Q7(b). Tetragonal distortion and the square-planar limit

Why the distortion occurs

1. Jahn–Teller theorem: a non-linear molecule in an orbitally degenerate electronic state distorts to remove the degeneracy and lower its energy.
2. In Oₕ, d⁹ is t_(2g)⁶e_g³: the third e_g electron can sit in d_(z²) or d_(x²-y²) (degenerate state). Lengthening the two axial (z) bonds lowers d_(z²) and raises d_(x²-y²), each by δ₁/2.
3. Two electrons go into the lowered d_(z²) and one into the raised d_(x²-y²): net gain 2(δ₁/2)-δ₁/2=δ₁/2. The t_(2g) set splits too (d_(xz),d_(yz) down, d_(xy) up) but, being full, gains nothing.
4. The effect is large for uneven e_g occupancy (d⁹, high-spin d⁴, low-spin d⁷), because e_g orbitals point at the ligands; it is small for uneven t_(2g) (d¹, d²).

Example

1. [Cu(H₂O)₆]²+ and most Cu(II) complexes have four short and two long bonds (elongated octahedron); Cr(II) (d⁴ high spin) behaves the same way. The single broad visible band of [Cu(H₂O)₆]²+ is broadened by the split levels.

Extreme distortion

1. As the axial ligands move away completely the complex becomes square planar (ML₄, D₄ₕ).
2. Splitting: d_(x²-y²) (b_(1g)) points at the four ligands: highest. Then d_(xy) (b_(2g), in the plane between ligands), then d_(z²) (a_(1g), only its torus in the plane), and lowest d_(xz),d_(yz) (e_g). Order: x²-y²≫ xy>z²>xz,yz
3. The large gap below d_(x²-y²) makes square planar the rule for low-spin d⁸: [Ni(CN)₄]²-, [PdCl₄]²-, [PtCl₄]²-, [AuCl₄]⁻. All eight electrons fill the four lower orbitals and the complexes are diamagnetic.

Application. the Jahn–Teller elongation explains why Cu(II) stands at the top of the Irving–Williams stability order and favours four short bonds, and why the fifth and sixth ligands of Cu(II) proteins and of CuSO₄.5H₂O are weakly held.