Category Archives: Topology

A hitherto unrecognized 2-periodic net derived from a circle packing

Recently, I have been immersed in the vast world of intermetallic phases, in particular those with an ordered superstructure of either cubic-closest packed (ccp) or body-centred cubic (bcc) parent structure. They caught my interest, because some of the researchers in that field are paying particular attention to how the densest-packed layers/planes look like and how they are organized, i.e., in which direction they are stacked and if they build an cubic (ABC…) or hexagonal (AB…) stacking sequence. They also partly give network-like descriptions of the atomic configurations of the densest layers.

Since densest layers are considered, one enters the field of circle packings. And the densest circle packing of equally sized circles is, of course, the hexagonal packing. If we draw lines from each circle midpoint to every midpoint of the next-neighboring circles, we get the well known hxl net, a uninodal 6-c net. The vertex symbol is 36.

In those densest layers of ordered alloys the atoms within one densest plane are of different kind. If we ignore this – as it is usually the case, when we analyze the topology of an atomic arrangement – then we obtain again the hxl net. But what happens, if we focus only on a subnet, for instance, the net that is built by the major component?

Let’s look at an example.

In the ordered binary intermetallic phase Cu3Au (space group Pm-3m), the Cu atoms in the densest layers, (111) planes, form a continous net as shown in Fig. 1, in which the Au atoms are a kind of decoration, located in the centres of the hexagons. The Cu subnet is the well-known, again a uninodal net, the kagome net kgm. The vertex symbol is

Fig. 1: Kagome net of Cu atoms in the densest layers of Cu3Au, decorated by Au atoms.

Two different binodal nets based on circle packings are realized in the subnets of densest layers in TiAl3 and MoNi4, respectively, shown in Fig. 2 and 3. In TiAl3 the Al atoms form a bew net, and the Ni atoms in MoNi4 form a krh net. The vertex symbols are given in the figures.

Fig. 2: The subnet formed by Al atoms in the binary intermetallic phase TiAl3.

Fig. 3: The subnet formed by Ni atoms in the binary intermetallic phase MoNi4.

Probably, it is hard to find a new uninodal or binodal net that is based on circle packings as it is believed that the RCSR contains all uninodal and binodal (stable) circle packings. In stable circle packings all tiles are strictly convex (angles between adjacent edges < 180°).

But on the other hand, this also means that one or the other higher-nodal net based on circle packings can still be discovered. Indeed, it turned out that the trinodal net that is formed by the Al atoms (see Fig. 4) in the ordered intermetallic phase ZrAl3 is a hitherto unrecognized net, which, of course, is only due to the fact that no one has cared about it yet.

Fig. 4: The subnet of Al atoms in the binary intermetallic phase ZrAl3.

Who knows if there are more undiscovered networks when looking at densest layers of further intermetallic phases. Maybe there is even a previously undiscovered binodal one 🙂

PS: I would like thank Davide Proserpio for confirming that this a new net by carrying out an appropriate analysis with ToposPro, and I would like to thank Michael Fischer for the inspiring exchange we had concerning this topic.


Lecture about topology

Topology concepts and the deconstruction of MOFs into their underlying nets

Two weeks ago I gave a talk at a nice little workshop about “Structure and Topology – at the heart of MOF chemistry”, which was organized by the Technical University of Munich and was held in the framework of the priority programme COORNETs of the German Research Foundation DFG, initiated by Prof. Roland A. Fischer.

I decided to make a screen-cast of it. It is now available on YouTube:




Creating images of augmented nets

Making beautiful pictures of beautiful nets in augmented version

There is one requisite in order to be able to create and make great looking images of augmented nets: A software package that has the option to make and show customizable coordination polyhedra. Two of the most frequently used (commercial) packages for this purpose are Diamond and CrystalMaker. Although the gorgeous freeware VESTA has also some capabilities in that respect it is, unfortunately, not suitable.

Okey, let’s begin. In this tutorial we want to make an image of the net reo-a.


1. In general, you need the structural information of the net-a net, i.e. the space group of the net and the fractional coordinates of the nodes. There are two possible sources:

a) RCSR (please checkmark the “including augmented version” option) – for reo the search is unsuccessfull

b) ToposPro is delivered with a database called “idealnets.cmp” (search with your File Explorer to find this file). Open this database, which contains approx. 3000 nets, within ToposPro and look-up, if you can find the desired net. Again, for reo-a there is no entry. But, if the net from which you want to make a nice-looking figure is present, proceed with step # 15.

2. Even if the coordinates of the augmented version of a net is not in your database “idealnets.cmp”, it is very likely that the coordinates of the basic net are available in the ToposPro database “idealnets.cmp”.

3. Create a new database in ToposPro. Name it “net-a.cmp”.

4. Copy the entry of choice (here reo) from the “idealnets.cmp” into the newly created database. Now we can create the augmented variant within ToposPro.

5. Go to “Compound – Auto Determine – Bond Midpoints”. Repeat this procedure, choose “Compound – Auto Determine – Bond Midpoints” again.

6. Run ADS in the standard simplification mode (Save Simplified Net/Standard), but with the option “Contract Atom” under the tab “Topology” checked.

– select all atoms except carbons
– select all nitrogen atoms
– select all carbon atoms

7. You will get a new database “net-a_c” containing one entry. This entry has to be modified a little bit.

8. Remove all 2-c boron atoms: Choose “Compound – Auto Determine – Simplify Adjacency Matrix” – remove all 0-, 1- and 2-coordinated vertices.

In the case of the reo net the result will look like this:

9. It looks quite okey, but two things are not correct. There are some additional bonds and it is probably not the most symmetrical representation. Remove the additional bonds by modifying the corresponding adjacency matrix. Turn the additional bonds into “No bond”. This looks better:


10. Now, try to maximize the symmetry with the help of Systre (which is part of the package Gavrog). Export the file from ToposPro into the Systre format (*.cgd).

11. Run Systre and open the file net-a.cgd. Systre tries to relax the nodes so that a) the average bond length is 1.000 Å and b) that the symmetry of the net is maximized.

12. After a few seconds Systre has finished and you can save the net with the highest possible symmetry by choosing “Save as…” and by specifying the format as “”Embedded nets type (*.cgd)”. Name it for instance “net-a_systre_out.cgd” to distinguish between the input and output file.

13. Import the Systre file into a new database in ToposPro. Run IsoCryst to check your results:

That looks good.

14. Export this net as a CIF. Now, CrystalMaker comes into play…

15. Run CrystalMaker and Import the file net-a.cif. It will look like this:

16. Expand the view so that all polyhedra are clearly visible, remove additional bonds by adjusting the bond length. Then select all atoms of one kind of polyhedron and choose “Selection – Make Polyhedron…”. A new Centroid is created, here coloured in blue:

17. Now, only some slight setting modifications have to be done – choose “Model – Model Inspector…”. In the Atom tab all new centroids (Zz and so on), i.e. the polyhedra have to be rendered as “Rendered solid”, all atoms as “Stick bonds”.

18. In the Bond tab, all bonds in which centroids are involved should be blanked out, and for the bonds between atoms choose an eye-catching colour:

19. A last step is necessary – choose “Model – Polyhedral” (and adjust the stick percentage…).

  1. Done!

Cycles, Rings, Strong rings

In chemical graph theory or chemical-related topology a distinction is made between a cycle, a ring, and a strong ring. It is relatively easy to define a cycle:

A cycle is a continuous path (along edges) in which the start and end vertex are the same.

So this means a cycle (also sometimes called a circuit) can be of any size, regardless if there are shorter (or also longer) paths to get back to the ‘home vertex’. We can also say that a cycle is a closed path.

What might be a little more difficult to understand is the formal definition of a ring:

A ring is a cycle that is not the sum of any two smaller cycles.

It is important to emphasize both attributes: two and smaller. The first conclusion is: Okey, in a graph or net there might be rings that are the sum of three, four, five (smaller) cycles, but a cycle is not a ring if it is the sum of exactly two (smaller) cycles.

In the following figure the graph of a cube is shown and some cycles (highlighted in red) within this graph:

Which of these cycles are rings? I think it is clear, that the two cycles in the top row are rings. Furthermore, the cycle in the middle in the bottom row is a ring, while the left and right cycle are not rings.  Why? Well, the 6-cycle on the left is the sum of two smaller 4-cycles. The 6-cycle in the middle is the sum of three(!) smaller cycles but not the sum of two smaller cycles. And the 8-cycle on the right can be considered as the sum of a 4-cycle and a 6-cycle (the latter can be formed as the sum of two smaller 4-cycles, the top-most and the central one):

Well, but shouldn’t it be possible to build the 6-cycle in the middle as the sum of a 4-cycle (top-most) and a 6-cycle (which itself be the sum of the central and right 4-cycles)? Yes, that is correct, but the definition states that the cycles of a sum of cycles have to be smaller! This is not the case here!

Interestingly, there is an alternative definition of a ring which is, at least for me, much more intuitive and comprehensible:

A ring is a cycle that contains along its path no potential short-cut (of the length of exactly one edge) to the ‘home vertex’.

It is easy to see that the cycle on the left has one short-cut and the cycle on the right even two (dashed edges):

So, what are now strong rings? They can be defined as follows:

Strong rings are rings that are not the sum of any number of smaller cycles.

From that definition It follows that only the two cycles in the top row are strong rings.

I would like to thank Michael Fischer (@ZeoliteMiFi) for helpful discussions.



Basic and derived nets

There is an ongoing discussion on the question how certain metal-organic frameworks (MOFs) should be deconstructed in the most appropriate way in order to obtain their underlying nets.

The majority of cases of uncertainties are dealing with different ways of deconstructing certain classes of polytopic linkers. For example, an elongated planar tetracarboxylate linker (~ C2h symmetry) can be regarded as one 4-coordinated vertex (square or rectangular local geometry) or as two 3-c vertices joined by an extra edge (two triangles in the same plane), see the following figure:


The most prominent example is MOF-505 [DOI: 10.1002/anie.200462787] (identical with NOTT-100 [DOI: 10.1002/anie.200601991]) based on the 4-c square-planar Cu paddle-wheel motif as inorganic secondary building units (SBU), and the question arises if the underlying net is nbo (cubic, 4-c uninodal) or fof (trigonal, (3,4)-c binodal):


Likewise tetratopic linkers with two pairs of connection points twisted by ~ 90° (~ D2d symmetry) can be regarded as one 4-c vertex ([distorted] tetrahedral local geometry) or as two 3-c vertices (two triangles joined by an extra edge and being perpendicular oriented to each other), as shown in the following figure:


One example is DUT-11 (DOI: 10.1002/ejic.201000415). Here, the underlying net can be described either as pts or sur:


There are arguments for and against a far-reaching compartimentalization of the linker, i.e. to explicitly take into account the additional branching points within the linker or not. It has been argued, for example, that tetratopic linkers should always regarded as a single 4-c node, because this would reflect its chemical nature more appropriately. However, there are at least two arguments against the single 4-c consideration: (i) the local geometry and the shape of the linker is represented very badly; (ii) a considerable amount of topological information would be lost, if two 3-c vertices are fused into one single 4-c vertex; if we take the example of the (4,4)-c net pts (“tetrahedra plus squares”) there are no less than six different topologies that can be derived by splitting either the 4-c tetrahedron into two 3-c vertices or the 4-c square into two 3-c vertices and there are no clear reasons why all these should be indiscriminately classified as their parent net pts, see the following figure:


(All nets are shown in their augmented versions – see here.)

What’s your opinion about that issue?

Augmentation of nets

An augmented net, which is indicated by an appendix -a to the three letter code of the basic net, is obtained from its underlying basic net by replacing all vertices with their respective coordination figure, i.e. polyhedron or polygon. For instance, all the vertices of the bcu net are replaced with cubes, and the (two different kinds of) 4-coordinated vertices of the pts net with squares and tetrahedra, respectively (see Figure below).


The first obvious advantage of such representations is that it is much easier to identify and to perceive the actual coordination environment of the vertices; not only the coordination number and type, but also their relative orientation to each other, so the local geometry of the vertices is explicitly included and visualized. In this sense it is also helpful in terms of a more intuitive or visual way of classification of nets, for instance, if we investigate all the possible nets that are assembled by a combination of certain node-types, say, tetrahedra and triangles, or trigonal prisms with squares and so on. However, augmentation is beneficial in particular in order to elucidate relationships between nets that are similar to each other: Note that the vertices of the augmented version of a given net have different coordination numbers, coordination sequences and vertex symbols; and these specifications – the topology – may be identical with another basic net that has been derived by the deconstruction of the chemical compound leading to this specific net description, for instance bcu-a is identical with the net of polycubane (pcb).