Structural types of TCNTs

A few weeks ago, I was informed by my colleague about a paper by Beuerle et al.[1], in which they identified eight structural types of high-symmetry achiral toroidal carbon nanotubes (TCNTs) based on two of our papers in J. Chem. Inf. Model.[2,3]
These eight structural types of TCNTs are

A. Type I, Parallel Prism; outer-rim: zigzag, inner-rim: zigzag: T240, another example, detailed construction
B. Type I, Parallel Prism; outer-rim: armchair; inner-rim: armchair: Two more models
C. Type III, Parallel anti-Prism; outer-rim: zigzag; inner-rim: zigzag: T120, Another T120, T240, T240
D. Type III, Parallel anti-Prism; outer-rim: armchair; inner-rim: armchair: T140, model 2
E. Type II, Anti-parallel Prism; outer-rim: zigzag; inner-rim: armchair: T250,
F. Type II, Anti-parallel Prism; outer-rim: armchair; inner-rim: zigzag: T240
G. Type IV, anti-Parallel anti-Prism; outer-rim: zigzag; inner-rim: armchair T240, T360
H. Type IV, anti-Parallel anti-Prism; outer-rim: armchair; inner-rim: zigzag: T240, T240, detailed construction

Here, we used zigzag and armchair patterns along lattitude coordinates, instead of using chiral vectors, (1,0) and (1,1). We have all these eight structural types in our first paper without giving them names though. In addition to these TCNTs with well-defined latitudes, we also included some high-symmetry TCNTs without latitudes (not in these eight types) in our papers. They seem to ignore them. Here is an example of TCNT (with the detailed construction) that does not belong to these eight structural types. Chuang and I are working on a comment to clarify these three extra types of TCNTs and the correct relationship (transformations) among these 11 types of TCNTs.


1. F. Beuerle, C. Herrmann, A. C. Whalley, C. Valente, A. Gamburd, M. A. Ratner, J. F. Stoddart, Optical and Vibrational Properties of Toroidal Carbon Nanotubes. Chem. Eur. J. 2011, 17, 3868-3875. See also Hot topic on carbon from Wiley-VCH.


2. C. Chuang, Y.-C Fan, B.-Y Jin, "Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 361-368.
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3. C. Chuang, Y.-C Fan, B.-Y Jin, "Dual Space Approach to the Classification of Toroidal Carbon Nanotubes." J. Chem. Inf. Model. 2009, 49, 1679-1686.
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