 potentials. Here is a small animation of a molecule that is very much periodic. This small aliphatic hydrocarbon will keep going around and around and around. It's a butate and that can actually happen. There will certainly be an energy barrier roughly there. When the two molecules are superimposed right on top of each other it's going to be better when it's stretched out but it can occur in both those states. You actually have two more torsions here because carbon atom one and two here. If that bond rotates that's going to correspond to the three hydrogens rotating and that's an even lower barrier. And of course here too the exact form of this potential can be complicated if we calculate it from quantum chemistry but if you want something that is periodic on the unit circle I hope that you would all say trigonometric functions right and that's what we're going to use. So the simplest possible trigonometric function would essentially be a sine or cosine in this case. I like to have the lowest energy value to be zero rather than having to worry about both positive and negative values so we're just adding a constant here to lift it up so that the baseline here is zero. Depending on this rotation then I will have an energy here that starts out at a high value then it goes to a low value and then it becomes high again and then it's low again and high again. This is an even simpler molecule. It's actually an ethane so you just have one bond you're rotating and then we have three hydrogen atoms here and three hydrogen atoms here. The reason why that is important once you rotated a third of a turn here you're back in a state that's exactly identical to your starting state. I'm not sure if you can see the x y axis here maybe not but as I already hinted the reason why this is important is that the energy levels involved here are much much lower than distortions in bond or torsions. So here we might be talking about a handful of kilojoules or k-cals and that's low enough that it's going to happen normally at room temperature which is of course also the reason that we need a potential that will realize that we were back in an equivalent periodic state. But I started with butane so I need to show you what this would look like in butane. So if I remove that one and move to a slightly more realistic potential there are different names trans, cis and gosch that I might not have time to go through here but if you look at that butane molecule again by far the best state here is the one we had straight in the middle here and that would be when the four hydrogen atoms are placed like that sorry for carbon atoms are placed like that the lowest energy by far the worst state we're gonna have is if they are placed like that and that's going to correspond to the peak here around zero degrees and that is a hint if you're going to look at biochemistry versus polymeric emissions. Now these two other states correspond to things where things are not quite as bad as the cis states so things have rotated roughly a third of a turn away from that one so that means that things are not directly superimposed but there is a not quite clashes it's not quite as good as the fully extended states. Those are called gosch the history behind that is not important but you see that you have two local minima here the local minima are definitely better than the peaks but they're not as good as the best possible minima between these two called gosch states and the fully trans state you have additional barriers those barriers are not good they're high but they're not as high as the barrier in the cis state and now things are getting pretty complicated for something as simple as butane that I have a very good low state and I have a pretty good low state I have a very bad barrier and I have a somewhat bad barrier. The way we represent this is pretty much by using two periodic function two cosine functions but they have slightly different periods one that corresponds to a period with 360 degrees and one that corresponds to a third of that. You're going to need to have a rough idea roughly what these energies are it's going to turn out that this is super important for proteins and the torsional degrees of freedoms are the ones that are going to describe the entire shape of the chain because once you start rotating around the bond if you have a long chain of connected molecules well in butane not much happens but if you had 500 other atoms bound at the end of that chain when you rotate this that would mean that the entire rest of the molecule would rotate with it so when we're rotating around bonds in a long chain that tends to change the global confirmation of the entire molecule and now I'm almost getting ahead of myself and talking about protein folding because this is going to be important but that's for later