II. Coordination of the atoms

The coordination number [N] of an atom is given by the number of coordinating atoms: the definition of `coordinating atoms' depends on the bonding model, the nature of the problem and the calculation methods. The coordination polyhedron of an atom is the polyhedron that has vertices coincident with the centres of coordinating atoms. In structures which contain lone electron pairs and in which volumes ascribable to these lone electron pairs are comparable with the volumes of individual atoms, coordination polyhedra can be considered that include lone electron pairs as some of the vertices.

Several methods have been proposed to define coordination numbers in complicated structures. Most yield a `weighted' coordination number resulting from a form of weighting the coordinating atoms according to their distances from the central atom, e.g. by means of Voronoi polyhedra (Wirkungsbereiche , Dirichlet domains) with or without consideration of the atomic radii (Frank & Kasper, 1958, 1959; Bruzzone, Fornasini & Merlo, 1970; Brunner & Schwarzenbach, 1971; Fischer, Koch & Hellner, 1971; Brunner, 1977; O'Keeffe, 1979; Hoppe, 1979).

This Subcommittee proposes, as an improvement of earlier notations (Machatschki, 1947; Donnay, Hellner & Niggli, 1964; Lima-de Faria & Figueiredo, 1976), a specific set of symbols for the most commonly observed types of coordination polyhedra. These coordination symbols are added as trailing superscripts to the symbols used for the chemical elements in crystal-chemical formulae and, preferably, are placed between square brackets.[*]

For another use of these symbols, see the Bauverband approach ($\S$ IV and Hellner, 1965).

Two levels of symbols are proposed, complete and simplified.

(1) Each complete symbol gives the total number of atoms coordinated to a central atom A. The type of coordination polyhedron is indicated by lower-case letters. The most common coordination polyhedra are listed in Table 2; for example,

\begin{displaymath}
\textrm{Ca}^{[8cb]}\textrm{F}_2^{[4t]}.\end{displaymath}


 
Table 2: Symbols for common coordination polyhedra
Coordination polyhedron around atom A Complete symbol   Simplified symbols   IUPAC symbol
Single neighbour [1l]    [1]    
Two atoms collinear with atom A [2l]   $\bigr\}$[2]   L-2
Two atoms non-collinear with atom A [2n]     A-2
Triangle coplanar with atom A [3l]   $\bigr\}$[3]   TP-3
Triangle non-coplanar with atom A [3n]     TPY-3
Triangular pyramid with atom A in the centre of the base [4y]   $\Bigr\}$[4]    
Tetrahedron [4t] [t] t T-4
Square coplanar with atom A [4l]* [s] s SP-4
Square non-coplanar with atom A [4n]     SPY-4
Pentagon coplanar with atom A [5l]   $\biggr\}[5]$    
Tetragonal pyramid with atom A in the centre of the base [5y]     SPY-5
Trigonal bipyramid [5by]     TBPY-5
Octahedron [6o] [o] $\biggr\}[6]$ o OC-6
Trigonal prism [6p] [p] p TPR-6
Trigonal antiprism [6ap] [ap] ap  
Pentagonal bipyramid [7by]   $\bigr\}[7]$   PBPY-7
Monocapped trigonal prism [6p1c]     TPRS-7
Bicapped trigonal prism [6p2c]       TPRT-8
Tetragonal prism [8p]   $\Biggr\}[8]$    
Tetragonal antiprism [8ap]      
Cube [8cb] [cb] cb CU-8
Anticube [8acb] [acb] acb SAPR-8
Dodecahedron with triangular faces [8do] [do] do DD-8
Hexagonal bipyramid [8by]     HBPY-8
Tricapped trigonal prism [6p3c]   [9]   TPRS-9
Cuboctahedron [12co] [co] $\Bigr\}[12]$ co  
Anticuboctahedron (twinned cuboctahedron) [12aco] [aco] aco  
Icosahedron [12i] [i] i  
Truncated tetrahedron [12tt]      
Hexagonal prism [12p]      
Frank-Kasper polyhedra with:          
  14 vertices [14FK]   [14]    
  15 vertices [15FK]   [15]    
  16 vertices [16FK]   [16]    

*Also [4s].


Some of the principles which have been adopted for creating coordination symbols and which can be used for possible extension are as follows:

[Nn] denotes an N-sided non-coplanar coordination polygon around atom A;

[Nl] denotes an N-sided coplanar (collinear for N < 3) coordination polygon around A;

[Np] denotes an N/2-sided coordination prism around A;

[Ny] denotes an (N - 1)-sided coordination pyramid around A;

[Nby] denotes an (N - 2)-sided coordination bipyramid around A.

In addition to these systematically derived symbols, Table 2 also contains the corresponding symbols recommended by IUPAC (1990).

(2) The simplified symbol requires only the coordination number [N], without specifying the polyhedron type. On the other hand, for the most common coordination polyhedra, a simplified letter notation can be used as a trailing superscript with or without square brackets (t for tetrahedron, o for octahedron, cb for cube etc., as in Table 2); for example,

\begin{displaymath}
\textrm{Ca}^{[8]}\textrm{F}_2^{[4]}\textrm{ or Ca}^{[cb]}\textrm{F}_2^{[t]}\textrm{ or Ca}^{cb}\textrm{F}^{t}_2.\end{displaymath}

Coordination polyhedra which include one or several lone electron pairs as vertices can be denoted as follows: [$\psi - 4t$] (equivalent to [3n]), [$\psi 
- 6by$], [$\psi - 6o$] (equivalent to [5y]) and [$2{\psi} - 6o$] etc.

The notation must be able to describe coordination by different sets of atoms, or coordination at different (sets of) distances, or self-coordination and coordination polyhedra composed of several distinct atomic species, in addition to giving the shape of coordination polyhedra and/or the number of coordinating atoms. The notation should also be flexible and able to express either the complete coordination or only the desired limited amount of information. The notation rules recommended by this Subcommittee are outlined in the following paragraphs.

For normal oxycompounds, a simple coordination notation such as that for perovskite, CaTiO3:

\begin{displaymath}
\textrm{Ca}^{[12co]}\textrm{Ti}^{[6o]}\textrm{O}_3,
\textrm{ Ca}^{[12]}\textrm{Ti}^{[6]}\textrm{O}_3\textrm{ or }\end{displaymath}

\begin{displaymath}
\textrm{Ca}^{[co]}\textrm{Ti}^{[o]}\textrm{O}_3\textrm{
(=Ca}^{co}\textrm{Ti}^o\textrm{O}_3)\end{displaymath}

will always be interpreted as coordination of Ca and Ti by oxygen.

However, in the general case such simplification results in substantial loss of information and ambiguity of interpretation. The coordination of atom A in compound AaBbCc for such a case is written as

A[m,n;p]

where m and n denote the numbers of atoms B and C (i.e. always in the sequence given in the formula) which are coordinated to atom A. These coordination numbers are separated by commas. The self-coordination number p of A by atoms A follows the semicolon. The coordination of atom B is then written as

\begin{displaymath}
B^{[m^{\prime},n^{\prime};p^{\prime}]}\end{displaymath}

where $m^{\prime}, n^{\prime}$ and $p^{\prime}$ denote the numbers of atoms A, C and B around atom B, respectively, etc.

For example, more complete information on perovskite reads:

Ti O Ca Ca O Ca Ti O
$\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$ $\downarrow$
Ca[8cb,12co;6o] Ti[8cb,6o;] O3[4l,2l;8p].

The element symbols above the superscripts merely indicate the species to which the given coordination symbol applies and are not part of the coordination terminology. The crystal-chemical formula can be simplified as follows:

\begin{displaymath}
\textrm{Ca}^{[8,12;6]}\textrm{Ti}^{[8,6;]}\textrm{O}_3^{[4,2;8]}\end{displaymath}

and, if only the information on coordination by oxygen atoms is required, it can be further simplified to

\begin{displaymath}
\textrm{Ca}^{[,12]}\textrm{Ti}^{[,6]}\textrm{O}_3^{[;8]}\textrm{ or Ca}^{[12]}\textrm{Ti}^{[6]}\textrm{O}_3^{[;8]}.\end{displaymath}

Further simplifications for perovskite have been given above.

As another example, atom A may be considered to be tetrahedrally coordinated by three B atoms and one C atom. In addition, atom A may be coordinated by a cuboctahedron of twelve A atoms. Its coordination can then be described in the following ways:

A[(3,1)t;12co], A[3,1;12], A[(3,1)t], A[;12], A[t;], A[t], A[;co].

For pyrite, cubic FeS2, the coordination is described as

\begin{displaymath}
\textrm{Fe}^{[6o;12co]}\textrm{S}_2^{[(3;1)t]}\end{displaymath}

where each sulfur atom is coordinated by three Fe and one S atoms,

\begin{displaymath}
\textrm{Fe}^{[6o]}\textrm{S}_2^{[(3;1)t]}\textrm{, Fe}^{[6o]}S_2^{[;lt]}\textrm{ or Fe}^o\textrm{S}_2^t.\end{displaymath}

If in a compount AaBb, B is coordinated to A at two different distances or at two distinct distance ranges (1 and 2), then the coordination is written as

A[(m1+m2);(p1+p2)].

For example, $\alpha$-SnS:

\begin{displaymath}
\textrm{Sn}^{[(3+3);2]}\textrm{S}^{[(3+3);2]}\textrm{, simplified to Sn}^{[3+3]}\textrm{S}^{[3+3]}.\end{displaymath}

NaCu4(AsO4)3: if only coordination of cations by oxygen is of interest,

\begin{displaymath}
\textrm{Na}^{[,,(2+4+2);]}\textrm{Cu}^{[,,4l;]}\textrm{Cu}_3^{[,,(4+2)by;]}\textrm{As}_3^{[,,4t;]}\textrm{O}_{12}\end{displaymath}

or, in short form,

\begin{displaymath}
\textrm{Na}^{[2+4+2]}\textrm{Cu}^{[4l]}\textrm{Cu}_3^{[(4+2)by]}\textrm{As}_3^{[4t]}\textrm{O}_{12}.\end{displaymath}


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