### Boron and Buckyballs

#### by Derrick Stolee

Recent news from the world of chemistry: the result is really the experimental observation of a special new molecule. People call it a “Boron Buckyball”, but this irritates me since we know that the buckyball is a specific fullerene on 60 vertices. In particular, it is the smallest fullerene that satisfies the *independent pentagon rule*. This new Boron molecule is called “Borospherene” and is quite interesting in its own right, as seen below:

The structure is very interesting when you consider it as a spherically-embedded (i.e. planar) graph: there are many triangular faces which create a cube-like structure, two of the faces of this cube are 6-faces, and the other four are 7-faces! These interesting *heptagonal* structures are particularly interesting. In the figure above, it appears to be a unit-distance embedding, and this creates a rotation in the opposing 6-faces.

Let’s dig more into the structure of this object, but also into the computational part of the experiment.

### Structure of Borospherene

Here, I will include many features of the Borospherene graph. However, you can explore them yourself! Just download this Sage worksheet to have access to the Borospherene graph and all of the plots below. You can also download SVG versions of each plot by clicking on the image.

First, observe from the picture above that the graph is embedded on a sphere. This can be flattened to the plane, so we will use that embedding.

While the picture is no longer as symmetric, you can perhaps see that the two 6-faces have are locally symmetric. That is, if you cut the graph in half, you will get two pieces that look like this:

By pasting two halves together on the red edges, we will get Borospherene (as long as you perform a quarter rotation before doing so!).

The graph has chromatic number 4.

The graph has domination number 8.

The graph has independence number 12.

It is Hamiltonian. In fact, it is pancyclic (has cycles of every possible length).

### The Computational Experiment

Let us investigate their computational method, to the extent that we can. The press release version states that they used a computer to examine “about 10,000” graphs, looking for structures that may exist in physical form. What computational experiment did they actually run?

It is difficult for me to fully determine their method, since I am not an expert in computational chemistry (and not even good at chemistry in general). However, what I can gather from attempting to read the paper are the following points:

- Their search is stochastic (using “stochastic surface walking”).
- They use local-search methods to improve the random search (“basin hopping”).
- They are trying to optimize the “stability” of the molecule (measured by “density functional theory methods”).
- They used existing software (TGmin and NWCHEM 6.3) in addition to their own code.

From this information, I gather that they performed several random searches to create a large variety of possible molecules. These were reduced to “locally minimal” molecules through the basin hopping mechanism. Then, these minima were examined to find the best possible molecules. Borospherene was the best one out of these options.

*“In total, 3,027 minima were evaluated and the D _{2d} fullerene was found to be the global minimum.”*

After finding these minima, they computed their photoelectron spectra. Thus, the researchers did not probe the molecule to discover the structure, but instead determined that the molecules have this structure by hitting it with a jet of helium and measuring the spectra. This signature was determined computationally, and so they knew which molecule structure matched which other structure. In fact, after mixing a bunch of boron together and isolating the molecules with 40 atoms, they found two types of structure, seen below.

### Combinatorial Questions

There is a lot of advanced chemistry going on with their search for this molecule. As a discrete mathematician, I would like the chemical questions reduced to mathematical language. Answers to the following questions would help tremendously:

- Does planarity have anything to do with the stability of a molecule?
- What features are important in a stable molecule?
- What determines when there is a double bond versus a single bond?
- How exactly is the energy determined?

While the researchers used random methods, perhaps a complete search could be done if the structure was limited to planar graphs with a lot of triangle faces. Do there exist graphs that “look like” Borospherene? Can we generalize this to larger molecules?

### Related Posts

### References

H.-J. Zhai, Y.-F. Zhao, W.-L. Li, Q. Chen, H. Bai, H.-S. Hu, Z.A. Piazza, W.-J. Tian, H.-G. Lu, Y.-B. Wu, Y.-W. Mu, G.-F. Wei, Z.-P. Liu, J. Li, S.-D. Li, L.-S. Wang, Observation of an all-Boron fullerene, *Nature Chemistry* (2014).

Space Daily Staff Writers, Researchers discover boron “buckyball”, *Space Daily* online July 15th, 2014.

Boroshperene, *Wikipedia*.

Shang, C. & Liu, Z. P. Stochastic surface walking method for structure prediction and pathway searching. *J. Chem. Theory Comput.* 9, 1838–1845 (2013).

Wales, D. J. & Scheraga, H. A. Global optimization of clusters, crystals, and biomolecules. *Science* 285, 1368–1372 (1999).

Valiev, M. *et al.* NWChem: a comprehensive and scalable open-source solution for large scale molecular simulations. *Comput. Phys. Commun.* 181, 1477–1489 (2010).