Transition Metals and the f-Block: Coordination Chemistry, Crystal Fields, Spectra, Magnetism and Mechanisms

A complex is described by its coordination number and geometry: 2 (linear, Ag(NH₃)₂⁺), 4 (tetrahedral for d¹⁰ and weak-field ions, square planar for d⁸ Pd²⁺, Pt²⁺, Au³⁺ and strong-field Ni²⁺), 5 (trigonal bipyramidal or square pyramidal, close in energy) and 6 (octahedral, by far the commonest). Structural isomers differ in which atoms are bonded: ionisation ([Co(NH₃)₅Br]SO₄ and [Co(NH₃)₅SO₄]Br), hydrate ([Cr(H₂O)₆]Cl₃, violet, and [CrCl(H₂O)₅]Cl₂·H₂O, green), linkage (nitro –NO₂ and nitrito –ONO; S- and N-bound thiocyanate) and coordination isomers ([Co(NH₃)₆][Cr(CN)₆] and [Cr(NH₃)₆][Co(CN)₆]). Stereoisomers differ only in arrangement.

Valence-bond theory describes octahedral complexes as inner-orbital d²sp³ (low-spin, [Fe(CN)₆]⁴⁻) or outer-orbital sp³d² (high-spin, [FeF₆]³⁻), and square planar ones as dsp²; it explains geometry and magnetism but not colour. Crystal-field theory treats the ligands as point charges. In an octahedral field the d orbitals split into a lower t₂g set (d_xy, d_xz, d_yz, at −0.4Δₒ) and an upper e_g set (d_z², d_x²−y², at +0.6Δₒ), the barycentre preserved. In a tetrahedral field the order inverts — e below t₂ — and Δ_t ≈ (4/9)Δₒ, so tetrahedral complexes are almost always high-spin. In a square planar field d_x²−y² rises far above the rest, which makes d⁸ ions with strong-field ligands diamagnetic and square planar.

The crystal-field stabilisation energy of an octahedral configuration t₂gˣe_gʸ is CFSE = (−0.4x + 0.6y)Δₒ, with a pairing energy P added for each pair forced together beyond the free-ion arrangement. When Δₒ > P a d⁴–d⁷ ion is low-spin, when Δₒ < P high-spin. Δₒ grows with the oxidation state (Co³⁺ > Co²⁺), down a group (4d and 5d complexes are nearly always low-spin) and along the spectrochemical series: I⁻ < Br⁻ < S²⁻ < SCN⁻ (S) < Cl⁻ < F⁻ < OH⁻ < C₂O₄²⁻ < H₂O < NCS⁻ (N) < NH₃ < en < bpy < phen < NO₂⁻ < CN⁻ < CO. The ligand-field (MO) picture explains the order: σ-donors raise e_g*, π-donors (halides, OH⁻) push t₂g up and shrink Δₒ, π-acceptors (CO, CN⁻, phosphines) pull t₂g down and enlarge it. CFSE explains the double-humped curve of hydration enthalpies across the 3d series (minima at d⁰, d⁵ high-spin and d¹⁰), trends in ionic radii, and site preferences in spinels.

The Jahn–Teller theorem: a non-linear molecule in an orbitally degenerate electronic state distorts to remove the degeneracy. The effect is strong when the degeneracy is in the σ-antibonding e_g set — high-spin d⁴ (Cr²⁺, Mn³⁺), low-spin d⁷ and d⁹ (Cu²⁺) — and usually shows as a tetragonal elongation, two long axial bonds and four short equatorial ones; t₂g degeneracies give only weak distortions. Metal–metal multiple bonds arise when d orbitals on adjacent metals overlap: in [Re₂Cl₈]²⁻ d_z² gives a σ bond, d_xz and d_yz two π bonds, and d_xy a δ bond, so the configuration σ²π⁴δ² gives a quadruple bond (bond order 4, Re–Re 224 pm). The δ overlap needs the two ReCl₄ units eclipsed, which is why the ion is eclipsed despite the steric cost.

For first-row complexes the orbital angular momentum is largely quenched by the ligand field, and the moment is close to the spin-only value μ_s = √(n(n + 2)) BM for n unpaired electrons (1.73, 2.83, 3.87, 4.90 and 5.92 BM for n = 1 to 5). Orbital contributions survive when the ground term is a T term (for example octahedral high-spin Co²⁺, t₂g⁵e_g², measured 4.7–5.2 BM against 3.87 spin-only), but not for A or E ground terms (Ni²⁺ octahedral, Cr³⁺, Mn²⁺, which match the spin-only value closely). The susceptibility of a magnetically dilute paramagnet follows the Curie law χ = C/T, and μ_eff = 2.828√(χ_M T) BM.

The free-ion ground terms of dⁿ are ²D (d¹, d⁹), ³F (d², d⁸), ⁴F (d³, d⁷), ⁵D (d⁴, d⁶) and ⁶S (d⁵). In an octahedral field a D term splits into T₂g + E_g, an F term into A₂g + T₂g + T₁g, and there is a higher P term of the same multiplicity for d², d³, d⁷ and d⁸. Selection rules: the spin rule ΔS = 0 and the Laporte rule (g ↔ u only). d–d transitions in an octahedral complex are g → g and Laporte-forbidden, so they are weak (ε ≈ 1–100 L mol⁻¹ cm⁻¹) and gain intensity only through vibronic coupling; tetrahedral complexes, lacking a centre of symmetry and with d–p mixing, are more intense (ε ≈ 100–1000). Spin-forbidden bands are weaker still, which is why high-spin d⁵ [Mn(H₂O)₆]²⁺, which has no spin-allowed d–d transition at all, is almost colourless.

Orgel diagrams plot the spin-allowed terms against Δ for weak fields: d¹, d⁴, d⁶ and d⁹ (D ground terms) show one band — [Ti(H₂O)₆]³⁺ absorbs at about 20 000 cm⁻¹ (500 nm), which is Δₒ itself — while d², d³, d⁷ and d⁸ (F ground terms) show three; for octahedral d³ Cr³⁺ and d⁸ Ni²⁺ the lowest band ⁴A₂g → ⁴T₂g (or ³A₂g → ³T₂g) equals Δₒ directly. Tanabe–Sugano diagrams plot E/B against Δₒ/B for all terms, including spin-forbidden ones, and show the high-spin to low-spin crossover as a vertical line for d⁴–d⁷. The Racah parameter B measures interelectronic repulsion; in a complex it is smaller than in the free ion because the d electrons spread onto the ligands — the nephelauxetic ("cloud-expanding") effect — with β = B(complex)/B(free ion) < 1 and a nephelauxetic series F⁻ > H₂O > NH₃ > en > Cl⁻ > CN⁻ > Br⁻ > I⁻ (β decreasing, i.e. more covalency to the right).

Charge-transfer bands move an electron between metal and ligand and are Laporte- and spin-allowed, so they are intense (ε ≈ 10³–10⁵). LMCT (ligand to metal) dominates for metals in high oxidation states with filled-π ligands: the intense purple of MnO₄⁻ and yellow of CrO₄²⁻ are O → M transitions in d⁰ ions, and the colours of the silver halides darken as the halide becomes easier to oxidise. MLCT (metal to ligand) occurs for electron-rich metals with low-lying π* ligands, as in the orange [Ru(bpy)₃]²⁺ and [Fe(phen)₃]²⁺.

Thermodynamic stability (formation constants, the chelate effect) and kinetic lability (the rate of ligand exchange) are independent. Taube called a complex labile if it exchanges ligands within about a minute at room temperature and inert otherwise. [Ni(CN)₄]²⁻ is very stable yet exchanges CN⁻ rapidly; [Co(NH₃)₆]³⁺ is thermodynamically unstable in acid yet survives for days. CFT explains the pattern: octahedral d³ and low-spin d⁴–d⁶ ions (Cr³⁺, Co³⁺, Rh³⁺, Pt⁴⁺) are inert because losing a ligand costs much ligand-field stabilisation; ions with e_g electrons (Cu²⁺, high-spin ions) and d⁰ or d¹⁰ ions are labile.

Substitution mechanisms are classed as dissociative (D), with a lower-coordinate intermediate; associative (A), with a higher-coordinate one; and the concerted interchange paths I_d and I_a. Octahedral substitution is usually dissociative in character: the rate depends little on the entering ligand, and ΔV‡ and ΔS‡ are positive. Base hydrolysis of [Co(NH₃)₅Cl]²⁺ is fast because OH⁻ deprotonates an NH₃ and the amido conjugate base loses Cl⁻ easily (the S_N1CB mechanism). Square planar d⁸ substitution is associative through a five-coordinate intermediate, and the ligand trans to the leaving group governs its rate — the trans effect: CO ≈ CN⁻ ≈ C₂H₄ > PR₃ ≈ H⁻ > CH₃⁻ > I⁻ > Br⁻ > Cl⁻ > NH₃ > H₂O. It is used synthetically: [PtCl₄]²⁻ + 2NH₃ gives cis-[PtCl₂(NH₃)₂] (the second NH₃ replaces a Cl trans to Cl), while [Pt(NH₃)₄]²⁺ + 2Cl⁻ gives the trans isomer.

Tris-chelate complexes M(AA)₃ can racemise without breaking a bond by twist mechanisms through a trigonal-prismatic intermediate: the Bailar (trigonal) twist rotates two opposite triangular faces about the real C₃ axis, and the Ray–Dutt (rhombic) twist rotates about one of the three pseudo-C₃ axes. Electron transfer between complexes is outer-sphere when both coordination shells stay intact and the electron tunnels between them — the rate follows Marcus theory and is fast when the reorganisation energy is small ([Fe(CN)₆]⁴⁻/³⁻) — or inner-sphere when a ligand bridges the two metals. Taube’s proof: [Co(NH₃)₅Cl]²⁺ + [Cr(H₂O)₆]²⁺ gives [CrCl(H₂O)₅]²⁺, with the chloride transferred to chromium, because the substitution-inert Cr(III) product can only have gained Cl while it was still labile Cr(II) bound to the bridge.

Recovery. The lanthanides occur together in minerals such as monazite (a phosphate, with thorium) and bastnäsite (a fluorocarbonate). After digestion and removal of thorium, cerium is often separated first by oxidation to Ce⁴⁺ and europium by reduction to Eu²⁺. The rest, all Ln³⁺ of nearly identical chemistry, are separated by ion-exchange chromatography or counter-current solvent extraction (tributyl phosphate or organophosphoric acids), which exploit the small, steady fall in ionic radius: with a complexing eluent the smaller, more strongly complexed heavy lanthanides come off a cation-exchange column first.

Periodic properties. The 4f orbitals are buried inside the 5s and 5p shells and shield the nuclear charge poorly, so Ln³⁺ radii shrink steadily from La³⁺ (about 103 pm) to Lu³⁺ (about 86 pm) — the lanthanide contraction. Its consequences: Y³⁺ falls among the heavy lanthanides; the 5d elements that follow have almost the same radii as their 4d congeners (Zr/Hf, Nb/Ta, Mo/W), making them hard to separate; and basicity of the hydroxides falls from La to Lu. +3 is the characteristic oxidation state; Ce⁴⁺ (f⁰), Eu²⁺ (f⁷) and Yb²⁺ (f¹⁴) exist because they reach empty, half-filled or filled f shells. The actinides show a wider range of oxidation states early in the series (U +3 to +6, as in the linear uranyl ion UO₂²⁺; Np and Pu up to +7), because the 5f orbitals extend further and lie closer in energy to 6d and 7s; after americium +3 dominates as for the lanthanides.

Spectra and magnetism. Because the 4f electrons are shielded, ligand fields split f levels by only about 100 cm⁻¹, far less than spin–orbit coupling. So f–f absorption bands are sharp, weak (Laporte-forbidden) and almost independent of the ligand, and emission is line-like — the red of Eu³⁺ and green of Tb³⁺ in phosphors. Magnetic moments follow the free-ion J value, μ_eff = g_J√(J(J + 1)) BM with g_J = 1 + [J(J + 1) + S(S + 1) − L(L + 1)]/[2J(J + 1)], not the spin-only formula. Gd³⁺ (f⁷, ⁸S₇/₂, g = 2) gives 7.94 BM; Nd³⁺ (f³, ⁴I₉/₂) 3.62 BM. Sm³⁺ and Eu³⁺ deviate from this formula because their excited J levels lie within kT and are thermally populated.