Crystal Structures of Inorganic Oxoacid Salts Perceived as Cation Arrays: A Periodic-Graph Approach - Inorganic 3D Structures - page 59

where Z is the number of cations in the asymmetric unit of the array, V VP (i) is the
volume of the Voronoi polyhedron of the i th cation, R is the distance between the
cation and a point inside its Voronoi polyhedron;

is equal to the root-mean-
square error of the corresponding space quantizer [ 83 ] . The

< G 3 >

value is
computed irrespective of the structure topology; however, the topology of the ion
array with a given

< G 3 >

value can be unambiguously determined from the
corresponding packing graph. The least-distorted (i.e., the most uniform) array
has the minimum

< G 3 >

; in such an array the cation domains (space regions
being most close to the corresponding cations) cover the crystal space most
economically. The most uniform placement of points in three-dimensional space
is the b.c.c. (bcu-x) lattice with

< G 3 >

< G 3 > ¼ ...

0.0785433 , while close packings with

< G 3 > ¼ 0.07875 ... [ 83 ] are less uniform. This means that the b.c.c. lattice with 14

(8 þ 6) neighbors surrounding each cation in the array should also be important for
eutactic arrangements. According to [ 15 ], the value can be used to determine

the structure-forming role of the array: the most uniform array makes the main
contribution to the Coulomb energy of the ionic structure.
This approach was applied to all kinds of ion arrays in inorganic oxoacid salts
M y (LO 3 ) z [ 15 , 17 ] : cation (M), (L), (M + L), anion (O), and mixed cation-anion
(M

< G 3 >

O) arrays. It was shown that usually the most uniform array includes all
cations (M

þ

L), and the corresponding packing graph has the bcu-x (rarer fcu)
topology. In general, the uniformity of the cation array increases along with the
cation charge and size. For instance, in the series of alkaline carbonates M 2 CO 3 the
<

þ

¼
Cs) [ 15 ] . It was shown [ 17 ] that in high-temperature and high-pressure phases the
structure-forming cation arrays become more uniform because the ambient factors
promote an increase in uniformity and symmetry of the structure-forming array.
These results prove the model of cation array but focus the role of the b.c.c. packing
along with the close packing motifs.

G 3 > values for the (M) arrays decrease from 0.0876 (M ¼ Li) to 0.0834 (M

4 General Regularities in the Cation Arrays

The crystal structure data discussed above reveal the following general regularities
in the cation arrays perceived as underlying nets of the oxoacid salts.

1. The underlying net usually has rather simple topology corresponding to a well-
known topological type. Indeed, of the 569 crystal structures of oxoacid salts
considered in this review, only 157 have novel topologies of their cation arrays.
The remaining 412 cation arrays belong to 55 topological types of binary
compounds or other binodal nets. For a given g ratio there are a few (one or
two) preferable ( default [ 31 ]) underlying nets that frequently occur in nature; the
rare nets can be treated as topological distortions of the default nets. The
underlying nets can be of three types: (a) quite common (default) for different
classes of inorganic compounds (Table 13 ) ; (b) typical only for a given class or