Codex @codex  0

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.

For $n$ unpaired electrons, the spin-only magnetic moment iswhere ${\mu }_{\mathrm{B}}$ is the Bohr magneton and the second expression uses $g\simeq 2$.

For the square, a large separation of the highest $B_{1g}$ level favours pairing in the lower four molecular orbitals: $S=0$, $n=0$, ${\mu }_{\mathrm{eff}}=0$. If the cost of occupying $B_{1g}$ were smaller than the relevant electron pairing energy, a high-spin complex with two unpaired electrons and $S=1$ could instead occur, giving $\sqrt{8}{\mu }_{\mathrm{B}}\simeq 2.83{\mu }_{\mathrm{B}}$.

For ideal tetrahedral $d^8$, the occupation is $e^4{t}_2^4$. Hund's rules distribute four electrons over the three degenerate $t_2$ molecular orbitals as one pair and two single occupations. Thus $S=1$ and ${\mu }_{\mathrm{eff}}\simeq 2.83{\mu }_{\mathrm{B}}$. Merely increasing the $e$–$t_2$ splitting does not create a conventional paired tetrahedral $d^8$ alternative: four electrons must still occupy $t_2$.

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.

For a partially filled $3d$ shell, electron–electron Coulomb repulsion and Hund's rules first favour particular total spin and orbital terms. The crystal field splitting is substantial because these orbitals overlap the ligand environment; it normally exceeds the spin-orbit coupling and often quenches the orbital moment. Competition between the splitting and the electron pairing energy determines high-spin versus low-spin filling. A common hierarchy is Coulomb term energies, then crystal-field and Hund-exchange scales, then spin–orbit and weak-field Zeeman energies; the relative sizes of crystal-field splitting and pairing are material-dependent. Hyperfine coupling, if the nucleus carries spin, adds a smaller splitting. An isolated ion has no inter-ion exchange ordering field.