Crystal Field Splitting & Jahn-Teller Distortion

Octahedral (O_h): d-orbitals split into t₂g (dxy,dxz,dyz, lower) and eg (dz²,dx²-y², upper) by 10Dq=Δ. t₂g lowered by 4Dq, eg raised by 6Dq.
Tetrahedral (T_d): Reversed splitting, Δ_tet = −(4/9)Δ_oct. e orbitals (dz²,dx²-y²) lower, t₂ (dxy,dxz,dyz) upper.
Jahn-Teller theorem: Any non-linear molecule with degenerate electronic state will distort to break degeneracy. For d⁹ (Cu²⁺) in octahedral field: eg is 1-fold occupied → strong tetragonal elongation, splitting eg into dz² and dx²-y².
Crystal field stabilization energy (CFSE) = Σ nᵢ·εᵢ (shown as colored bar). Strong-field limit → spin pairing; weak-field → maximum spin (Hund's rules).