Ligand field

A ligand field describes how ligands split and mix the central metal orbitals. Sigma donation, pi donation and pi backbonding explain trends that a purely electrostatic picture misses.

A parameter in a compact description of intra-atomic electron repulsion within a configuration. $B$ and $C$ control relative d-shell multiplet energies; $B$ is not the octahedral splitting or a rotational constant.

A plot of many-electron ligand-field term energies relative to the ground state against a normalized ligand-field strength. Axes commonly use $E/B$ and ${\Delta }_o/B$, with the ratio of other interelectronic parameters stipulated. A change of ground-state term can create a break in the plotted reference.

A model in which six donor orbitals of an octahedral complex form $a_{1g}+e_g+t_{1u}$ symmetry-adapted combinations. metal s, p and two d orbitals supply matching symmetry partners; the other three d orbitals remain nonbonding $t_{2g}$ orbitals. The model neglects ligand pi donation, backbonding and departures from octahedral symmetry.

The degeneracy-weighted mean energy of a specified group of orbitals, $\overline{\epsilon }={\sum }_i{g}_i{\epsilon }_i/{\sum }_i{g}_i$. Keeping it fixed separates level splitting from a common energy shift.

A qualitative d-orbital splitting in a square-planar environment, with the strongly in-plane antibonding $d_{x^2-y^2}$ orbital highest. Ordering among the remaining orbitals can depend on the ligand set.

The energetic cost assigned to pairing electrons in an orbital rather than placing them in distinct orbitals within a specified ligand-field model.

Spin crossover interconverts electronic states with different total spin, often in response to temperature. For octahedral iron(II), $t_{2g}^4{e}_g^2$ and $t_{2g}^6$ correspond to $S=2$ and $S=0$. Removing electrons from metal–ligand antibonding $e_g$ orbitals shortens bonds in the low-spin state. Crystal packing and hydrogen bonds can affect transition temperatures.

The electron pairing energy represents the energetic penalty for putting electrons into already occupied orbitals rather than suitable separate orbitals. A simple comparison with octahedral splitting explains high-spin and low-spin alternatives.

A high-spin complex retains more unpaired electrons when the energy required for pairing exceeds the relevant orbital splitting. An octahedral high-spin $d^6$ center has $t_{2g}^4{e}_g^2$ and $S=2$.

The Jahn-Teller effect is an instability of a nonlinear molecular arrangement with an orbitally degenerate electronic ground state toward a symmetry-lowering distortion. An octahedral high-spin $d^4$ ion has unequal occupation of its $e_g$ orbitals and often undergoes tetragonal elongation.

For a real nonmagnetic electronic multiplet, a vibrational coordinate can couple linearly through a real symmetric electronic operator only if its representation occurs in the symmetric square. Remove the fully symmetric component when asking about symmetry-lowering distortion. In $T_d$, both ${\mathrm{Sym}}^2{T}_1$ and ${\mathrm{Sym}}^2{T}_2$ equal $A_1+E+T_2$, so $E$ and $T_2$ modes are symmetry-eligible for either triplet; coupling strengths are not determined by this decomposition.

A symmetry-lowering change separating two axial metal–ligand distances from four equatorial ones. Occupancy of the split orbitals determines the electronic energy gain, while elastic cost and pre-existing ligand asymmetry affect the observed structure.

Crystal field theory models the effect of surrounding ligand charges or dipoles on metal atomic orbitals. Directional repulsion splits the $d$ levels. Unlike molecular orbital theory, the simplest model does not explicitly construct covalent metal–ligand bonds.

Separation of metal d-orbital energies by the electrostatic field of surrounding coordination ligands. An octahedral field puts the three $t_{2g}$ orbitals below the two $e_g$ orbitals.

In the ideal tetrahedral point-charge model, the five metal d orbitals split into a lower twofold $e$ set and an upper threefold $t_2$ set. Relative to their weighted barycentre their energies are $-3{\Delta }_t/5$ and $+2{\Delta }_t/5$, where ${\Delta }_t>0$ is the separation. For occupancies $e^{n_e}{t}_2^{n_t}$, the crystal field stabilization energy is $(-0.6n_e+0.4n_t){\Delta }_t$. Under weak-field high-spin filling it is $-0.6{\Delta }_t$, $-1.2{\Delta }_t$, $-0.8{\Delta }_t$ and zero for $d^6,d^7,d^8,d^{10}$ respectively. Shorter metal–ligand distances generally increase the splitting, so orbital filling can superimpose nonmonotonic bond-length changes on an ionic-size trend. The model alone does not determine an absolute bond length.

The change in d-electron energy relative to the unsplit barycentre in a crystal-field model, before separately accounting for pairing and other bonding contributions.

A low-spin complex has the smaller-spin occupancy favored when ligand-field splitting outweighs the cost of pairing electrons in the lower orbitals. An octahedral low-spin $d^6$ center has $t_{2g}^6{e}_g^0$ and total spin zero.

The spectrochemical series orders ligands empirically by the splitting they produce in comparable complexes. Pi-donor ligands tend toward smaller splittings and pi-acceptor ligands toward larger ones. Metal, geometry and oxidation state still matter; a sigma-basicity scale is not an interchangeable substitute.

In an ideal octahedral molecular geometry, the metal $d$ orbitals divide into lower $t_{2g}$ and higher $e_g$ sets. Their separation is ${\Delta }_o$. Whether electrons pair in the lower set depends on that separation relative to pairing costs; observed many-electron transition energies need not equal ${\Delta }_o$ exactly.