Klein bottle

Three students in my class, Molecular Aesthetics, constructed a bead model of Klein bottle.



My contribution to their work is to tell them that this model can be done with mathematical beading. Especially, I showed them how to systematically construct the interpenetration of a graphene surface and a carbon nanotube, which is essential part of this model. In addition to my part, the TA of this course, Hou-Xun Ho (何厚勳), who developed a systematic strategy to build smooth tori with large rotational number, plays an important role for successful construction of this model with beads.

Ping-Pong Valence Sphere Model

Valence sphere models, a qualitative chemical bond model that includes the influence of the electron pair repulsion among valence electron pairs and attraction between positive atomic core and negative electron pairs, can be constructed with Ping-Pong balls easily. Here, I made three pairs of linked ping-pong balls and used them to create a tetrahedral sp³-hybridized AX₄ system and a octahedral d²sp³ hybridized AX₆ system.

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.

Prismane

Prismane with 3-, 4-, and 5-fold rotational symmetry:

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.

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.

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.

Bead VSM of cubane

I just made a bead VSM (valence sphere model) of cubane (C₈H₈) by myself. The structure looks neat to me. Every valence electron pair is faithfully represented by a big bead, purple for CC bond and pink for CH bond. Small beads which have no chemical meaning are used to bind the pink beads to the central carbon cube. I didn't distinguish CC bonds from CH bonds. In principle, electron pairs responsible for these two types of chemical bonds should have different momenta. So they should have different sizes of charge clouds.

Bead VSM of methane, ammonium, water, and hydrogen fluoride

There is no doubt that tetravalent molecules such as methane occupy an important place in the chemical bond theory.
In 1865, German chemist August Wilhelm von Hofmann made the first stick-and-ball molecular models of methane in lecture at the Royal Institution of Great Britain. It is planar!

Then, 1872, van't Hoff, then a graduate student, learned of a possible tetrahedral arrangement of the valence bonds of carbon, proposed by the Russian chemist Alexander Butlerov in 1862. He later made a set of 3-D paper models of tetrahedral molecules.

Following Prof. H. Bent's recipes, Qing Pang (龐晴) of TFGH (北一女) made several bead valence sphere models (bead VSM) for tetravalent molecules with the formula AX_nE_m, where n+m=4, n is the number of bond electron pairs and m is the number of lone electron pairs. Here she didn't use the Windsor's knot to end the Nylon thread, instead she used simply tiny beads to cap the terminal beads of this kind of tree-like structures. Also, she use blue beads to represent bond electron pairs and yellow beads long electron pairs. You can see bead model exactly realizes the valence sphere model of Bent. I will show other bead VSM (made by Qing Pang) of molecules with several centers later.

Tangent sphere model of ethane

Using beads, one can make a faithful representation of the so-called valence sphere model (VSM) or tangent sphere model for a molecule proposed by Prof. Henry Bent in the 60s. In this model, each valence electron pair in a molecule is represented by a sphere. Its diameter is determined by the de Broglie wavelength of the corresponding electron, λ = h/p, where p is its momentum and h is the Planck constant.

Here is the first bead VSM (BVSM) of ethane (C₂H₆) made by Qing Pang (龐晴) of the Taipei First-Girls High School (北一女). She used the so-called Windsor-knot technique (雙活結) to bind beads that are not parts of loops in a molecular graph. For simplicity, she used beads of the same size to build the BVSM of ethane. This is equivalent to making the assumption that all valence electrons have the same momentum. The paper below the bead model is from the manuscript entitled "Approximate Molecular Electron Density Profiles. I. Construction" that I got from Prof. Bent last month.

BTW, Prof. Bent has just published a new book entitled "Molecules and the chemical bond" which is the first book-length sequel of his early articles on tangent sphere model last year. If you want to know more about the tangent sphere model, you should read the book or the original articles published in J. Chem. Edu.
You can read parts of this book at the google book.

1. Bent, H. A. J. Chem. Edu. 1965, 40, 446.
2. Bent, H. A. J. Chem. Edu. 1965, 40, 523.
3. Bent, H. A. J. Chem. Edu. 1967, 42, 308.
4. Bent, H. A. J. Chem. Edu. 1967, 42, 348.
5. Bent, H. A. J. Chem. Edu. 1968, 44, 512.
6. Bent, H. A. J. Chem. Edu. 1967, 45, 768.

Four face-sharing pentagonal dodecahedra

E. A. Lord, A. Mackay, and S. Ranganathan described in their book, "New geometries for new materials", a simple cluster consisting of four face-sharing pentagonal dodecahedra arranged in a tetrahedral configuration (pp.48). Here is a bead model of this cluster.

In their book, there are more clathrate structures that one might be able to construct with beads.

Clathrate cluster

I bought some more green rice-shape beads last week and managed to finish this interesting clathrate cluster of 60 dodecahedra in the last weekend. One can still see deformation of many dodecahedra in this clathrate cluster though.

Buckyball made of 60 dodecadedra

I made this structure with beads last weekend. Still unfinished. The finished structure should have 60 dodecahedra arranged like a buckball. One has two ways to interpret this structure:
1. If every dodecahedron represents a carbon atom, we have a standard C₆₀.
2. If we still use beads to represent CC bonds, then we have a giant molecule, C₇₅₀. In this molecule, 450 carbon atoms are sp³ hybridized or tetra-valent and 300 atoms are sp² hybridized or trivalent. But I suspect these sp² hybridized carbon atoms are not energetically favorable, so it is better to have hydrogen atoms connected to these sp2-carbons. Then we get C₇₅₀H₃₀₀!



The bead model of this structure (see here) might be first constructed by Emilie. She asked me to comment about this structure in my blog long time ago (I couldn't find the exact location though).

I decided to make one from beads after I saw the same structure made by a toy designer, Dick Esterle, who actually invented this kind of toys, in the Bridges conference last week.

Diamondoids

Qian-Rui and I discussed the possibility of using beads to build molecular models for compounds that contain hybridizations other than sp² yesterday. We both agree that diamondoids are the best candidates. Although I knew this class of compounds for a long time. I have not thought about constructing these molecules with beads.

Here are a few beaded models of Diamondoids I just made.


Adamantane:

Space-Filling Polyhedra Based on a Truncated Octahedron

I made this structure during the spring break. Chuang has made a similar structure with several truncated octahedron before. But his structure has three unit cells separated connected. So the first unit cell is not connected to the third unit cell directly. My original goal is to make a structure with six unit cells connected to each other, something like a 2x2x2 cluster. However, the structure seems to be sensitive the small deviation at each local connection. Although truncated octahedron can fill the space. the structure I made seems to have pretty large strain and distorion.