 Okay, so if we're going to make sense of my assertion that the symmetry elements of a molecule form a group, we need not just the elements, but the operation with which to combine them. So we need to talk about how to combine symmetry operators or symmetry elements together. And the operation that we use to combine symmetry operations is called composition. So that's just a fancy word for saying do one of the operations and then do another one afterward. So in some cases, composition of these operators is easy. Let's imagine, let's say we have some object, I'll just draw some three dimensional looking object. There's a rotational axis and if I imagine what would happen if I did a C2 operation followed by a C2 operation in particular remembering that these operators act to their right. If I do the C2 operation and then do a second C2 operation on the result of the previous operation. So this is doing two C2 operations one after the other. A C2 operation is just rotation by half of a full circle. So if I rotate by half of a circle and then rotate by the rest of the circle, then what I get back is the same as if I had done nothing. So like I said, sometimes understanding what is meant by the composition of two operators is relatively straightforward. Two half rotations or a full rotation is exactly the same as doing nothing. And this begins to show you why we needed this identity operator in our list of symmetry operators so that we can understand when nothing has been performed to an object. Likewise, if I did a reflection followed by another first reflection followed by a second reflection, if I reflect in a mirror, wherever I draw this mirror, if I reflect the object in a mirror and then I reflect it back, I'm going to get back the same object I started with, which is the same as doing nothing. But sometimes composition is a little bit trickier. If I first rotate by a half circle and then I reflect, what do I get then? It's not the same as if I did nothing, but it takes a little effort to figure out what is it in fact that you get when you rotate something by 180 degrees and then reflect it through a mirror. So to understand how to do composition of symmetry operations like that, the more tricky cases, I'll point out a couple of things. Number one, it doesn't matter what you apply the operations to. Any object you can use to visualize the rotations and reflections is fine. If I rotate this object by 180 degrees and then another 180 degrees, it comes back to where it started. But it doesn't matter whether I did that to this object or to this object. Rotation by 180, rotation by 180 brings it back to where it started. That's true for any object. It doesn't matter what the target of these operations is. It's just the operations themselves that define what you get when you combine them together. And along those same lines, somewhat counter-intuitively, it probably does not make sense to think about how to combine these operations by applying them to molecules, even though that's the goal we want to understand eventually. So for example, if I write down the geometry of a water molecule, which we've already seen has a C2 rotational axis by selecting this angle, meaning that if I rotate it by 180 degrees, I get something that looks the same as what I started with. But because of the symmetry of that molecule, so if I imagine doing a C2 rotation to that molecule, what I've gotten back is looks the same as where I started, even though it's only been rotated by half a circle. If I take an asymmetric object and I rotate it by 180 degrees, I can see the difference. But if I take a symmetric object, one that has C2 as a symmetry element, and I perform a C2 rotation, it looks exactly the same, I would need to label these molecules H1 and H2. And then after the rotation, see that I have H1 over here and H2 over there, before I realize that it's not exactly the same as performing an identity operation. So the symmetry of this object makes it a little bit difficult to distinguish the difference between, say, a rotation and a reflection or something like that. So it's actually much easier to visualize the results of these symmetry operations when you're doing it on a less symmetrical and asymmetrical object. So there's a couple of different ways to go about this. What I'll do is I will sketch a cube and a cube by itself, and I'm sketching a cube just because it's a relatively easy three-dimensional object to draw, but a cube is quite symmetric. So what I'm going to do is I'm going to label one face of this cube, I'm going to label the top face of the cube with the letters TOP. So now I have an asymmetrical object. If I rotate it or reflect it or do something to it, I'm also going to reflect or rotate those letters so I'll be able to see the change that's happened to the cube. So now let's imagine doing several of these symmetry operations to the cube. So I'm going to define a C2 axis, a rotational axis through the middle of the cube pointing out the top and the bottom through which I can rotate it by 180 degrees. I can also define a reflection plane. I should have bisected this molecule with my reflection plane. So there's a plane that cuts the molecule in half. I'm going to call that reflection plane sigma v, a vertical reflection plane, and I'll define another reflection plane. It's also a vertical reflection plane, it stands up vertically, but that reflection plane I'll call sigma v prime. So that reflection plane is sort of in the plane of the glass board here. The sigma v reflection plane cuts back and forth through the board. So I've defined three different symmetry operations and if I'm being careful my C2 axis should point through the same, all these should intersect in the middle of the cube. So I've got a C2 rotation and two different reflection planes. Now let's try to imagine what happens to this cube with the letters TOP on the top as I do the symmetry operations. So first let's do the C2 operation. So if I take this cube and I rotate it by 180 degrees, it's still going to look like a cube, but the right face will have been rotated around to the left, the front face will be rotated around to the back, the top face will still be on the top, but the letters will be rotated by 180 degrees. So what that will look like if I rotate the top face is hopefully you can see that that rotation will end up having rotated the writing on the top of the cube. So that's the result of what I get by doing a C2 rotation operation. We can also consider what happens with these two reflections. Let's write over here my cube after I've done a reflection. So let's do the sigma v prime reflection. So if I do the sigma v prime, so I'm reflecting through this plane that cuts the molecule dividing into a front half and a back half. So I'm reflecting the front to the back and the back to the front. The cube still looks like a cube, but the letters get reflected front to back. So instead of having TOP written from left to right with the letters right side up, when I reflect that P front, the T front to back, it looks like this. The O looks like this. The P looks like that. So I've reflected those letters front to back and notice that this writing doesn't look the same as this writing. A 180 degree rotation is clearly not the same as reflecting in this front to back mirror. How about our other mirror, the sigma v reflection? So the sigma v reflection is this reflection that divides the cube into left halves and right halves. So I'm reflecting the left half to the right and vice versa. After that operation, the left has become the right. But the front remains in the front and the back remains in the back. So these letters TOP, if I reflect the T from the left side over to the right side. The O remains in the middle. And then the P with its spine towards the center of the box remains with its spine toward the center of the box, but now it's facing the left side of the cube instead of the right side of the cube. So again, this writing is not the same as this writing. It's reversed front to back from this writing. So what that means is, if I want to make this cube look like this cube, I could in fact reflect it through the sigma v prime plane. If I reflect it front to back, then it's going to end up looking like this. And vice versa. If I take this cube and reflect it front to back, I'm gonna end up with this cube. So now we begin to see, have we done this one yet? If I do first a C2 rotation, no, we haven't done this one yet. So let's take a look at this case. If I want to make this cube look like this cube, where the upside down T is on the left, I want to make the upside down T on the right. I just need to reflect it left to right. So that corresponds to the sigma v operation and vice versa. And then to finish this off, if I want to make this cube with its upside down writing look like this cube, which is right side of writing just right to left instead of left to right, I can do that by doing a C2 rotation. If I take this cube and rotate it by 180 degrees, perhaps you can see that the writing would end up looking like this writing. Although it's hard because they're both backwards in different ways. So I can turn any one of these cubes with this writing written strangely into the others by some sort of rotation. But if we now want to understand what do I get if I do a C2 rotation followed by a sigma v? If I first do the C2 rotation and then do a sigma v, I get this cube. That's exactly the same as if I had first just done a sigma v prime. So now we have our answer to this question. C2 followed by, so it's actually preceded by when I write it down, but the action of the sigma v comes after the action of the C2. C2 followed by a sigma v is exactly the same as if I had done a sigma v prime. We could write out similar equations for all sorts of other combinations. If I first do a sigma v and then a sigma v prime, that's the same as doing a C2 and lots of other combinations. So this has hopefully been a useful example for showing us how to perform combinations or composition of multiple different symmetry operations. If you don't like this technique of drawing cubes, another handy thing to do is to use hands. So we can repeat this example with your hands. Your hands are chiral, one is the mirror image of the other. So as you watch this next video, do the same things I do and see if you get the same result. If I take one of my hands, so notice that I'm starting with my hand, back of my hand facing me, the front of my hand facing you, it's my right hand. Actually, it's gonna look like my left hand to you. So I'll avoid saying right and left. It's this hand without the watch on it, palm facing you, back of my hand facing me. If I do a C2 rotation, it's gonna look like this. If I then do a Sigma V, the one I was calling Sigma V cuts it in half this way. If I reflect it through a mirror, I can't literally reflect my hand through a mirror in this direction. But what I'll get is exactly the same as if I replace it with my other hand. If you ignore the fact that this one has a watch and a wedding ring on it. So if I take this hand and reflect it through a mirror, it's gonna look like this hand. So once I've done that, I've got my C2 rotated hand, followed by a mirror reflected hand. What is the hand that I got? That is exactly the mirror image of the hand that I started with. But reflected in, I'm sorry, when I started I had my back of my hand facing me, front of my hand facing you. This starting hand is, again, a mirror image reflection of this hand. But the mirror is now a mirror that bisects them in this direction. So in other words, the Sigma V prime mirror. So you can use your hands without bothering to write diagrams if you don't feel comfortable with the 3D sketching. You can use your hands to perform the symmetry operations and find that a C2 rotation, followed by a mirror reflection, gives you the same result as some other symmetry operation. So now that we know how to combine symmetry operators, we'll take a look and see if we can make them form a mathematical group.