Coordination number and geometry

Result

Coordination number and geometry

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Statement

Predicting the shapes of metal complexes.

Why it matters

A coordination compound's chemistry — its colour, its magnetism, which isomers it can form — all follow from two related facts: how many ligands surround the metal, and how those ligands are arranged in space. Coordination number and geometry are therefore the entry point for the rest of this unit: crystal-field-splitting only makes sense once a geometry (octahedral, tetrahedral, square planar) has fixed which set of d orbitals point at the ligands and which point between them, spectrochemical-colour inherits that splitting pattern directly, chelate-effect concerns how a ligand's shape constrains the geometry it can adopt, and complex-isomerism depends entirely on which geometries even admit cis/trans or optical isomers in the first place.

Unlike a simple organic molecule, where VSEPR alone fixes the shape, a transition-metal complex's geometry is set by a genuine competition between steric packing of the ligands and the electronic preference of the metal's own d electrons, which is what makes this topic distinct from (and a natural sequel to) main-group shape prediction.

Hypotheses

Ligands are treated, to a first approximation, as point charges or dipoles that arrange themselves to minimise mutual repulsion around the central metal ion.This is the same electrostatic logic as VSEPR, applied to a metal centre instead of a main-group atom: for a fixed coordination number, the ligand arrangement that keeps the ligands farthest apart is preferred purely on steric/electrostatic grounds, before any d-electron effects are considered. The metal's d-electron count and the ligand field strength can override the purely steric preference for a given coordination number.Coordination number four is the clearest case: four points minimising mutual repulsion in isolation would be tetrahedral, yet many four-coordinate d8 complexes are square planar instead, because a square-planar arrangement lets the d8 configuration achieve a lower total energy under a strong ligand field (Proof, Step 3) than the sterically preferred tetrahedron does. Idealised geometries (a perfect octahedron, a perfect tetrahedron) are a starting approximation; real complexes routinely distort from these ideal shapes, most famously via the Jahn–Teller effect for octahedral complexes with an asymmetrically filled eg set (covered in crystal-field-splitting), and this page's geometries should be read as the reference shapes from which real structures deviate, not as universally exact predictions.

Proof

\text{Coordination number 2} \Rightarrow \text{linear geometry (180}^\circ\text{ ligand–metal–ligand angle)}

With only two ligands, electrostatic repulsion is minimised by placing them on directly opposite sides of the metal; two points admit no other arrangement that separates them further. Linear coordination is common for d10 metal ions in a +1 oxidation state with weak-field ligands, e.g. \([\text{Ag(NH}_3)_2]^+\). A

\text{Coordination number 4} \Rightarrow \text{tetrahedral (109.5}^\circ\text{) or square planar (90}^\circ\text{), depending on the metal}

Purely sterically, four point charges minimise repulsion in a tetrahedral arrangement, and this is indeed the default for d0, d10, and most weak-field four-coordinate complexes. Square planar geometry is instead adopted preferentially by d8 metal ions (e.g. Pt(II), Pd(II), Ni(II), Au(III)) under a moderate-to-strong ligand field, because that specific electron count achieves a large electronic stabilisation in the square-planar d-orbital splitting pattern that outweighs the steric cost of forcing four ligands into a plane. B

\text{Coordination number 6} \Rightarrow \text{octahedral geometry (six 90}^\circ\text{ ligand–metal–ligand angles)}

Six point charges minimise mutual repulsion when placed at the vertices of an octahedron; this is overwhelmingly the most common coordination geometry across the transition series; because it is both the sterically optimal six-point arrangement and (per crystal-field-splitting) the geometry whose d-orbital splitting most transition-metal ions can accommodate favourably across a wide range of d-electron counts. A

\text{Actual coordination number} = f(\text{metal ion radius, metal charge, ligand steric bulk, ligand field strength})

A given metal does not have one fixed coordination number: a small, highly charged metal ion with small ligands (e.g. F−) can accommodate a higher coordination number than the same metal with bulky ligands (e.g. substituted phosphines), since steric crowding around the metal centre sets a practical upper limit on how many ligands can be packed in, independent of the purely electrostatic argument of Steps 1–3. B

Result

\text{CN 2: linear} \qquad \text{CN 4: tetrahedral or square planar} \qquad \text{CN 6: octahedral}

Reading. The three common coordination numbers each correspond to a small set of characteristic geometries, chosen by balancing minimisation of ligand–ligand repulsion against, for coordination number four specifically, the electronic stabilisation a particular d-electron count can gain from adopting the square-planar alternative.

Scope. These are the dominant, textbook geometries; coordination numbers five (trigonal bipyramidal or square pyramidal, often close in energy and interconverting), seven, eight and beyond occur for larger metal ions (particularly lanthanides and actinides) and are treated as extensions of this same reasoning rather than separate cases.

Corollaries & converses

Whether a complex can show geometric or optical isomerism (complex-isomerism) follows directly from its geometry here: octahedral and square-planar complexes readily show cis/trans isomerism for appropriately substituted ligand sets, while simple tetrahedral complexes with monodentate ligands generally cannot.
crystal-field-splitting's entire d-orbital energy-level diagram is geometry-specific — the octahedral, tetrahedral, and square-planar splitting patterns are genuinely different diagrams, each derived from the specific ligand positions fixed by this result.
chelate-effect ligands, by spanning two or more coordination positions with a fixed bite angle, can favour or restrict particular geometries (e.g. certain tetradentate ligands strongly favour square-planar coordination) beyond what a set of independent monodentate ligands would.

Fails without

Drop the electronic-override hypothesis and predict every four-coordinate complex sterically, as tetrahedral: this wrongly predicts \([\text{Ni(CN)}_4]^{2-}\) as a paramagnetic tetrahedron, when it is in fact a diamagnetic square-planar complex — the strong-field d8 electronic preference of Step 2 is essential to get this class of complex right at all.
Treat the octahedron as a rigid, undistorted shape in every case (ignore the t3 caveat): this fails to predict the pronounced tetragonal (elongated or compressed) distortion characteristic of many Cu(II), d9 octahedral complexes, a real, measurable structural feature the idealised-geometry picture alone cannot account for.

Common errors

Assuming every four-coordinate complex is tetrahedral, forgetting the d8 square-planar preference (Step 2) that governs so much of platinum and palladium chemistry specifically.
Treating coordination geometry as fixed for a given metal regardless of the ligands attached, rather than recognising it as an outcome of the specific metal–ligand combination (Step 4).
Confusing coordination number with oxidation state; the two are independent quantities, and a metal in a fixed oxidation state can still adopt different coordination numbers with different ligand sets.
Predicting geometry from steric considerations alone (a VSEPR-style argument) without checking whether the specific d-electron count creates an electronic preference for a non-sterically-optimal shape, as with square-planar d8 complexes.

Discussion

Alfred Werner proposed coordination theory in 1893, the first framework to distinguish a metal's primary valence (oxidation state) from its secondary valence (coordination number), and to insist that the ligands attached by secondary valence occupy specific, predictable positions in space around the metal, rather than being attached in some unstructured way. Werner supported this radical proposal by correctly predicting and then isolating distinct isomers of cobalt(III) ammine chloride complexes whose existence was explicable only if the complexes had a definite three-dimensional geometry — work for which he received the Nobel Prize in Chemistry in 1913.

The competition between steric and electronic preferences described in Step 2 recurs throughout inorganic chemistry: it is the same underlying tension responsible for the existence of high-spin/low-spin equilibria and for many of the structural distortions treated later in this unit, all cases where a purely geometric, ligand-repulsion argument alone is insufficient and the specific electron configuration of the metal has to be brought in explicitly.

Common misconception: that coordination number and geometry are simply "read off" the metal's identity, the way an element's typical oxidation states might be. In reality, both depend jointly on the metal ion, its oxidation state, and the specific ligands present, which is exactly why the same metal ion can be found in genuinely different coordination environments across different compounds.

Worked examples

[\text{NiCl}_4]^{2-}\ (\text{tetrahedral, paramagnetic}) \quad\text{vs.}\quad [\text{Ni(CN)}_4]^{2-}\ (\text{square planar, diamagnetic})

Both complexes contain Ni(II), a d8 ion, and both are four-coordinate, yet they adopt different geometries because Cl− is a weak-field ligand and CN− a strong-field ligand. With the weak-field chloride ligands, the steric preference for a tetrahedral arrangement dominates and the d8 configuration remains high-spin (two unpaired electrons, paramagnetic); with the strong-field cyanide ligands, the electronic stabilisation available from adopting square-planar geometry is large enough to override the steric cost, and all eight d electrons pair up in the lower-lying orbitals (diamagnetic). B

\text{Same d}^8\text{ ion, same coordination number, different ligand field} \Rightarrow \text{different geometry}

Reading. Coordination number alone does not fix geometry; the ligand field strength interacting with the metal's specific d-electron count is what decides between the tetrahedral and square-planar alternatives for a four-coordinate d8 complex.

Scope. The same logic, with the spectrochemical series (spectrochemical-colour) used to judge whether a given ligand counts as weak- or strong-field, predicts the geometry of any four-coordinate d8 complex.

Problems

Predict the coordination geometry of \([\text{Co(NH}_3)_6]^{3+}\), and state which coordination number this corresponds to.
Solution
Six ammine ligands give coordination number six, and by the Result, six-coordinate complexes overwhelmingly adopt octahedral geometry, the sterically optimal arrangement for six point ligands around a central ion; \([\text{Co(NH}_3)_6]^{3+}\) is indeed octahedral.
\([\text{PtCl}_4]^{2-}\) is square planar rather than tetrahedral. Identify the metal ion's d-electron count and explain the observed geometry in those terms.
Solution
Pt is in the +2 oxidation state here (four Cl− ligands each carrying −1 charge, overall charge −2, so Pt is +2); Pt(II) is a d8 ion. By Step 2, d8 metal ions under a sufficiently strong ligand field preferentially adopt square-planar geometry over the sterically simpler tetrahedron, because the electronic stabilisation gained is large enough to outweigh the steric cost — exactly the pattern illustrated for Ni(II) in the Worked example.
Explain, using only the electrostatic point-charge argument of the Hypotheses (not any d-electron effects), why a linear geometry is the only reasonable arrangement for a coordination number of two.
Solution
Treating the two ligands as point charges repelling one another, the arrangement that maximises their mutual separation around a central point places them at exactly \(180^\circ\) to each other — directly opposite — since any other angle would bring them closer together than this maximum-separation configuration. With only two points to place, there is no alternative arrangement that reduces repulsion further, so linear geometry is the unique steric optimum for coordination number two.