Counting the neighbours
By now you have met Werner's central insight — that a metal ion holds a fixed number of [[inorg-ligand|ligands]] in a tight inner shell, a number he called the secondary valence and we now call the [[inorg-coordination-number|coordination number]] (CN). The coordination number simply counts how many donor atoms touch the metal. For [Co(NH3)6]3+ it is six, because six ammonia nitrogens reach in; for a chelate like [Co(en)3]3+ it is still six, because each ethylenediamine grips with two nitrogens. Count donor atoms, not ligand molecules — a subtlety you already saw when denticity entered the story.
Here is the surprising part: although coordination numbers from 2 all the way to 12 are known, the chemistry of the d-block crowds overwhelmingly into just a few. Six is by far the most common, four is the runner-up, and two shows up for a small special club. Each number, in turn, prefers a particular shape — a [[coordination-geometry|coordination geometry]] — the arrangement that spreads the ligands out so they jostle each other as little as possible. This is the same instinct behind VSEPR from the bonding rung, where electron domains pushed apart to give bent, trigonal, and tetrahedral molecules. Coordination geometry is VSEPR's grown-up cousin: ligands repel, and they settle into the roomiest pattern they can find.
Two and four: the small crowds
Start small. Coordination number two is rare and almost a private club for the d10 ions of Group 11 in their +1 state — Cu+, Ag+, Au+ — and for Hg2+. Two ligands push as far apart as they can, which means straight across the metal at 180 degrees: a linear geometry. The silver-ammonia complex [Ag(NH3)2]+ that dissolves silver halides, and the [Au(CN)2]- ion that carries gold through a cyanide leach, are both simple H3N-Ag-NH3 and NC-Au-CN sticks. These metals favour so few neighbours partly because their large, electron-rich d10 shells are content and partly because of a relativistic stiffening of their s and p orbitals that makes two strong, collinear bonds especially favourable.
Coordination number four is where things get interesting, because four ligands have two genuinely different ways to arrange. The natural, low-repulsion choice is tetrahedral: four ligands at the corners of a tetrahedron, every angle 109.5 degrees, electrons maximally spread — exactly the shape methane takes for the same reason. Tetrahedral complexes are favoured when the central ion is large enough to need no help spreading the ligands, when the ligands are bulky and want elbow room, and especially when the metal has no electronic reason to prefer otherwise — d0 ions like the permanganate [MnO4]- and chromate [CrO4]2- ions, and d10 ions like [Zn(OH)4]2- and tetrahedral nickel(II) species, all sit happily as tetrahedra.
The other four-coordinate shape, square planar, is the strange one. Take the same four ligands and flatten them into a square, all in one plane at 90 degrees, leaving the spots directly above and below the metal empty. Pure repulsion would never choose this — the ligands are crowded closer than in a tetrahedron. So why does it happen at all? The answer is electronic, and it is the headline reason square planar exists: it is overwhelmingly the geometry of d8 metal ions — Ni2+, Pd2+, Pt2+, Au3+, Rh+, Ir+. The classic case is Vaska-style platinum chemistry and the anticancer drug cisplatin, cis-[Pt(NH3)2Cl2], a flat square. We will see in a moment why d8 abandons the tetrahedron for the flat square; it is one of the most satisfying payoffs of electronic-structure thinking.
Six and beyond: the octahedron rules
Coordination number six is the workhorse of the entire d-block, and its shape is the octahedron: six ligands sitting on the six points of a three-dimensional plus sign — one above, one below, and four around the equator — all at 90 degrees to their neighbours. Despite the intimidating name, an octahedron is simply 'ligands on the +x, -x, +y, -y, +z, -z axes'. It is the roomiest way to pack six things around a centre, so VSEPR alone already predicts it; what is remarkable is how astonishingly many complexes adopt it. From [Co(NH3)6]3+ to [Fe(H2O)6]2+ to [Cr(en)3]3+, octahedral is the default you should assume for a six-coordinate complex unless told otherwise.
The octahedron's importance is not just statistical — it is the stage on which the next rung's whole theory is set. When six ligands approach along the axes, the metal's five d orbitals stop being equal in energy. The two that point their lobes straight down the axes, right at the incoming ligands (the dz2 and dx2-y2 pair, called eg), are shoved up in energy because an electron there sits nose-to-nose with a ligand lone pair. The three that point into the gaps between the axes (dxy, dxz, dyz, called t2g) sink down, because an electron there dodges the ligands. That energy gap is the famous [[crystal-field-theory|crystal field splitting]], written delta-o, and almost everything you will learn next — colour, magnetism, the high-spin/low-spin choice — flows from it.
Octahedral d-orbital splitting (the stage for crystal field theory):
eg (dz2, dx2-y2) <- point AT the ligands, raised
___ ___ by +0.6 * delta_o
|
| delta_o (the crystal field splitting)
|
___ ___ ___ lowered by -0.4 * delta_o
t2g (dxy, dxz, dyz) <- point BETWEEN the ligands
A tetrahedral field inverts this (e below t2), and is smaller:
delta_t = (4/9) * delta_o -> never enough to force low-spinWhy d8 goes flat, and the three pressures
Now the promised payoff: why does a d8 ion choose the crowded square plane over the roomy tetrahedron? Imagine starting from an octahedron and slowly pulling the two ligands on the z-axis away to infinity. As they retreat, every d orbital with a z-component drops in energy — and the dz2 orbital, which pointed straight at them, plummets the furthest. What is left behind is the [[square-planar-field-splitting|square-planar splitting]]: four orbitals fairly low and one orbital, the dx2-y2 aimed at the four ligands still in the plane, pushed dramatically high. For a d8 ion you have exactly eight electrons; they fill the four lower orbitals completely (paired up) and leave that lone, very high dx2-y2 empty. The complex pays no penalty for the empty high orbital and pockets all the stabilization of the dropped ones. That energetic prize — large only when the splitting is large — is why square planar belongs to d8 and grows ever more dominant down a group, where Pt2+ is almost always square planar while the lighter Ni2+ can sometimes still go tetrahedral.
Step back and the whole picture resolves into three pressures fighting over every metal. The first is size and charge: a small, highly charged ion can only fit a few ligands but holds them very tightly, while a big ion can host more — this is why early, high-charge ions and the larger lanthanides reach high coordination numbers, and why CN tends to rise as you go down a group. The second is ligand bulk: fat, demanding ligands take up room and push the count down, the steric idea captured by a ligand's cone angle; slim ligands let more crowd in. The third, and the one unique to the d-block, is the [[d-electron-count|d-electron count]] acting through the field splitting — the electronic preference we just used to explain square planar, and which more generally biases a metal toward whichever geometry gives its d electrons the lowest total energy.
Higher numbers, honest limits, and what to remember
Coordination does not stop at six. Five-coordinate complexes exist and flicker between two near-equal shapes — the trigonal bipyramid (think [Fe(CO)5]) and the square pyramid — and they interconvert so easily that five-coordination is a watchword for floppiness. Seven, eight, and higher are the territory of the big ions: the early transition metals in high oxidation states and, above all, the spacious lanthanides and actinides, where eight-coordinate (square antiprism, dodecahedron) and nine-coordinate species are routine. The record-holders climb to twelve. The trend is exactly what the size pressure predicts — the larger the central ion, the more neighbours it can seat.
Two honest cautions before you move on. First, do not over-trust the rules. Geometry is a balance of competing pressures, and the winner can be a near-thing — Ni2+ is the famous fence-sitter, square planar with strong-field ligands like cyanide but tetrahedral with weak-field, bulky ones like chloride. Predictions of geometry are good guides, not guarantees. Second, beware the old story that high coordination numbers happen because the metal 'uses its d orbitals to expand the octet.' That language is now seen as largely wrong: the bonding is better described by spreading a few electrons over many ligands (and, for the d-block, by the metal's own d electrons), not by promoting electrons into high-energy d orbitals. It is the same correction that retired the d-orbital story of hypervalency back in the bonding rung. Coordination geometry is about packing and electronics, not about magically enlarging an octet.
Put it together and a complex's shape is a story with three characters. Count the donor atoms to get the coordination number; let size and charge set the rough budget, ligand bulk trim it, and the d-electron count cast the deciding vote among the shapes that fit. The d-block lives mostly in octahedra, with d8 metals slipping into the flat square and d0 and d10 ions relaxing into tetrahedra. Hold that map in your head, because the very same axes and orbitals you have just been picturing are about to become the whole language of colour and magnetism in the crystal-field rung that follows.