Tag Archives: Topology

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:




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?

Chapter 7 will be activated today!

Dear Students,

we will shortly make Chapter 7 of this course available. This will be the final chapter of this course and it will be all about topology.

We will show you how to use three programs for the analysis of framework structures:TOPOS, Systre and 3dt.

In the last chapter, we have shown you that framework structures have an underlying net. This week you will learn how to classify these underlying nets with TOPOS. We will reduce the crystal structure of HKUST-1 – a MOF that you may remember from the last chapter – to its underlying net. Systre helps us to find the most symmetrical representation of the net. Finally, 3dt allows displaying the topology of a net as tilings, meaning that the empty space of the net is shown by space-filling polyhedra, resulting in very beautiful pictures of three-dimensional patterns.

We hope that you will upload some of the pictures you will have created upon completion of this course to our flickr group. This can be done by sending an eMail with the picture attached to:


The subject of the mail automatically serves as the title and the body as the description of the picture.You can find our flickr group at


Once more we invite you to like our facebook page

Even after this course ends we will provide interesting, crystallographic news on this page. And we also would like to keep you informed about our future plans for another online course. Do you have a question concerning crystallography? We would like to hear from you there!

For one more time: see you in class!

Michael and Frank


PS: All students, who are enrolled at the 17th of June 2014, will have access to all course materials at least until October 2014. This will also be the deadline for achieving the 80 % level of personal progress in order to obtain the statement of participation.

Chapter 6 starts today!

Dear Students,

in this week we will introduce a very special class of crystalline materials, which are called Metal-Organic Frameworks or short MOFs. Research interest in this kind of materials has intensified immensely over the last decade. As MOFs are also the field of study of our research group, telling you something about these very special crystals is a matter of heart for us.

MOFs are comprised of inorganic and organic secondary building units. We will take a look at different variations of how these secondary building units can be assembled together and we will introduce the principles of classification of these network-like assemblies.

We provide several crystal structures of MOFs as VESTA files, in order for you to be able to get familiar with the structural features of MOFs at the atomic level. (You can find a tutorial on how to obtain and install VESTA in Unit 2.7)

At the end of this chapter we will be prepared for the next week, in which we will introduce software (called TOPOS and Systre) that allows investigating, determining and classifying the topology of Metal-Organic Frameworks in a systematic way, i.e. to answer the question what are the underlying nets (= linked vertices) and how can we precisely describe them?

Enjoy the beauty of porous crystals!
Michael and Frank