# Coordination number

**Coordination number** is the number of other atoms directly linked to a specified atom in a chemical species, or, in crystallography, the number of nearest neighbors of an atom in a crystal. The IUPAC definition gives two senses: for a general chemical species it is the count of atoms directly linked to the specified atom (carbon has a coordination number of four in methane and five in protonated methane, \( \ce{CH5^{+}} \)), while in an inorganic coordination entity it is the number of \( \sigma \)-bonds between ligands and the central atom, with \( \pi \)-bonds not counted.<sup>[1](https://goldbook.iupac.org/terms/view/C01331)</sup>

| Key fact | Detail |
|---|---|
| Molecular definition | The number of other atoms directly linked to a specified atom; carbon is 4 in methane and 5 in \( \ce{CH5^{+}} \)<sup>[1](https://goldbook.iupac.org/terms/view/C01331)</sup> |
| Coordination entity definition | The number of \( \sigma \)-bonds between ligands and the central atom; \( \pi \)-bonds are excluded<sup>[1](https://goldbook.iupac.org/terms/view/C01331)</sup> |
| Common values | The most common coordination numbers are two, four, and six, but examples of all coordination numbers from 1 to 15 are known<sup>[2](https://pressbooks-dev.oer.hawaii.edu/chemistry/chapter/coordination-chemistry-of-transition-metals/)</sup> |
| Chelation | Three bidentate ethylenediamine ligands in \( \ce{[Co(en)3]^{3+}} \) give cobalt(III) a coordination number of six<sup>[2](https://pressbooks-dev.oer.hawaii.edu/chemistry/chapter/coordination-chemistry-of-transition-metals/)</sup> |
| Ionic crystals | NaCl has 6:6 coordination and CsCl has 8:8 coordination; the difference follows from \( \ce{Cs^{+}} \) being considerably larger than \( \ce{Na^{+}} \)<sup>[3](https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_%28CK-12%29/08%3A_Ionic_and_Metallic_Bonding/8.08%3A_Coordination_Number)</sup> |
| Radius ratio rule | Cation:anion radius ratios of 0.225–0.414, 0.414–0.732, and 0.732–1.000 correspond to tetrahedral (CN 4), octahedral (CN 6), and cubic (CN 8) sites<sup>[4](https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Chemistry_%28Averill_and_Eldredge%29/12%3A_Solids/12.3%3A_Structures_of_Simple_Binary_Compounds)</sup> |
| Crystallographic ambiguity | The definition of "coordinating atoms" depends on the bonding model, the nature of the problem, and the calculation methods<sup>[5](https://www.iucr.org/who-we-are/commissions/commission-on-crystallographic-nomenclature/published-reports/inorganic)</sup> |

## In coordination complexes

In a coordination compound, a central metal atom or ion is bonded to one or more ligands by coordinate covalent bonds, in which both bonding electrons are contributed by a donor (Lewis base) to an electron acceptor (Lewis acid). The coordination number counts the coordinate covalent bonds to the central metal, so a monodentate ligand contributes one and a bidentate ligand such as ethylenediamine (en, \( \ce{H2NCH2CH2NH2} \)) contributes two.<sup>[2](https://pressbooks-dev.oer.hawaii.edu/chemistry/chapter/coordination-chemistry-of-transition-metals/)</sup><sup> • </sup><sup>[6](https://chem.libretexts.org/Courses/Lakehead_University/CHEM_1110_1130/19%3A_Transition_Metals_and_Coordination_Chemistry/19.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup> The heme ligand in hemoglobin illustrates a polydentate case: a single ligand with four nitrogen donor atoms coordinates to iron.<sup>[2](https://pressbooks-dev.oer.hawaii.edu/chemistry/chapter/coordination-chemistry-of-transition-metals/)</sup>

For transition metal complexes, the coordination number determines the geometry around the central metal ion:<sup>[7](https://chem.libretexts.org/Courses/University_of_Minnesota_Rochester/genchem2/7%3A_Transition_Metals_and_Coordination_Chemistry/7.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup>

| Coordination number | Molecular geometry | Example |
|---|---|---|
| 2 | linear | \( \ce{[Ag(NH3)2]^{+}} \) |
| 3 | trigonal planar | \( \ce{[Cu(CN)3]^{2-}} \) |
| 4 | tetrahedral (\( d^{0} \) or \( d^{10} \), low oxidation states) | \( \ce{[Ni(CO)4]} \) |
| 4 | square planar (\( d^{8} \)) | \( \ce{[NiCl4]^{2-}} \) |
| 5 | trigonal bipyramidal | \( \ce{[CoCl5]^{2-}} \) |
| 5 | square pyramidal | \( \ce{[VO(CN)4]^{2-}} \) |
| 6 | octahedral | \( \ce{[CoCl6]^{3-}} \) |
| 7 | pentagonal bipyramid | \( \ce{[ZrF7]^{3-}} \) |
| 8 | square antiprism | \( \ce{[ReF8]^{2-}} \) |
| 8 | dodecahedron | \( \ce{[Mo(CN)8]^{4-}} \) |
| 9 and above | more complicated structures | \( \ce{[ReH9]^{2-}} \) |

**Coordination number four** is the smallest case with a real choice of geometry: tetrahedral complexes such as \( \ce{[Zn(CN)4]^{2-}} \) have ligand pairs at 109.5°, while square planar complexes such as \( \ce{[Pt(NH3)2Cl2]} \) place two ligands at 90° (cis positions) and one at 180° (trans). Unlike main group elements, where [VSEPR theory](https://www.edgepedia.org/vsepr-theory) predicts the shape, deciding which \( d \)-electron configurations give tetrahedral versus square planar complexes requires crystal field theory.<sup>[7](https://chem.libretexts.org/Courses/University_of_Minnesota_Rochester/genchem2/7%3A_Transition_Metals_and_Coordination_Chemistry/7.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup> Octahedral complexes, with six donor atoms at 90° bond angles, are the most common structures of coordination compounds.<sup>[7](https://chem.libretexts.org/Courses/University_of_Minnesota_Rochester/genchem2/7%3A_Transition_Metals_and_Coordination_Chemistry/7.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup>

The coordination number also constrains isomerism. An octahedral complex with two different ligands, such as \( \ce{[Co(NH3)4Cl2]^{+}} \), can place the two chloride ligands adjacent (cis) or opposite (trans); the two isomers of \( \ce{[Co(NH3)4Cl2]NO3} \) differ in color, the cis form violet and the trans form green, and also in dipole moment, solubility, and reactivity.<sup>[7](https://chem.libretexts.org/Courses/University_of_Minnesota_Rochester/genchem2/7%3A_Transition_Metals_and_Coordination_Chemistry/7.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup>

## In ionic and metallic crystals

In the crystallographic sense, the coordination number is the number of nearest neighbors of an atom or ion in the lattice.<sup>[6](https://chem.libretexts.org/Courses/Lakehead_University/CHEM_1110_1130/19%3A_Transition_Metals_and_Coordination_Chemistry/19.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup> For ionic solids it is usually quoted as the number of nearest ions of opposite charge. [Sodium chloride](https://www.edgepedia.org/sodium-chloride) has 6:6 coordination, each \( \ce{Na^{+}} \) surrounded by six \( \ce{Cl^{-}} \) and each \( \ce{Cl^{-}} \) by six \( \ce{Na^{+}} \); cesium chloride has 8:8 coordination, with the \( \ce{Cs^{+}} \) ion at the center of a cube of eight chloride ions. The two 1:1 compounds adopt different packing because \( \ce{Cs^{+}} \) is considerably larger than \( \ce{Na^{+}} \).<sup>[3](https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_%28CK-12%29/08%3A_Ionic_and_Metallic_Bonding/8.08%3A_Coordination_Number)</sup>

The coordination numbers in a compound must be consistent with its formula unit. In rutile \( \ce{TiO2} \), each \( \ce{Ti^{4+}} \) is surrounded by six \( \ce{O^{2-}} \) ions while each oxide ion is surrounded by three titanium ions, a 6:3 ratio that matches the 1:2 stoichiometry.<sup>[3](https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_%28CK-12%29/08%3A_Ionic_and_Metallic_Bonding/8.08%3A_Coordination_Number)</sup> In the perovskite \( \ce{CaTiO3} \), the Ti and Ca atoms have coordination numbers of 6 and 12, respectively.<sup>[4](https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Chemistry_%28Averill_and_Eldredge%29/12%3A_Solids/12.3%3A_Structures_of_Simple_Binary_Compounds)</sup>

**Radius ratio.** The size relationship between cation and anion predicts which site the cation occupies and hence its coordination number. Cation:anion radius ratios of 0.225–0.414 give tetrahedral holes (CN 4), 0.414–0.732 give octahedral holes (CN 6), and 0.732–1.000 give cubic holes (CN 8); in general, larger cations have higher coordination numbers than small cations.<sup>[4](https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Chemistry_%28Averill_and_Eldredge%29/12%3A_Solids/12.3%3A_Structures_of_Simple_Binary_Compounds)</sup> The zinc blende form of \( \ce{ZnS} \), with a radius ratio of about 0.40, places \( \ce{Zn^{2+}} \) in tetrahedral holes with a coordination number of 4.<sup>[4](https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Chemistry_%28Averill_and_Eldredge%29/12%3A_Solids/12.3%3A_Structures_of_Simple_Binary_Compounds)</sup> In covalent network structures the same count applies to nearest neighbors: in the diamond-type structure each atom is tetrahedrally coordinated with four nearest neighbors at distance \( a\sqrt{3}/4 \).<sup>[8](https://cmp.df.unipi.it/roddaro/SSPnotes/ch01-bravais.html)</sup>

## Determining coordination numbers in crystals

The IUCr nomenclature report states that the coordination number [N] of an atom is given by the number of coordinating atoms, and that the definition of coordinating atoms depends on the bonding model, the nature of the problem, and the calculation methods.<sup>[5](https://www.iucr.org/who-we-are/commissions/commission-on-crystallographic-nomenclature/published-reports/inorganic)</sup> In structures where bonding is not well established, such as intermetallic compounds, only the metric information from [X-ray diffraction](https://www.edgepedia.org/x-ray-diffraction) is experimentally available, so coordination numbers must be extracted from interatomic distances.<sup>[9](https://journals.iucr.org/a/issues/2025/03/00/pen5011/index.html)</sup>

Several methods exist. Distance-string approaches look for the first large gap in the sequence of decreasing interatomic distances; the Brunner–Schwarzenbach gap method is used in the Pearson's Crystal Data database. These methods have known limitations: more than one gap of similar size may appear (the \( \alpha \)-Fe structure is a classical example, with CN = 6 + 8), and the resulting coordination numbers are often not pairwise symmetric, so that for the same A–B distance atom A may be counted in the coordination sphere of B but not vice versa.<sup>[9](https://journals.iucr.org/a/issues/2025/03/00/pen5011/index.html)</sup> The Voronoi–Dirichlet construction instead counts atoms whose Voronoi polyhedra share a face; the number of faces is uniquely defined and gives the maximum possible value of the so-evaluated coordination number, usually larger than the chemically expected one. Weighted schemes, in which each neighbor receives a weight below one that sums to an effective coordination number, have been developed using solid angles and radial overlaps.<sup>[9](https://journals.iucr.org/a/issues/2025/03/00/pen5011/index.html)</sup>

## Related counting concepts

Coordination number is distinct from several neighboring quantities. Hapticity, written \( \eta^{n} \), counts the number of atoms of a single ligand involved in the metal bonding interaction, as with the cyclopentadienyl anion coordinating in \( \eta^{1} \), \( \eta^{3} \), or \( \eta^{5} \) modes.<sup>[10](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_%28LibreTexts%29/13%3A_Organometallic_Chemistry/13.02%3A_Nomenclature_Ligands_and_Classification)</sup> Denticity counts the donor atoms of one ligand bound to a metal, so a hexadentate ligand forms six coordinate bonds.<sup>[6](https://chem.libretexts.org/Courses/Lakehead_University/CHEM_1110_1130/19%3A_Transition_Metals_and_Coordination_Chemistry/19.2%3A_Coordination_Chemistry_of_Transition_Metals)</sup> In organometallic chemistry, the covalent bond classification (CBC) method, in use since the late 1990s, classifies ligands as X-, L-, or Z-type and was introduced as an alternative to concepts based on formal oxidation state and coordination number.<sup>[10](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_%28LibreTexts%29/13%3A_Organometallic_Chemistry/13.02%3A_Nomenclature_Ligands_and_Classification)</sup><sup> • </sup><sup>[11](https://www.sciencedirect.com/science/article/pii/0022328X9500508N)</sup> In IUPAC stereochemical notation for coordination entities, the coordination number of the central atom appears as an arabic numeral in the polyhedral symbol.<sup>[12](https://iupac.qmul.ac.uk/bibliog/RedBook2005.pdf)</sup>

## References

1. [IUPAC Gold Book: coordination number (C01331)](https://goldbook.iupac.org/terms/view/C01331)
2. [Coordination Chemistry of Transition Metals, Chemistry (OpenStax via Pressbooks)](https://pressbooks-dev.oer.hawaii.edu/chemistry/chapter/coordination-chemistry-of-transition-metals/)
3. [8.8: Coordination Number, Chemistry LibreTexts (CK-12)](https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Introductory_Chemistry_%28CK-12%29/08%3A_Ionic_and_Metallic_Bonding/8.08%3A_Coordination_Number)
4. [12.3: Structures of Simple Binary Compounds, Chemistry LibreTexts (Averill & Eldredge)](https://chem.libretexts.org/Bookshelves/General_Chemistry/Book%3A_Chemistry_%28Averill_and_Eldredge%29/12%3A_Solids/12.3%3A_Structures_of_Simple_Binary_Compounds)
5. [Nomenclature of inorganic structure types, IUCr Commission on Crystallographic Nomenclature](https://www.iucr.org/who-we-are/commissions/commission-on-crystallographic-nomenclature/published-reports/inorganic)
6. [19.2: Coordination Chemistry of Transition Metals, Chemistry LibreTexts (OpenStax)](https://chem.libretexts.org/Courses/Lakehead_University/CHEM_1110_1130/19%3A_Transition_Metals_and_Coordination_Chemistry/19.2%3A_Coordination_Chemistry_of_Transition_Metals)
7. [7.2: Coordination Chemistry of Transition Metals, Chemistry LibreTexts](https://chem.libretexts.org/Courses/University_of_Minnesota_Rochester/genchem2/7%3A_Transition_Metals_and_Coordination_Chemistry/7.2%3A_Coordination_Chemistry_of_Transition_Metals)
8. [Crystals and periodicity, Solid State Physics notes, Università di Pisa](https://cmp.df.unipi.it/roddaro/SSPnotes/ch01-bravais.html)
9. [Topological coordination numbers and coordination reciprocity from electron-density distributions, Acta Crystallographica A (2025)](https://journals.iucr.org/a/issues/2025/03/00/pen5011/index.html)
10. [13.2: Nomenclature, Ligands, and Classification, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_%28LibreTexts%29/13%3A_Organometallic_Chemistry/13.02%3A_Nomenclature_Ligands_and_Classification)
11. [A new approach to the formal classification of covalent compounds of the elements, Journal of Organometallic Chemistry](https://www.sciencedirect.com/science/article/pii/0022328X9500508N)
12. [Nomenclature of Inorganic Chemistry, IUPAC Recommendations 2005 (Red Book)](https://iupac.qmul.ac.uk/bibliog/RedBook2005.pdf)

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