The spin-only magnetic moment of $[\text{Mn(CN)}_6]^{3-}$ is approximately:

Crystal Field Theory (CFT), developed by Hans Bethe and John van Vleck in the 1930s, is a model that explains the bonding, colour, and magnetic properties of transition metal coordination compounds by treating the interaction between the central metal ion and the surrounding ligands as a purely electrostatic (ionic) interaction. In CFT, the ligands are treated as negative point charges (or point dipoles for neutral ligands) that approach the central metal cation and exert an electrostatic field (the "crystal field") on the metal d-orbitals. In an octahedral complex, six ligands approach along the Cartesian axes (+x, -x, +y, -y, +z, -z). The five d-orbitals, which are degenerate (equal energy) in the free metal ion, split into two sets in the octahedral crystal field: the t2g set (dxy, dyz, dxz) pointing between the Cartesian axes, away from the ligands — lower energy (stabilised, less repelled by ligands). The eg set (dx2-y2, dz2) pointing directly toward the ligands along the axes — higher energy (destabilised, more repelled by ligands). The energy gap between t2g and eg is the octahedral crystal field splitting energy, delta_o (also written 10Dq). The actual energies: t2g is stabilised by -0.4 delta_o per electron (below barycentre), eg is destabilised by +0.6 delta_o per electron (above barycentre). The stabilisation of d-electrons by the crystal field (relative to the barycentre) is the Crystal Field Stabilisation Energy (CFSE).

When there are more than three d-electrons in an octahedral complex (d4-d7 configurations), a choice arises: electrons can either enter the eg orbitals (against the crystal field splitting) or pair up in the already singly occupied t2g orbitals (against the pairing energy P). If delta_o > P (strong field ligands): it is energetically cheaper to pair electrons in t2g than to place them in the high-energy eg → low spin configuration with minimum number of unpaired electrons. If delta_o < P (weak field ligands): it is cheaper to place electrons in eg (maintaining Hund's rule) → high spin configuration with maximum unpaired electrons. The pairing energy P is a property of the metal ion and varies across the series but is typically in the range 15,000-25,000 cm-1 for first-row transition metals. The magnitude of delta_o depends on: the ligand (spectrochemical series), the metal (higher oxidation state → larger delta_o; 4d > 3d metals for same ligand), and the geometry (tetrahedral delta_t ≈ 4/9 delta_o, so most tetrahedral complexes are high spin). The crystal field spin states determine: magnetic moment (more unpaired = higher mu), colour (different splitting = different absorption wavelength), and reactivity (low-spin complexes often kinetically inert).

The magnetic properties of transition metal complexes arise from the spin and orbital angular momenta of d-electrons. In first-row transition metals, the orbital contribution to the magnetic moment is largely quenched by ligand field effects and spin-orbit coupling is relatively small, so the spin-only formula provides a good approximation: mu_spin-only = sqrt(n(n+2)) BM, where n = number of unpaired electrons and BM = Bohr magneton (1 BM = 9.274 × 10^-24 J/T). Compounds with unpaired electrons are paramagnetic (attracted to a magnetic field) and their paramagnetism can be measured by the Gouy balance method (measuring the attraction of a sample to an inhomogeneous magnetic field) or by modern superconducting quantum interference device (SQUID) magnetometers. Compounds with no unpaired electrons are diamagnetic (slightly repelled by magnetic field). Ferromagnetism: in certain metal alloys and compounds (Fe, Co, Ni, and their alloys), the electron spins of adjacent atoms align parallel throughout large magnetic domains even without an external field, giving permanent magnetism. This occurs when the exchange interaction between unpaired electrons on adjacent atoms is ferromagnetic (prefers parallel alignment). Ferrimagnetism: mixed-oxide spinels (ferrites, e.g., Fe3O4 which is FeO·Fe2O3) have antiparallel alignment of unequal magnetic moments, giving a net magnetic moment.

The spectrochemical series arranges ligands in order of increasing crystal field splitting delta_o that they produce for a given metal: I- < Br- < S2- < SCN- < Cl- < F- < OH- < C2O4^2- < H2O < NCS- < pyridine < NH3 < en < bipy < phen < NO2- < PPh3 < CN- < CO. The position of a ligand in the spectrochemical series depends on two types of interaction with the metal: Sigma donation: all ligands donate a lone pair to the metal through a sigma bond; this raises the energy of all d-orbitals but affects the d-orbital splitting less. Pi interactions: (a) pi donor ligands (I-, Br-, Cl-, F-, OH-, H2O, and other ligands with lone pairs in p-orbitals): these interact with the t2g orbitals by pi donation from filled ligand pi orbitals to empty metal t2g orbitals, RAISING the t2g energy and therefore DECREASING delta_o → these ligands are lower in the spectrochemical series (weak field). (b) pi acceptor ligands (CN-, CO, NO2-, PPh3): these accept electron density from filled metal t2g orbitals into empty ligand pi* antibonding orbitals (pi backbonding), LOWERING the t2g energy and INCREASING delta_o → these ligands are highest in the spectrochemical series (strong field). Pure sigma donors (NH3, en): intermediate, no pi interaction.

Alfred Werner's coordination theory (1893) was a remarkable achievement of chemical intuition and careful experimental observation — he proposed the octahedral geometry and the existence of primary (ionic/oxidation state) and secondary (coordination number) valences for transition metals before X-ray crystallography or quantum mechanics existed. His resolution of chiral cobalt complexes in 1911 and his synthesis of hundreds of coordination compounds provided the experimental foundation for coordination chemistry. However, Werner's theory is essentially a classical theory that cannot explain why certain ligands are stronger than others or why complexes are coloured. Crystal field theory, developed in the 1930s, successfully explained colour (d-d transitions) and magnetic properties (high spin vs low spin) of coordination compounds, but was criticised for being purely electrostatic and ignoring covalent bonding. Ligand field theory (a molecular orbital approach that incorporates both electrostatic and covalent contributions) provides the most complete and accurate description: it explains the spectrochemical series in terms of sigma and pi interactions (as described above), predicts more subtle aspects of spectra like charge-transfer bands, and provides a quantum mechanical foundation for all the empirical observations of coordination chemistry.

Coordination chemistry is not merely an academic exercise but underlies a vast range of practical applications of enormous economic and medical importance. In medicine: Cisplatin [cis-Pt(NH3)2Cl2] is a square planar Pt(II) complex that is one of the most important anticancer drugs; it works by forming crosslinks between adjacent guanine bases on the same DNA strand, blocking DNA replication and triggering apoptosis in rapidly dividing cancer cells. Carboplatin and oxaliplatin are second and third-generation platinum anticancer drugs with improved side-effect profiles. Bleomycin (a natural coordination compound containing Fe2/3+) is another anticancer drug that cleaves DNA by generating reactive oxygen species. EDTA (ethylenediaminetetraacetic acid) chelate therapy is used to treat heavy metal poisoning (Pb, Hg, As) by forming very stable, water-soluble complexes with the toxic metals that are rapidly excreted by the kidneys. Gadolinium(III) complexes are used as contrast agents in MRI (magnetic resonance imaging) — the paramagnetic Gd3+ ion (7 unpaired f-electrons, maximum paramagnetism) shortens the T1 relaxation time of water protons in nearby tissue, enhancing contrast. In industry: Wilkinson's catalyst [RhCl(PPh3)3] is a homogeneous catalyst for hydrogenation of alkenes. Ziegler-Natta catalyst [TiCl4/Al(C2H5)3] for stereospecific polymerisation. SHOP (Shell Higher Olefin Process) uses Ni phosphino-enolate catalyst. Gold cyanide complexes [Au(CN)2]- in gold extraction (hydrometallurgy). Vitamin B12 (cyanocobalamin) contains Co(III) in a corrin ring (similar to porphyrin) and is the most complex cofactor known, essential for DNA synthesis and neurological function.