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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.

Six inward-directed donor orbitals form symmetry-adapted linear combinations of $a_{1g}+e_g+t_{1u}$ symmetry. They mix respectively with metal $4s$, $(3d_{z^2},3d_{x^2-y^2})$ and $4p$ orbitals. Thus the sigma framework contains six occupied bonding molecular orbitals and six corresponding antibonding molecular orbitals. The metal $(d_{xy},d_{xz},d_{yz})$ orbitals have no sigma-compatible partner and remain nonbonding $t_{2g}$ orbitals in this model. The important metal-dominated octahedral splitting isThe $e_g^{\ast }$ orbitals point directly toward the donors; the $t_{2g}$ orbitals point between them. The original diagram shows the entire sigma framework, rather than treating the five metal d levels as the whole molecular-orbital diagram.

For the same donor set and approximately the same geometry, ${\Delta }_o(\mathrm{C}{\mathrm{o}}^{\mathrm{III}})>{\Delta }_o(\mathrm{C}{\mathrm{o}}^{\mathrm{II}})$. The higher metal charge usually shortens the metal–donor distances and strengthens the interaction that raises $e_g^{\ast }$. This comparison holds other chemical variables fixed: changing the ligand at the same time could reverse a comparison between two unrelated complexes.

A Co(III) centre has $d^6$, and charge balance requires one closed-shell catecholate ligand and one $S=1/2$ semiquinonate ligand. For low-spin Co(III), $t_{2g}^6(e_g^{\ast }{)}^0$, the metal has $S=0$, giving an independent-spin magnetic moment ofThus $A^1$ contains low-spin Co(III), $S_{\mathrm{Co}}=0$, with its observed moment residing on the free radical ligand. High-spin Co(III), $t_{2g}^4(e_g^{\ast }{)}^2$, would have $S=2$ and give $\sqrt{27}=5.20$ under the stated independent-spin approximation, so it is inconsistent with this measurement.

For Co(II), $d^7$, charge balance instead requires two $S=1/2$ semiquinonate ligands. High-spin Co(II) has $t_{2g}^5(e_g^{\ast }{)}^2$ and $S=3/2$, soTherefore $A^2$ contains high-spin Co(II), $S_{\mathrm{Co}}=3/2$. Low-spin Co(II), $t_{2g}^6(e_g^{\ast }{)}^1$, would give $\sqrt{9}=3.00$ when the two ligand spins are included. These four checks distinguish both oxidation state and metal spin.

The larger octahedral splitting at Co(III) exceeds the effective pairing energy and favours a low-spin complex; the smaller octahedral splitting at Co(II) favours a high-spin complex with occupation of both $e_g^{\ast }$ orbitals before further pairing. Interconversion also transfers an electron between cobalt and a dioxolene ligand, so this is valence tautomerism, rather than a metal-only spin change. Squared moments add here because the supplied model treats the centres as independent; adding the three spins first would be a different, exchange-coupled model.

cisplatin has square planar molecular geometry and a $5d^8$ metal centre. Choose $x$ and $y$ along the four Pt–donor directions. The strongly antibonding $d_{x^2-y^2}$ orbital is highest and remains empty; the other four d orbitals are doubly occupied. The square-planar splitting is represented by the conventional qualitative orderThe ordering among the lower occupied levels can depend on the ligand set, and the cis arrangement removes exact fourfold symmetry. The robust result is the filled lower four levels and empty highest level.

satraplatin has approximately octahedral molecular geometry around Pt(IV), $5d^6$. Its large octahedral splitting favours a low-spin complex:All three $t_{2g}$ orbitals are paired. Mixed ligands remove exact degeneracies, but do not change this qualitative occupancy.

For an octahedral complex, strong axial sigma donation raises the metal–ligand antibonding $e_g$-derived orbitals. A hydride is a very strong sigma donor: its occupied $1s$ orbital overlaps effectively along a metal–ligand axis and has no filled perpendicular p orbitals to act as a conventional pi donor. It can therefore produce a large octahedral splitting by raising $e_g$ relative to $t_{2g}$.

Triphenylphosphine also gives sigma donation, but its useful acceptor orbitals additionally receive metal pi backbonding. This lowers the metal $t_{2g}$-derived levels and enlarges their separation from $e_g$. The acceptors have substantial P–C antibonding character; invoking a dominant phosphorus $3d$ acceptor is unnecessary. Thus hydride is strong-field principally through sigma bonding, while PPh${}_3$ combines sigma donation with pi acceptance. The spectrochemical series describes the resulting splitting, not just a ranking of ligand basicity.

For comparable octahedral environments with the same donor set, the expected qualitative order isAcross comparable first-row ions, the increasing effective attraction and changing metal–ligand distances tend to increase octahedral splitting, giving Mn(II) below Fe(II). Raising the charge on the same metal gives Fe(III) above Fe(II), because the ligand interaction generally strengthens and metal–ligand distances contract. Moving from $3d$ to more radially extended $4d$ orbitals gives larger metal–ligand overlap, placing Mo(III) above Fe(III); Rh(III) lies further along the same $4d$ series and typically gives the larger splitting. The $5d$ ion Pt(IV) combines the heavier row with high charge and gives the largest splitting here.

This is a matched-environment trend, not an intrinsic list of constants for bare ions. Actual $\Delta$ also depends strongly on donor identity, geometry and distance. It would be meaningless to compare an unspecified weak-field tetrahedral compound with a strong-field octahedral compound solely through this ordering.

iron remains Fe(II), $d^6$, in both states; the change is a spin crossover, rather than a change in oxidation state. At the higher temperature the high-spin complex has $t_{2g}^4{e}_g^2$: four unpaired electrons, $S=2$, andThis matches the observed moment. At low temperature a low-spin complex with $t_{2g}^6{e}_g^0$ has $S=0$ and shows diamagnetism. Cooling switches high-spin $d^6$ to low-spin $d^6$. Emptying the metal–ligand antibonding $e_g$ orbitals strengthens and shortens the Fe–N bonds, accounting for the contraction by roughly $0.2\text{A˚}$.

The octahedral splitting and electron pairing energy must be sufficiently close for both states to be accessible. The low-spin state gains electronic energy, while the high-spin state has greater spin degeneracy and usually greater vibrational entropy because of its softer bonds. The resulting Gibbs free energy balance favors high spin at higher temperature. The two endpoint observations do not by themselves specify hysteresis or transition cooperativity; those require a temperature series. Coupled structural and magnetic measurements distinguish this explanation from a mere bond-length anomaly.

The relevant spectrochemical series order is O-bound nitrito $<$ ammonia $<$ N-bound nitro. A pi-donor ligand tends to raise the metal orbitals derived from $t_{2g}$, reducing the octahedral splitting; a pi-acceptor ligand stabilizes those orbitals, increasing the splitting. Since photon energy is $E=hc/\lambda$, the stronger-field complex absorbs at the shorter wavelength. ThusThe corresponding spectroscopic wavenumbers are approximately $21834$, $21097$ and $20576\mathrm{c}{\mathrm{m}}^{-1}$. These are the observed band energies, not three exact independent determinations of ${\Delta }_o$: low-spin $d^6$ transition energies also contain electron repulsion contributions, and replacing one ligand lowers the ideal octahedral symmetry. The relative assignment is nevertheless robust.

Both complexes contain cobalt(III), hence a $d^6$ metal ion. The approximately octahedral molecular geometry, relatively high metal oxidation state and five ammonia donors make the octahedral splitting large enough that pairing is favorable. Their low-spin configuration isAll six metal $d$ electrons are paired, and the ligands in coordination chemistry have closed shells, so neither complex has a permanent spin magnetic moment: both exhibit diamagnetism. The difference between N and O attachment is insufficient to change this low-spin assignment.

The octahedral splitting ${\Delta }_o$ separates the metal $t_{2g}$ and $e_g$ orbital sets. The Racah parameter $B$ characterizes interelectronic repulsion and hence the spacing of many-electron d-shell terms; it is not another name for ${\Delta }_o$. A Tanabe-Sugano diagram plots term excitation energies $E/B$ against ${\Delta }_o/B$, for specified values of the other repulsion-parameter ratios. Each ordinate is measured relative to whichever term is the ground state at that field strength. The original many-electron treatment diagonalizes the relevant d-shell energy matrices rather than identifying every absorption with a one-electron $t_{2g}\to e_g$ gap.^[ref]

A pi-acceptor ligand has a suitable unoccupied orbital for pi backbonding. Stabilization of metal $t_{2g}$-derived levels increases the usual octahedral splitting.

A pi-donor ligand has an occupied orbital that can overlap sideways with a metal orbital. In the standard octahedral splitting picture it raises the metal $t_{2g}$-derived levels, tending to reduce their separation from $e_g$.

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.

The Tolman electronic parameter compares ligand electronic influence using the symmetric CO stretching band of a matched metal-carbonyl probe, classically $\mathrm{Ni}(\mathrm{CO}{)}_3\mathrm{L}$. Lower frequency commonly indicates greater net electron release into the remaining CO ligands. It combines donation and acceptance and is not a direct measurement of octahedral splitting.