 with an interaction involving two atoms, then we had an interaction involving three atoms, you can probably guess what's coming next, an interaction involving four atoms. So we pick these same four central atoms, what if we look at some sort of relative motion that includes all four of them. This is getting a bit trickier to visualize, I'm going to try it anyway. I can use my pens here, so here's one bond, here's the second bond and here's the third bond, so the atoms would be at the end of the pointers here. So if you look at this, I can turn this around, so I'm not changing any angle here. The angle around this one you see, the angle there is fixed, and if I instead do the same motion here, but if I follow it there, you see that the angle between that, that and that atom, that doesn't really change either. So what is changing here, if you're looking along the middle bond, what is changing is a rotation along the middle bond. There are many names we use for this, actually no, not many, but there are two. We occasionally call this a torsion angle and we occasionally call it a dihedral angle, and sorry they are used interchangeably in the field. The way we define this is by each triplet of atoms here, i, j, k for the first three atoms and then the last three atoms that would be j, k and l. Each such triplet defines a plane and then we can take the blue plane here and the red plane and there is then an angle between those two planes. That angle is the torsion angle. What gets a bit complicated here now is that we're going to need to find a way to describe this and this gets complicated in two ways. First if you look at the red and blue planes here, there are two ways to describe this. Do you pick in this case the small angle or the large angle between them? Because that's going to be basically, one angle phi here is going to be 180 minus the other angle, so it matters which one you're choosing. The great thing is that we're scientists, so that we like to have a standard to define things and in this case we have chosen to define two standards. There is a convention that you sometimes call the polymer convention and there is a convention that you call the biochemistry convention. If I tell you what these conventions are, you're going to skim through that, so I'm not going to tell you. That's a homework task now for lecture two. Go out in Wikipedia, look at torsion angles and check what is the biochemistry convention, biochemistry and what is the polymer convention. You will typically not have to worry because we're normally going to stick to biochemistry, but if you start defining and calculating these angles yourself, you're going to need to know which one you're picking. Because you pick the wrong one, things will be pretty garbled. And now you might think that you know exactly how you're going to treat this interaction. This is a sort of complex interaction that we would like to describe around local equilibrium and we don't know exactly how they work. And I think you're going to get this wrong. Because you would pick the simplest potential is proportional to a force constant multiplied by the square of the displacement of the angle, first second order approximation. The problem with that is if I take this torsion angle here, if I start out here, and then I wrote this an entire turn, 360 degrees 2 pi. Well, the deviation is now 2 pi, but I'm back in exactly the same state as I started. And in particular for small molecules, small molecules, these barriers are low enough that small molecules can occasionally rotate an entire turn here. So for torsion, it's not going to work to simply have this very simple second order harmonic description of it. So we're going to need something else. We're going to need a potential that is periodic. And I bet you know about periodic potentials. So let's move on to the next slide and see what they look like.