One idea, carried to a new shape
The previous guide handed you the whole engine of [[crystal-field-theory|crystal field theory]]. Treat each ligand as a small negative point charge; let those charges push on the five d orbitals; the orbitals that point straight at the charges are shoved up in energy, and the ones that point into the gaps sink down. In an octahedron the six ligands sit on the x, y, and z axes, so the two orbitals aimed along the axes — dz2 and dx2-y2, the eg pair — rise, while the three that point between the axes — dxy, dxz, dyz, the t2g set — fall. The gap between them is delta-o, and a few patterns followed: t2g lies below eg, the splitting is read against the [[electron-pairing-energy|pairing energy]] to decide high-spin versus low-spin, and the colour you see is complementary to the light the complex absorbs.
Here is the liberating part: nothing in that recipe was special to six ligands on the axes. The machine works for any arrangement of charges. Move the ligands to new positions and the same five d orbitals simply feel a different push — some that used to be raised are now lowered, and vice versa. So we do not need a new theory for a tetrahedron or a flat square; we only need to ask, for each new shape, which d orbitals now point at the ligands and which now point into the gaps. The answer changes the whole [[d-orbital-splitting|splitting]] pattern. That single question is this guide.
The tetrahedral field: the pattern inverts
Put four ligands around a metal in the most symmetric way and you get a tetrahedron. The neat way to picture it: take a cube, put the metal at its centre, and place the four ligands on four alternating corners — every other corner, so no two are adjacent. Now look carefully at where those corners sit. A cube corner lies along none of the x, y, z axes; it points into the space between them, off-axis in every direction. This is the exact opposite of the octahedral situation, and it flips everything.
Work out who feels the squeeze. The t2 orbitals — dxy, dxz, dyz — point into the regions between the axes, which is precisely where the cube-corner ligands now live. So they point more directly toward the ligands and are pushed up. The e orbitals — dz2 and dx2-y2 — point along the axes, toward the centres of the cube faces, where there is no ligand at all. So they point into the gaps and sink down. The names carry over from the octahedral case but the roles have swapped: in [[tetrahedral-field-splitting|a tetrahedral field]] the e pair sits low and the t2 set sits high. The little 'g' subscripts vanish, by the way, because a tetrahedron has no centre of inversion — a symmetry detail that will matter later for why tetrahedral complexes are often so vividly coloured.
OCTAHEDRAL (6 ligands ON axes) TETRAHEDRAL (4 ligands OFF axes)
eg (dz2, dx2-y2) t2 (dxy, dxz, dyz)
--------------- <- high --------------- <- high
| |
delta-o (large) delta-t (small)
| |
--------------- <- low --------------- <- low
t2g (dxy,dxz,dyz) e (dz2, dx2-y2)
delta-t is roughly (4/9) * delta-o for the same metal & ligandsTwo reasons make the tetrahedral gap, delta-t, much smaller than delta-o. First, there are only four ligands instead of six, so there is simply less charge pushing on the orbitals. Second — and this is the bigger effect — not one of the four ligands points straight at any d orbital; every interaction is glancing and off-axis, so even the orbitals that are 'raised' are only mildly raised. Add the geometry up honestly and, for the same metal and the same ligands, delta-t comes out to about four-ninths of delta-o, roughly 0.44 times as large. That number is not a coincidence of a single example; it falls out of the geometry itself.
Why tetrahedral complexes are almost always high-spin
Recall the contest from the last guide: when you add electrons to a split set of orbitals, they either spread out to keep their spins parallel (filling the upper level early, the high-spin choice) or crowd into the lower level and pair up (the low-spin choice). Which wins is a tug-of-war between the splitting and the pairing energy — the cost of forcing two electrons to share one orbital. If the gap is large, paying the gap to reach the upper level is the expensive move, so electrons pair up: low-spin. If the gap is small, climbing it is cheap, so electrons stay unpaired: high-spin.
Now the tetrahedral consequence writes itself. Since delta-t is only about four-ninths of delta-o, the gap in a tetrahedral complex is almost always smaller than the pairing energy. Climbing the gap is cheap; pairing is dear. So electrons stay spread out and unpaired, and [[tetrahedral-field-splitting|tetrahedral complexes are essentially always high-spin]]. Low-spin tetrahedral complexes are so rare they are curiosities. This is a genuinely useful shortcut: if a complex is tetrahedral, you can usually skip the high-spin/low-spin deliberation entirely and just fill the orbitals the high-spin way. Be honest about why, though — it is not a law, it is a near-certain consequence of that small four-ninths gap rarely beating the pairing cost.
The square-planar field: squash an octahedron flat
The cleanest way to reach the square plane is to start from a familiar octahedron and deform it. Picture six ligands on the x, y, z axes, then slide the two ligands on the z axis — the top and bottom ones — slowly outward, away from the metal, while the four in the xy plane stay put. As the two z ligands retreat, every d orbital with a z component feels less repulsion and falls in energy: dz2 drops a lot, dxz and dyz drop a little. Pull the two z ligands all the way off to infinity and you are left with just four ligands in a flat square in the xy plane. That is the square-planar field, and it is geometrically the octahedron's close cousin, not a stranger.
Track each orbital to its final home and you get a distinctive four-level pattern, very different from the clean two-tier split of the octahedron. The dxy orbital lies in the plane but points between the four ligands, so it stays relatively low. The dz2 and the dxz/dyz orbitals, having lost their axial partners, drop to the bottom — dz2 ends up surprisingly low, since the only ligands left are pinching it edge-on through its small doughnut, not head-on. And then one orbital is left badly exposed: dx2-y2 points its lobes straight along the x and y axes, directly into all four remaining ligands. With nothing pulled away from it, [[square-planar-field-splitting|dx2-y2 is shoved far above everything else]], sitting alone at the top with a large gap beneath it.
Stack the four levels from the bottom up and the square-planar ladder reads like this: lowest sit dxz and dyz together; just above them dz2; then dxy a step higher; and finally, after a large gap, dx2-y2 stranded alone at the top, the one orbital that aims its lobes straight down all four ligands. The exact heights of the lower three depend on the metal and ligands and shuffle a little from complex to complex — but the headline never changes, and it is the only feature you must remember: dx2-y2 is the lone high orbital, with a wide energy gap yawning beneath it.
Why d8 metals love the flat square
Now the payoff. Count out electrons into that four-level pattern for a [[d-electron-count|d8 metal]] — think Ni2+, Pd2+, Pt2+, or Au3+. Eight electrons go in, two to an orbital. They exactly fill the lowest four orbitals — dxz, dyz, dz2, and dxy — and stop. The one orbital left empty is dx2-y2, the single orbital that would have cost the most to occupy because it stares straight down all four ligands. The complex pays no penalty for the most repulsive orbital because it simply never puts an electron there. For a d8 ion, square planar is a near-perfect fit: every electron is housed below the big gap, and the expensive top orbital sits empty.
Compare the alternative. If that same d8 ion went octahedral, the eg pair (including dx2-y2) would have to hold two electrons, paying full price for the most repulsive orbitals. Flattening to a square plane is the metal's way of evicting two ligands precisely so it can leave dx2-y2 empty. The bigger the splitting, the bigger the prize for emptying the top orbital — which is why square planar is overwhelmingly favoured by the heavier d8 metals Pd2+ and Pt2+ and Au3+. Second- and third-row metals have larger, more diffuse d orbitals and produce much bigger splittings than first-row ions, so the energy saved is large and square planar is essentially guaranteed. Ni2+, a first-row ion with a smaller gap, sits closer to the fence: it is square planar with strong-field ligands like cyanide, in [Ni(CN)4]2-, but tetrahedral or octahedral with weaker ones.
Three geometries, one method
Step back and the three pictures rhyme. Octahedral: six ligands on the axes, a clean two-tier split with t2g low and eg high, a big delta-o. Tetrahedral: four ligands off the axes, the pattern inverted with e low and t2 high, and a gap only four-ninths as large — small enough that high-spin always wins. Square planar: an octahedron with its axial ligands stripped away, a four-level ladder with dx2-y2 stranded at the top, perfect for d8 metals that want that orbital empty. You did not memorise three unrelated diagrams; you ran one method — ask which d orbitals point at the ligands — three times.
There is a loose end worth flagging, because it links straight to the next guide. In the square-planar story we said a d8 metal flattens an octahedron to empty dx2-y2. But that very logic — an unevenly filled set of orbitals making a complex distort to lower its energy — is a general phenomenon with a name, the Jahn-Teller distortion, and it explains why many supposedly 'octahedral' complexes are quietly stretched or squashed. Square planar is in a sense the extreme limit of that distortion. So the geometries are not even fully separate boxes; they shade into one another, and the d-electron count is the dial that decides where on that continuum a given metal will settle.