A few bead models of diamondoid molecules

Mathematical beading can be used to construct any diamondoid molecule, also known as nanodiamonds or condensed adamantanes. Here, I show three such molecular systems:

Adamantane (C₁₀H₁₆);
Diamantane (C₁₄H₂₀) also diadamantane, two face-fused cages;
One of 9 isomers of Pentamantane with chemical formula C₂₆H₃₂.

Talk on the connection between bead models and chemical bonding

The three figures at the bottom are taken from the works of Su and Prof. W. Goddard III at Caltech. These figures show electron density profiles based on the so-called electron Force Field, which is essentially a simplified floating spherical Gaussian orbital method with an empirically fitted Pauli potential that takes care of the antisymmetric property of electrons.

Hyperbolic soccerball

I posted many hyperbolic bead models before. But most of them are periodic surface structures in 1- to 3-D dimensional spaces. Examples are various periodic minimal surfaces. In these models, one needs to pay attention to the subtle periodic conditions in the course of beading. Sometimes, it makes the beading quite difficult.
Here, I show a simple construction of hyperbolic soccerball (truncated order-7 triangular tiling) consisting of infinitely many heptagons (blue beads), each of them are connected to seven neighboring heptagons by only one carbon-carbon bond, which is represented by a yellow bead in the model shown below.

Following the spiral beading path by adding hexagons and heptagons, eventually one obtains the hyperbolic soccerball, or more exactly a hyperbolic graphitic snowflake. There is no need to worry about the periodic conditions among different parts of the structure.

From wiki: Truncated order-7 triangular tiling

In principle, one can also use kirigami (paper cutting) to make a model of the hyperbolic soccerball. But I found that the beading technique is much easier for making robust structure of this object due to the nature of mathematical beading. Also the bead hyperbolic soccerball should be able to model the local force field of hyperbolic soccerball to certain extent because the bead model not just gives the connectivity of the molecular graph right, but also mimics the microscopic repulsions among chemical bonds.

Two artworks for the Mathematical Exhibition of Joint Mathematical Meeting

Chern and I submitted two artworks to the Joint Mathematical Beading, which were accepted today: 1. Super Buckyball of Genus 31
2. Beaded Hilbert Curve, step two

In addition to the beadworks we submitted, we also noticed five Platonic bead models made by Ron Asherov. His bead models have multiple beads in an edge, which are similar to our works a few years ago. I labeled this type of bead models with Edge with Multiple Beads, where you can find all the posts. He doesn't seem to know our works along this direction, though.
He also mentioned that the Nylon string passes adjacent edges exactly once with carefully chosen path of string, which is simply the consequence of Hamiltonian path on the dual graph of the corresponding Platonic solids. We can view the whole beading process as a path through the face of polyhedron. Thus if there exists a Hamiltonian path through each face once (Hamiltonian path for the dual polyhedron), then the Nylon string will go through each beads exactly twice and only twice. Of course, you can also say the Nylon string will go through the adjacent edges exactly once. They are the same thing.

Here is a model made by Chern Chuang almost five years ago:

I was quite surprised by its rigidity when Chern showed me this model. At that time, people questioned me about the meaning of beads. I told some of my colleagues that spherical beads represent chemical bonds instead of atoms. Atoms are not shown in the bead model explicitly, instead, they are located at somewhere three beads meet. Most chemists feel uncomfortable with this connection. So Chern and I tried try to explore with the shape of beads and multiple beads and hope that they can better represent the shape of chemical bonds. So that is why we have these models in which multiple beads represent an edge.

But now, I have the valence sphere model of chemical bond as the theoretical foundation of bead models. Spherical beads are in fact the simplest possible approximation of electron pairs, in accord with the principle of Occam's razor. So to build a model of a molecule with only beads and strings is equivalent to performing a molecular analogue computation with beads. The result of computation is the approximate electron density of the corresponding molecule without referring to the Schrödinger equation or atomic orbitals (this is the comment I got from Prof. H. Bent). I have written an article on the connection between bead models and valence sphere model in Chinese for September issue of Science Monthly (科學月刊). I think I should write something about the molecular analogue computation with beads.

Of course, what I am saying above is to view bead models as molecular models. Results of mathematical beading do not need to have any connection to the molecular world. For instance, the beaded Hilbert curve accepted by the JMM mathematical art exhibition is a good example.

Styrofoam ball/rubber band model

Bead models can be viewed as a kind of valence sphere models (or tangent sphere models 價球模型或是切球模型), in which beads represent electron pairs. Similar idea using styrofoam balls and rubber band has been exploited in the 60s by L. Carrol King, a chemistry professor in the Northwestern university. Here is the first styrofoam ball/rubber band model of methane that I made.

Many people use this model to illustrate the valence sphere electron repulsion theory (VSEPR) for molecules with four valence electron pairs. We can easily see that the energy of the model in the square planar configuration is higher than that in the tetrahedral configuration. But one has to be careful in the interpretation because the energy difference in the total energy of these two configurations is from the elastic energy of the rubber band, instead of the repulsion energy among four hard spheres.

Twistane

I just learned a kind of molecule called twistane the other day. It looks so interesting and I decided to make a bead model for it.

The hydrated diacetates of rhenium

The cover of the inorganic chemistry by J. E. House shows the structure of the hydrated diacetates of molybdenum(II), chromium(II), and rhenium. One can make a very good bead valence sphere model of this molecule containing a metal-metal bond.

Bead valence sphere model of penta-prismane

The structure of a penta-prismane is similar to that of a cubane. Instead of 4-fold rotational symmetry, one has a five-fold rotational symmetry. So the shape of penta-prismane is just like a pentagonal prism.

Bead valence sphere model of tetra-t-butyl tetrahedrane

The tetrahedrane derivative with four tert-butyl substituents, tetra-t-butyl tetrahedrane, was synthesized by the Austrian chemist,Günther Maier, in 1978. Here is the bead valence sphere model of this interesting molecule I made this afternoon. I think the hardest part to make molecules with many sp3 centers how to control the force evenly in the whole weaving process.

Comments from Prof. Henry Bent

I mailed the reprint of my paper with Prof. Cuccia, "Molecular Modeling of Fullerenes with Beads" (J. Chem. Edu 2012 , 89 (3), 414–416) to Prof. Henry Bent two months ago. He is the person who proposed the tangent sphere model in the 60s. He made the following comments on the beaded molecules:

"... contributions to the literature on molecular modeling such sophisticated molecules with so elegantly simple methods: beads and string, that's all!"
"The saturation and directional character of chemical affinity falls out naturally from your bead models without any need to refer, e.g., to Schrödinger's equation and atomic orbitals when constructing approximate electron density profiles, even for molecules as complicated as the fullerenes." 

Pagodane （塔烷）

According to the wiki:

A pagoda （塔） is the general term in the English language for a tiered tower with multiple eaves common in Nepal, India, China,Japan, Korea, Vietnam, Burma and other parts of Asia. (source: wiki)

About twenty years ago, H. Prinzbach was able to synthesize an organic compound with a skeleton which resembles a pagoda by a 14-step sequence starting from isodrin. Thus he named the compound pagodane （塔烷）.

I just knew this molecule from the book "Molecules With Silly Or Unusual Names" by Paul W. May the other day. So here is the bead model of this interesting molecule.

Adamantane (金剛烷)

I made a bead valence sphere model of adamantane (chemical formula C₁₀H₁₆), which is a cycloalkane and also the simplest diamondoid.

Neopentane

Neopentane, also called dimethylpropane, is a double-branched-chain alkane with five carbon atoms. (from wiki)

Cyclohexane conformation

I made a bead valence sphere model of cyclohexane, C₆H₁₂, which seems to reproduce all important structural features of this molecule.

Fullerane: C60H60

Fullerane is any hydrogenated fullerene or fully saturated fullerene. For instance, the fully saturated C₆₀ is C₆₀H₆₀. One can make a faithful valence sphere model of C₆₀H₆₀ with beads. The 150 beads in this model represent 300 valence electrons (240 from carbon and 60 from hydrogen) in this molecule.

Bead VSM of tetrahedrane

I should forget another platonic alkane, the tetrahedrane. The shape of this molecule based on the valence sphere model is just like 10 spheres close packed in a tetrahedron. Which models, valence sphere model or ball-and-stick model, is closer to the true shape of a tetrahedrane molecule?

Bead VSM of dodecahedrane

In principle, we can make the valence sphere model for any molecule with beads. But in practice, it is a little bit hard to thread the Nylon cord through a bead structure with tetravalent bonds, which are common for most molecules though.
But anyway, I made a bead VSM of dodecahedrane, C₂₀H₂₀.

Two EMACs

Qian-Rui made these two bead models of EMACs, one with 3 metal cores and the other with 9 cores, last year. The ratio of two types of beads is so small such that the surrounding ligands stop to spiral around the central metal string.

Bead model of the longest EMAC

Qian-Rui made this bead model of the longest EMAC (Extended Metal Atom Chain) for Prof. J. McGrady (Univ. of Oxford) who is an expert on the electronic structures of this class of molecules and is going to visit our department today. In making this model, Qian-Rui chose 10mm beads for both metal-metal and metal-ligand bonds, and 8mm beads for chemical bonds of surrounding ligands. The pitch of the surrounding ligands is slightly larger than the true molecular structure of this molecule. Only a single nylon cord with about 7 meter long is used for this model.





Here is a photo for the giant structure of this molecule hung in the main lobby of our department.

Pentanuclear EMAC

There is a whole family of EMACs with different chain lengths and different metal atoms.
Qian-Rui made this first bead model of EMAC with five metal atoms.


The sizes of beads are 8mm and 6mm, respectively.