Analysis of Torsion multiplicity

Problem Statement

Background: When we assign force field parameters to a given molecule, every bond is assigned one or more torsional parameters. These describe how difficult it is to rotate that bond.  For example, see the attached image of biphenyl and imagine rotating that central single bond between the two rings. We can measure a torsion angle which describes whether the two aromatic rings lie in the same plane or are twisted out of plane with one another. There are several torsional parameters in the force field which actually end up being applied to that bond to govern how it rotates. Each torsional angle (and torsional parameter) actually involves four atoms, though, since it takes four atoms to define two planes and look at how they rotate relative to one another. So in our biphenyl example (see figure) if we are interested in rotation of the central C-D bond, we have to consider torsions A-C-D-X, B-C-D-X, A-C-D-Y, and B-C-D-Y — that’s four torsions. It’s four torsions in this case because the two carbons adjacent to that rotatable bond each have three connections, but if we were looking at the central bond in cyclohexylcyclohexane (second attached image, with no labels) we would be dealing with more torsions (in this case, the image does NOT show the hydrogen atoms, which are implied; here, each carbon has an additional hydrogen attached to it beyond what’s there in biphenyl). Specifically, now we’d have torsions involving elements C-C-C-C like in biphenyl, but also additional torsions involving H-C-C-C and H-C-C-H, so we’d have 9 torsions through the central bond rather than 4, I believe. This number, which we could call the multiplicity, is not terribly important, except that (a) it tells us something about the chemistry involved, and (b) the force field needs to be somewhat different depending on the multiplicity. Usually, any given torsional parameter in the force field will probably only be applied in molecules which have a single multiplicity (e.g. always for molecules with only four torsions passing through a bond, or always for molecules with nine, etc) but this may not always be the case.

Scope

Reference materials

https://drive.google.com/drive/folders/1FigXwRQfbM9p3QtDMAK_CoDkNS2pEHNN