Zeolites, Frameworks, Coordination Sequences & Encyclopedia of Integer Sequences

Ralf W. Grosse-Kunstleve


What have all these things to do with each other?

I love all of them!

I love zeolites, because they have all these nice 4-connected 3-dimensional frameworks. You can learn a lot about zeolites and frameworks if you make an excursion to the web site of the Structure Commission of the International Zeolite Association (IZA-SC). They also have a huge number of nice pictures showing zeolite frameworks, e.g. the framework of VPI-9.

Actually, that particular framework looks pretty ugly. But I like it anyway, because it was me who solved the crystal structure. I had to devise a whole new method (FOCUS), before I finally succeeded in solving the structure of VPI-9. All of this is documented in my dissertation, Zeolite Structure Determination from Powder Data: Computer-based Incorporation of Crystal Chemical Information, ETH Zurich, Switzerland, 1996.

In the FOCUS method, coordination sequences are used for identifying and sorting of frameworks. The following citation from a submitted paper about FOCUS gives a rough idea about the nature of coordination sequences (CS):

The notion of CS was formally introduced by Brunner & Laves (1971) in order to investigate the topological identity of frameworks and of atomic positions within a framework. The CS is a number sequence in which the k-th term is the number of atoms in "shell" k that are bonded to atoms in "shell"k-1. Shell 0 consists of a single atom, and the number of atoms in the first shell is the conventional coordination number.
Brunner, G.O. & Laves, F. (1971). Wiss. Z. Techn. Univers. Dresden 20, 387-390.
Go back to the VPI-9 page to find seven examples of coordination sequences. You can immediately see that coordination sequences are just integer sequences!

Coordination sequences can easily be computed by simply counting. This is also explained in the paper about FOCUS:

The CS determination algorithm used here can be described as a node counting algorithm or a coordination shell algorithm. The algorithm is started by selecting an initial node (k = 0). In the next step, all nodes bonded to the initial node are determined (k = 1). For k = 2, all characteristics of the algorithm become evident: those nodes, which are bonded to the "new nodes of the previous step (k-1)", but have not been counted before, are counted.
However, we did not only want to compute the sequences by counting, but also get a better understanding. We wanted to know formulae for the coordination sequences, which would allow us to compute them without having to do the cumbersome and time-consuming counting.

It took an awful lot of time and effort, but finally we had found recursive formulae for all our coordination sequences. For a compact description, look at our Poster. For a detailed discussion have a look at our paper Algebraic Description of Coordination Sequences and Exact Topological Densities for Zeolites.

This paper is an example of a success which was only possible because we have the Internet. Months before I started to work on the coordination sequences, I had seen a posting in the newsgroup sci.math.num-analysis. Somebody wrote about an e-mail server which accepts integer sequences and tries to find an algebraic description. Unfortunately, I had lost the message, but people in the newsgroup helped me finding Neil J. A. Sloane's program Superseeker. And indeed, for a relatively simple coordination sequence, superseeker produced a recursive formula, which then gave us the key to solve the more complex cases.

This means, finally, we had established a link between zeolites and the Encyclopedia of Integer Sequences, and these days you can find all the coordination sequences of all zeolite framework topologies approved by the IZA Structure Commission in Sloane's On-Line Encyclopedia of Integer Sequences.

Give it a try! Look up a coordination sequence of VPI-9: 4,11,23,39,63,93,126,170,210,255

[Revised by N. J. A. Sloane, Jan 03 2014, to repair broken links.]