Valence Bond & Crystal Field Theory — formula sheet

Coordinate (dative) bond: a two-electron, two-centre covalent bond in which both electrons came from the same atom — the ligand donor. Once formed, it is indistinguishable from any other single bond; the distinction is one of bookkeeping, not of physics.

Hybridisation: a mathematical recombination of atomic orbitals on one atom into an equal number of new, equivalent, directed orbitals whose lobes point towards the bonded neighbours. It is a construction on paper, not a physical process the atom undergoes.

Inner-orbital complex (also low spin, spin-paired, or in Pauling’s word hyperligated): the two d orbitals used in hybridisation come from the (n−1)d shell, so the d electrons must pair up to vacate them.Outer-orbital complex (also high spin, spin-free, hypoligated): the two d orbitals come from the higher nd shell, so the d electrons keep their maximum-multiplicity arrangement.

Electroneutrality principle (Pauling): the electron distribution in a stable molecule is such that the charge on any atom stays close to zero — conventionally within about ±1 unit. Bonds adjust their ionic character until this is satisfied.

The crystal field assumption: the ligands are treated as point negative charges (for anions) or point dipoles (for neutral ligands such as H₂O and NH₃, with the negative end towards the metal). The only interaction considered is the electrostatic repulsion between those charges and the electrons in the metal’s d orbitals. No orbital overlap. No electron sharing. No covalency whatever.

Barycentre: the mean energy of the five d orbitals, equal to their energy in a hypothetical spherical field of the same total charge. Any purely electrostatic rearrangement of that charge conserves the barycentre: the sum of the energy shifts, each weighted by the orbital’s degeneracy, is zero.

Δ_o (crystal field splitting parameter, also written 10 Dq): the energy separation between the t_2g and e_g sets in an octahedral field. It is measured, from the position of the d–d absorption band, not calculated. Typical first-row values are roughly 7 500–35 000 cm⁻¹.

The consequence to memorise: Δ_t is small — typically 3000–5000 cm⁻¹ for first-row ions — and always much smaller than the pairing energy P (15 000–25 000 cm⁻¹). Therefore tetrahedral complexes of first-row metals are essentially always high spin. There is no tetrahedral high-spin/low-spin question to answer.

Crystal field stabilisation energy (CFSE): the energy by which the actual d-electron configuration in the ligand field lies below the energy the same electrons would have in a spherical field of the same strength (i.e. at the barycentre). It is conventionally quoted as a negative number, or as a positive number described as a stabilisation — be explicit about which convention you are using.

Pairing energy P: the energy cost of putting a second electron into an orbital that already holds one, comprising the extra coulombic repulsion between them and the exchange stabilisation lost by no longer having the two spins parallel in different orbitals. For first-row ions P lies in the range 15 000–25 000 cm⁻¹.

Spectrochemical series: the ordering of ligands by the size of the crystal field splitting they produce. It is experimental, derived from electronic spectra, and it is essentially independent of the metal ion.

Jahn–Teller theorem (1937): any non-linear molecule in an orbitally degenerate electronic state is unstable with respect to a distortion that removes the degeneracy. Since the distortion also lowers the energy, the symmetric geometry cannot be the equilibrium structure.

Normal spinel: (A^2+)_tet[B^3+_2]_octO₄ — the divalent ion in the tetrahedral hole.Inverse spinel: (B^3+)_tet[A^2+B^3+]_octO₄ — half the trivalent ions swap into the tetrahedral hole and the divalent ion takes an octahedral one.

Octahedral site preference energy (OSPE): CFSE(octahedral) − CFSE(tetrahedral), expressed as how much an ion gains by occupying an octahedral rather than a tetrahedral hole. Computed with Δ_t = (4/9)Δ_o, so both terms are in the same units.