Jahn-Teller effect

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 linearly split orbital doublet occupied by one electron or one hole, the lower adiabatic branch has a cusp $-g|Q|$. A harmonic elastic cost produces equal minima at $Q=\pm g/K$. Orbital degeneracy therefore establishes instability of the regular geometry, but cannot alone distinguish opposite tetragonal distortions. A symmetry-allowed cubic elastic invariant in the two-component distortion can select one sign.

The positive energy reduction from a symmetry-lowering Jahn-Teller effect, measured from the undistorted reference configuration. In the harmonic linear-coupling model $E(Q)=-\alpha Q+\beta Q^2/2$, with $\alpha ,\beta >0$, completing the square gives $Q_0=\alpha /\beta$ and $E(Q_0)=-{\alpha }^2/(2\beta )$. Relating this local energy directly to a cooperative transition temperature is an extra approximation.

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.