Inchemistry,crystallography, andmaterials science, thecoordination number, also calledligancy, of a centralatom in amolecule orcrystal is the number of atoms, molecules or ions bonded to it. The ion/molecule/atom surrounding the central ion/molecule/atom is called aligand. This number is determined somewhat differently for molecules than for crystals.
For molecules and polyatomic ions the coordination number of an atom is determined by simply counting the other atoms to which it is bonded (by either single or multiple bonds).[1] For example, [Cr(NH3)2Cl2Br2]− has Cr3+ as its central cation, which has a coordination number of 6 and is described ashexacoordinate. The common coordination numbers are4,6 and8.
In chemistry,coordination number, defined originally in 1893 byAlfred Werner, is the total number of neighbors of a central atom in a molecule or ion.[1][3] The concept is most commonly applied tocoordination complexes.
The most common coordination number ford-blocktransition metal complexes is 6. The coordination number does not distinguish the geometry of such complexes, i.e. octahedral vs trigonal prismatic.
For transition metal complexes, coordination numbers range from 2 (e.g., AuI in Ph3PAuCl) to 9 (e.g., ReVII in [ReH9]2−). Metals in thef-block (thelanthanoids andactinoids) can accommodate higher coordination number due to their greater ionic radii and availability of more orbitals for bonding. Coordination numbers of 8 to 12 are commonly observed forf-block elements. For example, withbidentatenitrate ions as ligands, CeIV and ThIV form the 12-coordinate ions [Ce(NO3)6]2− (ceric ammonium nitrate) and [Th(NO3)6]2−. When the surrounding ligands are much smaller than the central atom, even higher coordination numbers may be possible. Onecomputational chemistry study predicted a particularly stablePbHe2+
15 ion composed of a centrallead ion coordinated with no fewer than 15 helium atoms.[4] Among theFrank–Kasper phases, the packing of metallic atoms can give coordination numbers of up to 16.[5] At the opposite extreme, steric shielding can give rise to unusually low coordination numbers. An extremely rare instance of a metal adopting a coordination number of 1 occurs in theterphenyl-based arylthallium(I) complex 2,6-Tipp2C6H3Tl, where Tipp is the 2,4,6-triisopropylphenyl group.[6]
Coordination numbers become ambiguous when dealing withpolyhapto ligands.For π-electron ligands such as thecyclopentadienide ion [C5H5]−,alkenes and thecyclooctatetraenide ion [C8H8]2−, the number of adjacent atoms in the π-electron system that bind to the central atom is termed thehapticity.[7] Inferrocene the hapticity,η, of each cyclopentadienide anion is five, Fe(η5-C5H5)2. Various ways exist for assigning the contribution made to the coordination number of the central iron atom by each cyclopentadienide ligand. The contribution could be assigned as one since there is one ligand, or as five since there are five neighbouring atoms, or as three since there are three electron pairs involved. Normally the count of electron pairs is taken.[8]
The coordination numbers are well defined for atoms in the interior of acrystal lattice: one counts the nearest neighbors in all directions. The number of neighbors of an interior atom is termed thebulk coordination number. For surfaces, the number of neighbors is more limited, so thesurface coordination number is smaller than the bulk coordination number. Often the surface coordination number is unknown or variable.[9] The surface coordination number is also dependent on theMiller indices of the surface. In abody-centered cubic (BCC) crystal, the bulk coordination number is 8, whereas, for the (100) surface, the surface coordination number is 4.[10]
A common way to determine the coordination number of an atom is byX-ray crystallography. Related techniques includeneutron orelectron diffraction.[11] The coordination number of an atom can be determined straightforwardly by counting nearest neighbors.
For example, α-Aluminium has a regular cubic close packed structure,fcc, where each aluminium atom has 12 nearest neighbors, 6 in the same plane and 3 above and below and the coordination polyhedron is acuboctahedron. α-Iron has abody centered cubic structure where each iron atom has 8 nearest neighbors situated at the corners of a cube.
The two most commonallotropes of carbon have different coordination numbers. Indiamond, each carbon atom is at the centre of a regulartetrahedron formed by four other carbon atoms, the coordination number is four, as for methane.Graphite is made of two-dimensional layers in which each carbon is covalently bonded to three other carbons; atoms in other layers are further away and are not nearest neighbours, giving a coordination number of 3.[12]
For chemical compounds with regular lattices such assodium chloride andcaesium chloride, a count of the nearest neighbors gives a good picture of the environment of the ions. In sodium chloride each sodium ion has 6 chloride ions as nearest neighbours (at 276 pm) at the corners of anoctahedron and each chloride ion has 6 sodium atoms (also at 276 pm) at the corners of an octahedron. In caesium chloride each caesium has 8 chloride ions (at 356 pm) situated at the corners of acube and each chloride has eight caesium ions (also at 356 pm) at the corners of a cube.
International Union of Crystallography, IUCR, states that the coordination number of an atom in a crystalline solid depends on the chemical bonding model and the way in which the coordination number is calculated.[13][14]
In some compounds the metal-ligand bonds may not all be at the same distance. For example in PbCl2, the coordination number of Pb2+ depends on which chlorides are assigned as ligands. Seven chloride ligands have Pb-Cl distances of 280–309 pm. Two chloride ligands are more distant, with a Pb-Cl distances of 370 pm.[15]
Some metals have irregular structures. For example, zinc has a distorted hexagonal close packed structure. Regular hexagonal close packing of spheres would predict that each atom has 12 nearest neighbours and atriangular orthobicupola (also called an anticuboctahedron or twinned cuboctahedron) coordination polyhedron.[12][16] In zinc there are only 6 nearest neighbours at 266 pm in the same close packed plane with six other, next-nearest neighbours, equidistant, three in each of the close packed planes above and below at 291 pm. The coordination number of Zn can be assigned as 12 rather than 6.[14] Similar considerations can be applied to the regular body centred cube structure where in addition to the 8 nearest neighbors there 6 more, approximately 15% more distant,[12] and in this case the coordination number is often considered to be 14.
InNickel arsenide (NiAs) and several related compounds, the coordination number of the metal is ambiguous. The metal is bound to six As ligands, but also has two Ni---Ni contacts that could qualify as bonds.[12]
Two other examples of commonly-encountered chemicals areFe2O3 andTiO2. Fe2O3 has a crystal structure that can be described as having a near close packed array of oxygen atoms with iron atoms filling two thirds of the octahedral holes. However each iron atom has 3 nearest neighbors and 3 others a little further away. The structure is quite complex, the oxygen atoms are coordinated to four iron atoms and the iron atoms in turn share vertices, edges and faces of the distorted octahedra.[12] TiO2 has therutile structure. The titanium atoms 6-coordinate, 2 atoms at 198.3 pm and 4 at 194.6 pm, in a slightly distorted octahedron. The octahedra around the titanium atoms share edges and vertices to form a 3-D network. The oxide ions are 3-coordinate in atrigonal planar configuration.[17]
Several propositions have been made to calculate a mean or « effective » coordination number (e.c.n. or ECoN) by adding all surrounding atoms with a weighting scheme, in that the atoms are not counted as full atoms, but as fractional atoms with a number between 0 and 1; this number is closer to zero when the atom is further away.[18] Frequently a gap can be found in the distribution of the interatomic distance of the neighboring atoms: if the shortest distance to a neighboring atom is set equal to 1, then often further atoms are found at distances between 1 and 1.3, and after them follows a gap in which no atoms are found.
According to a proposition of G. Brunner and D. Schwarzenbach[19] an atom at the distance of 1 obtains a weight 1, the first atom beyond the gap obtains zero weight, and all intermediate atoms are included with weights that are calculated from their distances by linear interpolation:
where is the distance to the closest atom, is the distance to the first atom beyond the gap and is the distance to thei-th atom in the region between and. This method is however of no help when no clear gap can be discerned.
A mathematically unique method of calculation considers the domain of influence (also called Voronoi polyhedron,Wigner-Seitz cell or Dirichlet domain). The domain is constructed by connecting the atom in question with all surrounding atoms; the set of planes perpendicular to the connecting lines and passing through their midpoints forms the domain of influence, which is aconvex polyhedron. In this way, a polyhedron face can be assigned to every neighboring atom, the area of the face serving as measure for the weighting. A value of 1 is assigned to the largest face. Other formulas have also been derived,[18] for example:
where = 5 or 6, is the distance to thei-th atom and is the shortest distance or the assumed standard distance.
The coordination number of systems with disorder cannot be precisely defined.
Thefirst coordination number can be defined using theradial distribution functiong(r):[20][21]wherer0 is the rightmost position starting fromr = 0 whereong(r) is approximately zero,r1 is the first minimum. Therefore, it is the area under the first peak ofg(r).
Thesecond coordination number is defined similarly:
Alternative definitions for the coordination number can be found in literature, but in essence the main idea is the same. One of those definition are as follows: Denoting the position of the first peak asrp,
Thefirst coordination shell is thespherical shell with radius betweenr0 andr1 around the central particle under investigation.[22][23]
| Coordination number 2 | |
|---|---|
| Coordination number 3 | |
| Coordination number 4 | |
| Coordination number 5 | |
| Coordination number 6 | |
| Coordination number 7 | |
| Coordination number 8 | |
| Coordination number 9 | |
