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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.

In mixed-valence manganites, mobile $e_g$ electrons hop between sites whose localized $t_{2g}$ core spins are coupled to them by strong Hund's coupling. Projecting onto local spin-aligned states gives the double-exchange mechanismAlignment maximizes the hopping bandwidth and lowers carrier kinetic energy, favouring ferromagnetism. This involves real charge transfer, unlike insulating superexchange. Heating disorders core spins and reduces coherent transport; a magnetic field aligns spins, increasing mobility and reducing resistivity. The large negative response near the magnetic transition is colossal magnetoresistance.

Double exchange supplies the conduction–magnetism link, but its simplest spin-only model is insufficient for the full resistivity change. Lattice distortions, Jahn-Teller effect and carrier localization can amplify the effect; materials need not obey a universal resistivity-versus-magnetization law.^[ref][ref]

At fixed $+2$ oxidation state, increasing effective nuclear charge across the first transition series generally contracts the ionic radius. The first expectation is therefore shorter average metal–nitrogen and metal–sulfur distances toward copper. This is a trend in comparable coordination environments, rather than a requirement that every individual contact shorten monotonically. The occupation of metal–coordination ligand antibonding orbitals, donor identity and lattice distortion all affect the observed distances.

The high-spin $d^5$ and $d^6$ ions both have two electrons in the strongly sigma-antibonding $e_g$ set, so adding the sixth electron to $t_{2g}$ does not itself introduce another sigma-antibonding electron. Copper(II), however, has $e_g^3$ occupation and is subject to the Jahn-Teller effect. A tetragonal Jahn-Teller distortion commonly gives four shorter and two longer contacts, so an elongated copper contact can oppose the overall size trend. Here the nitrogen and sulfur donors are already chemically inequivalent; the supplied schematic alone does not determine which trans pair becomes longest. One should compare actual M–N and M–S sets separately, rather than infer a unique axis or a numerical distance from the magnetic moments. The experimentally determined structures illustrate why the donor geometry and individual contacts matter.^[ref]

The metal is Cr(II), $d^4$. A chloride ligand produces a relatively weak ligand field, so the octahedral center is a high-spin complex with $t_{2g}^3{e}_g^1$, $S=2$. The singly occupied $e_g$ pair is orbitally degenerate: $d_{z^2}$ and $d_{x^2-y^2}$ cannot retain identical occupancies once the distortion selects one. This is a strong setting for the Jahn-Teller effect.

On tetragonal elongation, the two axial Cr–Cl distances increase. Repulsion is reduced most for $d_{z^2}$, so it falls relative to $d_{x^2-y^2}$; $d_{xz},d_{yz}$ also fall relative to $d_{xy}$. The single $e_g$ electron occupies the lower $d_{z^2}$-derived level. The electronic stabilization offsets an elastic cost until a distorted equilibrium geometry is reached. The high-spin $d^4$ occupancy drives an elongated, rather than regular, octahedron. The three lower-set electrons remain one per orbital in this simple diagram; it is primarily the unequal $e_g$ occupancy that produces the strong distortion.

The four shortest contacts are the three Cu–N bonds and Cu–OR bond. They define an approximately square equatorial set. The two remaining Cu–O contacts are longer and define a strongly elongated axial direction, as expected for the Jahn-Teller effect of $d^9$ Cu(II).

Choose $z$ along the long contacts and $x,y$ towards the short equatorial donors. The singly occupied, predominantly metal magnetic orbital is then $d_{x^2-y^2}$, with substantial Ligand in coordination chemistry admixture. All three pyrazine nitrogen donors lie in this equatorial set, so both ladder directions have strong metal–Ligand in coordination chemistry orbital overlap. Neighboring copper centers communicate through comparable pyrazine superexchange pathways. Virtual electron transfer between half-filled orbitals stabilizes the antiparallel arrangement, giving antiferromagnetic interactions in both directions with comparable magnitudes. The axial oxygen donors are much less effective exchange routes.