square-planar field splitting
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A third common shape for a four-coordinate metal is square planar: four ligands at the corners of a flat square, all in one plane, with nothing above or below. Picture an octahedron and then pull the two ligands on the z axis straight off to infinity. What is left is the square plane, the favorite geometry of platinum(II), palladium(II), nickel(II) in strong fields, and gold(III). The splitting pattern here is the most spread-out of the three.
Follow what happens to each orbital as the two z-axis ligands leave. With nothing pushing along z anymore, every orbital with a z component drops in energy: dz2 falls sharply (it lost its two head-on ligands), and dxz and dyz also fall. Meanwhile dx2-y2 still points straight at the four remaining in-plane ligands and is pushed very high — it ends up the highest orbital by far. The result is a four-level ladder: from bottom to top, roughly dz2 and the pair dxz, dyz low, then dxy in the middle, then dx2-y2 alone at the very top. The big gap between the next-highest orbital and the lonely dx2-y2 is large.
This pattern explains a famous fact: square-planar complexes are overwhelmingly d8 (eight d electrons) and low-spin and diamagnetic. Eight electrons fill the four lower orbitals neatly and leave the sky-high dx2-y2 empty, which is extremely stable. That is why cisplatin, the anticancer drug, is square-planar Pt(II) d8, and why nickel can be coaxed from tetrahedral (weak field) to square planar (strong field) by changing the ligands. It is also the natural endpoint of a strong Jahn-Teller elongation of an octahedron.
[Ni(CN)4]2- is square planar, diamagnetic, and yellow: the strong-field cyanide ligands push the splitting wide, the eight d electrons of Ni2+ pair up in the four lower orbitals, and the empty dx2-y2 sits far above. Swap cyanide for the weaker-field chloride and you get tetrahedral, paramagnetic [NiCl4]2- instead — same metal, different geometry, decided by the ligand field.
The same d8 nickel ion goes square planar with strong ligands and tetrahedral with weak ones.
The exact ordering of the middle orbitals (especially whether dz2 sits above or below dxy) depends on the metal and ligands and on whether the model includes covalency, so different textbooks draw the lower three levels in slightly different orders; what is robust is that dx2-y2 is by far the highest.