Coordination Chemistry

Werner’s primary valence (Hauptvalenz) corresponds to the modern concept of oxidation state — the number of ionic bonds or charges the metal satisfies. His secondary valence (Nebenvalenz) corresponds to the coordination number — the number of ligands directly bonded to the metal center. For example, in [Co(NH₃)₆]Cl₃, the primary valence is 3+ and the secondary valence (coordination number) is 6.

The most frequently encountered coordination numbers are 4 and 6. Coordination number 4 gives rise to either tetrahedral or square planar geometry, while coordination number 6 overwhelmingly adopts octahedral geometry. Tetrahedral is favoured by small, highly charged metals with large ligands, while square planar is characteristic of d⁸ metal ions (e.g. Pt²⁺, Pd²⁺, Ni²⁺ with strong-field ligands).

Ni(II) is a d⁸ ion. CN⁻ is a strong-field ligand that forces electron pairing, producing a diamagnetic, square planar complex with a large crystal field splitting. Cl⁻ is a weak-field ligand that cannot force pairing, so [NiCl₄]²⁻ adopts the tetrahedral geometry with two unpaired electrons (paramagnetic). The ligand field strength determines which geometry a d⁸ complex adopts.

In a d⁸ system, square planar geometry creates a large energy gap between the d_x²−y² orbital (strongly antibonding) and the other four d orbitals. With strong-field ligands, this gap is large enough that all eight electrons occupy the four lower-energy orbitals, leaving d_x²−y² empty. This produces a diamagnetic, 16-electron complex with substantial crystal field stabilisation energy (CFSE). Second- and third-row metals (Pd, Pt, Au) especially favour this geometry because their larger 4d/5d orbitals produce inherently larger splittings.

Denticity refers to the number of donor atoms a single ligand uses to bind one metal centre. Monodentate ligands (NH₃, Cl⁻, H₂O) donate through one atom. Bidentate ligands like ethylenediamine (en) use two nitrogen donors, forming a chelate ring. Polydentate ligands have three or more donor atoms — EDTA, for instance, is hexadentate with six donors. This classification is distinct from bridging, which describes a ligand connecting two or more metal centres.

The chelate effect is primarily entropic in origin. When a bidentate ligand like ethylenediamine (en) replaces two monodentate NH₃ molecules, the reaction [M(NH₃)₂]ⁿ⁺ + en → [M(en)]ⁿ⁺ + 2 NH₃ goes from 2 particles on the left to 3 on the right. This net increase in the number of free molecules produces a favourable entropy change (ΔS > 0), making ΔG more negative. The enthalpy change is typically similar since the M–N bond strengths are comparable, so entropy is the dominant driving force.

EDTA⁴⁻ coordinates through six donor atoms: two amine nitrogen atoms from the ethylenediamine backbone and four oxygen atoms, one from each of the four carboxylate groups. This allows a single EDTA molecule to occupy all six coordination sites around an octahedral metal centre, forming five chelate rings. The resulting 1:1 metal-EDTA complex is extremely stable, which is why EDTA is widely used in analytical chemistry and as a sequestering agent for metal ions.

HSAB (Hard-Soft Acid-Base) theory predicts that hard acids prefer hard bases and soft acids prefer soft bases. In SCN⁻, the nitrogen end is a harder donor (small, electronegative, low polarisability), while the sulfur end is a softer donor (large, polarisable, lower electronegativity). Therefore, hard metal centres like Cr³⁺ and Co³⁺ preferentially bind through nitrogen (isothiocyanato, M–NCS), while soft metals like Pt²⁺ and Pd²⁺ bind through sulfur (thiocyanato, M–SCN).

The stepwise stability constant K_n describes the equilibrium for adding a single ligand to a complex that already has (n−1) ligands: [ML_n−1] + L → [ML_n]. The overall stability constant β_n describes the complete formation from the aqua ion: M + nL → [ML_n]. These are related by β_n = K₁ × K₂ × … × K_n, which is the product (not sum) of all stepwise constants. Larger values of β_n indicate greater thermodynamic stability.

Both complexes contain four Cu–N bonds with similar individual bond strengths, so the enthalpy contributions are comparable. However, the reaction replacing four NH₃ with two en molecules releases two extra free particles into solution (4 NH₃ released, 2 en consumed). This net increase in disorder produces a favourable entropy change. Experimentally, log β for [Cu(en)₂]²⁺ is significantly larger than for [Cu(NH₃)₄]²⁺, directly demonstrating the chelate effect.

The Irving–Williams series reflects the combined effects of decreasing ionic radius across the period (increasing electrostatic attraction) and crystal field stabilisation energy (CFSE). From Mn²⁺ to Cu²⁺, CFSE generally increases. Cu²⁺ (d⁹) additionally benefits from a Jahn–Teller distortion that further stabilises the complex. Zn²⁺ (d¹⁰) has a completely filled d shell with zero CFSE, causing a sharp drop in stability despite its small ionic radius. This explains the characteristic “tent-shaped” plot.

The macrocyclic effect describes the observation that macrocyclic ligands (e.g. porphyrins, crown ethers, cyclam) form significantly more stable complexes than analogous open-chain polydentate ligands. This extra stabilisation has both enthalpic and entropic origins. Enthalpically, the macrocycle is pre-organised into a shape complementary to the metal ion, reducing the reorganisation energy upon binding. Entropically, the cyclic ligand is more rigid and loses less conformational freedom upon coordination. The macrocyclic effect thus goes beyond the chelate effect, which itself already exceeds monodentate ligand binding.

In an octahedral field, the six ligands lie along the x, y, and z axes. The d_z² and d_x²−y² orbitals (collectively called the e_g set) have their lobes pointing directly at the ligands, resulting in strong σ-antibonding interactions that raise their energy. The d_xy, d_xz, and d_yz orbitals (the t_2g set) point between the ligands and experience less repulsion, remaining at lower energy. The energy gap between these two sets is Δ_oct.

The standard spectrochemical series from weak field to strong field includes: I⁻ < Br⁻ < Cl⁻ < F⁻ < H₂O < NH₃ < en < NO₂⁻ < CN⁻ < CO. Halides are weak-field π-donors, water and ammonia are intermediate σ-donors, and CN⁻ and CO are strong-field ligands that act as π-acceptors. The ability to accept electron density into empty π* orbitals is what makes CO and CN⁻ produce the largest Δ_oct values.

CO binds to metals through carbon using a lone pair in a σ-donation step (C → M). Simultaneously, filled metal d orbitals (t_2g in an octahedral complex) donate electron density back into the empty π* antibonding orbitals of CO — this is π-back-bonding (M → CO). These two processes are synergistic: σ-donation increases electron density on the metal, which enhances π-back-donation, and vice versa. The π-back-bonding stabilises the t_2g set while the e_g set remains destabilised by σ-antibonding, resulting in a very large Δ_oct.