R.W. Grosse-Kunstleve & G.O. Brunner
Lab. of Crystallography, ETH Zurich, Switzerland
The coordination sequence, CS, was introduced by Brunner & Laves [1] to prove or disprove the topological identity of frameworks and of atomic positions within a framework. The CS consists of a series of k numbers, which indicate the number of atoms in "shell" k bonded to atoms of shell k-1. Shell 0 is, then, a single atom, and the number of atoms in the first shell is the conventional coordination number.
The CS is used routinely to characterize sphere packings and to tabulate zeolite frameworks. Further applications involve the use of the topological density, which is defined as the sum of a certain number of terms of the CS. These densities show correlations with other parameters such as lattice energy, Al distribution and catalytic activity.
For a number of examples, the progression of the CS terms was shown to be quadratic with k, but a comprehensive investigation was lacking. Now, in view of the increasing application of the CS, values for k up to 2000 have been calculated, and their algebraic structure analysed for all zeolites tabulated in [2] and for 11 selected "dense" SiO2 polymorphs.
All investigated CS can be described in two ways: (1) by a certain number of "initial terms" together with a recursion law [3], or (2) by a periodic set of quadratic equations. The mean quadratic coefficient of a full period gives the exact topological density. For a limited number of cases, the properties of the coefficients of the quadratic equations were examined, and sub- periods and mirror planes could be detected.
While the analysis of the CS in terms of a recursion required some 300 terms in the most complex case, period lengths for the description with quadratic equations can become extraordinarily long (>140 million for zeolite EUO, as deduced from recursion).
[1] G.O. Brunner & F. Laves; Zum Problem der Koordinationszahl.
Wiss. Zeitschr. Techn. Univ. Dresden 20 (1971) 387-390
[2] W.M. Meier & D.H. Olson; Atlas of Zeolite Structure Types1992
[3] N.J.A. Sloane, private communication