference (5 vs. 1) is directly detectable by magnetic susceptibility measurements.

The tetrahedral crystal field splitting Δ_tet is inherently only about 4/9 the magnitude of Δ_oct for the same metal and ligand. This reduced splitting is almost never large enough to exceed the electron pairing energy P. Since Δ_tet < P in virtually all cases, electrons occupy the higher e set before pairing in the lower t₂ set, producing a high-spin configuration. Low-spin tetrahedral complexes would require an unusually large Δ_tet that is rarely achievable.

Strong Jahn-Teller distortions occur when the e_g orbitals are unevenly occupied, because these orbitals point directly at the ligands and unequal occupation creates a large driving force for distortion. High-spin d⁴ (t_2g³e_g¹) and d⁹ (t_2g⁶e_g³) both have one e_g orbital singly occupied and the other empty or doubly occupied. Low-spin d⁷ also shows a strong effect. Uneven t_2g occupation produces weaker distortions.

Cu²⁺ is d⁹ with configuration t_2g⁶e_g³. The three electrons in the doubly degenerate e_g set cannot occupy both orbitals equally. In a tetragonal elongation, the axial ligands move away, lowering d_z² relative to d_x²−y². The extra electron occupies the now-lower d_z², removing the degeneracy and lowering the total energy. This is a textbook example of a strong Jahn-Teller distortion.

Starting from a regular octahedron, tetragonal elongation progressively lengthens the two axial bonds. In the extreme limit, the axial ligands are effectively removed, leaving four ligands in the xy plane — a square planar geometry. This causes extensive further splitting of the d orbitals: d_x²−y² becomes highest in energy (pointing directly at the four remaining ligands), while d_z² drops significantly. The resulting d orbital order from low to high energy is approximately d_xz,d_yz < d_z² < d_xy < d_x²−y².

Pd²⁺ and Pt²⁺ are d⁸ ions from the second and third transition rows, which means their d orbitals are spatially extended and produce very large crystal field splittings. In square planar geometry, the d_x²−y² orbital is pushed so high in energy that all eight d electrons fill the four lower orbitals, leaving d_x²−y² empty. This is strongly stabilized when Δ is large. First-row d⁸ ions like Ni²⁺ can be tetrahedral or square planar depending on the ligand field strength.

Crystal field theory treats ligands as point charges or dipoles — a purely electrostatic model that ignores covalency entirely. Ligand field theory (LFT) instead constructs molecular orbitals from metal d orbitals and ligand σ/π orbitals, capturing covalent bonding, π back-donation, and the true electronic structure of the complex. LFT explains the spectrochemical series, magnetic properties, and spectroscopic data that CFT cannot fully account for.

π-Acceptor ligands like CO and CN⁻ possess low-energy empty π* orbitals that interact with the filled metal t_2g orbitals. This π back-bonding delocalizes electron density from metal to ligand, stabilizing (lowering) the t_2g set. Since the e_g level is unaffected by this interaction, the gap Δ_oct increases. This is why π-acceptor ligands appear at the strong-field end of the spectrochemical series, and conversely, π-donor ligands (like halides) raise t_2g and reduce Δ_oct.

CO acts as both a strong σ-donor through its lone pair on carbon and a strong π-acceptor through its low-lying π* orbitals. The σ-donation pushes the e_g set up, while the π-acceptance stabilizes the t_2g set through back-bonding — both effects increase Δ_oct. In contrast, H₂O is a modest σ-donor with no significant π-acceptor capability; its lone pairs act as weak π-donors that slightly raise t_2g, reducing Δ_oct. This dual σ/π character is why CO sits at the very top of the spectrochemical series.