LinkTW and LinkTWM

Higher twisted (Möbius) annulenes can be characterised by topological invariants such as the linking number Lk and related by the Cãlugãreanu/Fuller theorem to twist and writhe by the expression

Lk = Tw + Wr

A program to compute these properties is available here. The source code is File:LinkTW.cpp and File:LinkTWM.cpp. The file File:Calculus.cpp, is also needed. The former code should be used for systems with an even linking number and the latter for systems with an odd linking number.

Compile the code using e.g. c++ LinkTWM.cpp -o TWM.

Input

Typical inputs are File:14-d2-H-iso-ribbon.xyz for an even linked system and File:15-d3-ribbon1c.xyz for an odd linked system. The ribbon constructed looks something like this. The first line of the input file corresponds to the number of segments in the ribbon (i.e. number of carbons in a [n] annulene). This is followed by 2n sets of XYZ cartesian coordinates (no element symbols!). The coordinates MUST be arranged in adjacent ordering, ie the ordered set of 14 coordinates describing the C positions of the [14]-annulene (in this example), with each carbon connected to the next, followed by a second circuit of the ring in the same direction as the first, corresponding to the normal to the (approximate) plane of the sp2 hydbridized carbons (i.e. mapping to the lobe of the 2p -AO attached to that carbon). If any coordinate is out of sequence, the program will return nonsense. The first set of n XYZ coordinates represents the backbone of the annulene. The second set of n XYZ coordinates is that of a dummy atom (arbitrarily) placed 1Å away from the first backbone atom, and orthogonal to the plane containing that (ideally sp2-hybrid) atom and its three ligands. In other words, it represents the orientation of the p-π orbital located on the carbon atom. For odd-link systems, TWO circuits of the backbone must be made to return to the original position.

Output

filename:	14-d2-H-iso-ribbon.xyz

# of Carbons=	14
Tw=	0.695283	( Phi=  -0.304717,	Alpha=  1)
Tw2=	0.695283	( Phi=  -0.304717,	Alpha=  1)
Wr_1=	0.304717	Wr_2=	0.304717
Wr_1+Tw2=	1	Wr_2+Tw2=	1
Lk_1=	1	Lk_2=	1

The units for this output are of 2π. It is more useful to think in terms of units of 1π, in which case the above system has a linking number Lk =2π, composed of 1.39π of twist and 0.61π of writhe. The value of Lk can be positive or negative, and as such it constitutes a chiral descriptor (for each enantiomer). --Rzepa 11:19, 17 February 2013 (UTC)