 So, to discuss an application of symmetry that's more relevant to actual chemistry, let's see if we can talk about the symmetry elements of some actual molecules. And we'll start with relatively simple molecules, like ammonia, NH3. So the key to finding the symmetry elements is to be able to first draw or visualize the object and then ask yourselves for each one of these different symmetry operators, which of them leave the molecule unchanged? So we need to be able to draw the molecule. One word of caution, it's not enough just to be able to draw the Lewis structure. If I draw the Lewis structure for ammonia like this, and then start treating this as a visualization of what the molecule looks like, I'll start making all sorts of mistakes because the molecule doesn't have 90 degree bond angles. Lewis structure doesn't tell us anything about the geometry, it just tells us about the connectivity of the molecule. So you have to remember your VSEBR theory and be able to draw these molecules the way you've done since general chemistry. So for ammonia, that's a pyramidal molecule. I'll draw that with this hydrogen in the plane of the board. This one coming out of the board towards you, this one with a dashed line going back into the board away from you. So it's a trigonal pyramidal molecule. So that's what the molecule looks like three dimensionally. So now we just need to ask ourselves if we want to know the symmetry elements. Does it have a symmetry element that corresponds to the identity operator? And of course, the answer is yes. So here's the list of symmetry elements. Has an identity operator. The identity operator leaves the molecule unmodified. Does it have any rotations? Does this molecule have any rotations that leave it unchanged? And in fact, it does. The axis that points out the top of this pyramid, that is a C3 axis, meaning that if I rotate the molecule by a third of a circle around that axis, one of these H's spins onto another H. This one spins onto that one. This one will spin onto that one. So that three fold rotation or one third of a circle rotation is a C3 axis. So the molecule has a C3 rotation symmetry element. And that's the only rotation we'll be able to find. Reflection planes. Does the molecule have any reflection planes? Can I bisect the molecule in some way that leaves the half on one side of the mirror identical? A mirror reflection of the half on the other side of the molecule? That might be a little bit difficult to see from this orientation. But if I imagine that same trigonal pyramidal molecule, if I imagine the top view, if I look down upon this molecule from the top, so top view, then that molecule is going to look like this. So again, this is not a three-dimensional structure. This is just the top-down view. And what that allows me to see is I can, in fact, bisect the molecule along this plane. And if I reflect the molecule in that plane, this H becomes that H and vice versa. Or, in fact, I have a symmetrically equivalent plane that looks like this in which this H reflects onto that one. And there's a mirror plane here in which this H reflects onto that one. So there's, because there's three identical H's in this molecule, if I found one mirror plane, there's going to be three that go through the end. And one of those H's, there's going to be equivalent ones that go through the other H's. So there are, in fact, three of these mirror planes. Inversions? No. Wherever the center of the molecule is, if I invert some of the H's through the center of the molecule and out the other side, there's no H there for it to land on. So inversion will modify the orientation of the molecule. So there's no inversion center and there's no improper rotations. If I, there's no way I can rotate the molecule and then reflect it through a perpendicular reflection plane and then up with an unmodified molecule. So I won't be able to find any of these improper rotations. So the, the list of symmetry elements for the ammonia molecule is an identity, a C3 rotation and three different mirror planes. If we consider one more example, and now I will consider an example where there's an improper rotation just so we can get a sense of what the improper rotations look like when they actually do show up, we'll consider a slightly more complicated molecule, CH4, methane. So I need to be able to draw the methane molecule. Geometry of methane is tetrahedral. So one way I could draw that molecule would be like this, carbon in the center, one hydrogen pointing straight up from the top of this pyramid, three hydrogens pointing down, forming the legs of a pyramid with three sides. It's going to turn out to be useful also to draw the molecule in a different orientation. So here I'll encourage you if you have, if you still have an organic, a model kit that you used as an organic chemistry student, go ahead and pause this video, dig out your organic model kit, build yourself a methane molecule. In fact, as you get used to these symmetry problems, build every molecule that you're trying to understand the symmetry properties of. It's, it's very handy thing to do. So if I draw methane with a different orientation, it's very useful to be able to say for these tetrahedral molecules, I'm actually going to draw a cube. And if I place a hydrogen, let's say at this corner, and this corner, and this corner, and this corner of the cube, so there's, there's some other edges of the cube that are obscured in the back. So there's my cube. There's also a carbon at the very center of this cube that's bonded to each of these four carbon atoms. So perspective makes that look a little strange. If I want to make it more clear, I can show some of them coming out of the board towards us and some of them going back into the board away from us. But putting the hydrogens on opposite corners of a cube can also be an interesting, a useful way of picturing the symmetry of this molecule as well. Those are both, both tetrahedral geometries. I've just drawn them a little bit differently. One having this pyramidal shape and one having rotated so that the hydrogens appear on the corners of a cube. So if I now ask myself what are the symmetry elements of methane, always start out with an E, doing the identity operation to any one of these structures is going to leave it unchanged. So the identity always leaves something unchanged. Let's actually start with the S. Maybe not. Let's, let's, we'll go ahead and in order. So the C rotations. Are there any rotational symmetries of this molecule? If I spin this molecule around some axis by some amount, does it leave it unchanged? And now you'll see why I wanted to draw this molecule in two different ways. Looking at this geometry, it's pretty clear that just like ammonia, there's some three-fold rotations. If I treat this as the rotation axis, if I rotate the molecule one-third of the way around a circle or around this axis, then this H will rotate into this one. This will rotate into this one. This one will rotate around back to the first one. And that's a C3 rotation. The difference with ammonia is because I have, because my rotation actually contains this hydrogen, I can have a C3 axis along this CH bond. It's harder to see from the picture, but you can convince yourself with your model kit that there's also a C3 rotation around that CH bond. If I take the molecule and I tip it so that that hydrogen is pointing straight up, it will have a C3 rotation just like when the C3 rotation was pointing upwards. Likewise, for that CH bond and for that CH bond. So there's a total of four of these C3 rotations. But we're actually not done. There's some additional rotational symmetries of this molecule. And those are easiest to see from this diagram. In this picture where I have two hydrons at opposite corners of the top surface of this cube, if I draw a rotation axis pointing out the center of the top of the cube, if I rotate by 180 degrees halfway around the circle, I'll just exchange the two places of these hydrogens. And I'll exchange the two positions of the lower hydrogens. So there's a C2 axis pointing out the top and bottom of the cube. Similarly, there's a C2 axis pointing out the sides of the cube where these two hydrogens will rotate into each other. And these two hydrogens will rotate into each other as I spin them by 180 degrees around this axis. And also out the front and back sides of the cube. In this case, these two hydrogens will exchange places when I spin. And these two hydrogens will exchange places as I spin. So those are, again, perhaps difficult to see on this two-dimensional diagram with my poor artistic abilities. But if you build this model with the model kit and can picture in three dimensions, it's easy to find that C2 rotation you can convince yourself that spinning the molecule by 180 degrees, there's three different ways you can do that to leave the molecule unchanged. So I have three C2 rotational axes. Next, we'll ask about reflection planes. Are there any ways? And perhaps I'll try to clean this diagram up a little bit so we can see the reflection planes. Are there any ways I can bisect this molecule so that the half on one side of the mirror is the mirror reflection of the other half? And probably the easiest one to spot in this diagram, we can spot some of them in either one of these diagrams, in fact. If I create any plane that contains an H and the central carbon and another one of the H's. So in this case, that would be this diagonal plane that cuts through the top of the box. So the full plane is, this is the front edge of the plane, this is the back edge of the plane, and down here is the bottom edge. So I've cut this cube in half diagonally. This H and this C and this H are all within the plane. This H, excuse me, is to one side of the plane. This H is on the other side of the plane, but they're mirror reflections of each other. So there's certainly a mirror reflection plane. We just have to count how many of that type of reflections there are. If I bisect the top face of this cube the other way, between the other pair of corners, now these two hydrogens are in the plane. This plane, cutting the cube in half diagonally that way, these two hydrogens are within the plane. Now these two hydrogens are in front of and behind the plane and they're mirror reflections of each other. So on the top face, I can slice it in half two different ways. But the same exact thing is going to be true on the left face and on the front face. So I can, in fact, I can cut this cube. Maybe I should redraw it to convince you of that fact. So here's my cube with hydrogens here and here and here and here, carbon in the middle. If I take this face and I cut it in half either this way. So in this case the plane would look like, so I cut the cube in half like that. The carbon and two of the hydrogens are within the plane. These two hydrogens are on opposite sides of the mirror plane, so they're mirror reflections of each other. Mirror plane, each of the faces I can cut in half in two different ways. So I've got a top, bottom, left, right, front, back, face of the cube, each of which I can cut in half two different ways and that leaves me six mirror reflections. So as you can see, the more symmetric a molecule gets, a molecule like CH4 with multiple different symmetry elements, it can be somewhat difficult to visualize and to see all of them. As our last, I don't know, inversion center. Does the molecule have an inversion center? That one's easy to answer, relatively easy to answer. If I take any hydrogen, invert it through the center of the molecule and out the other side, it doesn't land on another hydrogen. There's no hydrogen that corresponds to the inversion of this hydrogen, so there's not an inversion center. Methane is particularly interesting when it comes to its improper rotations. We found lots of rotations, three-fold rotations and two-fold rotations that leave the molecule unchanged. It turns out there's also a four-fold rotation, but it's an improper rotation. So let me draw one more time this cube that contains the methane with hydrogens here and here and here and here. So my four-fold rotation is going to be an axis like this one. So imagine I take this molecule, carbon in the middle, and I rotate it around this S4 axis. If I just do the rotation without the associated reflection, so if I just do the C4, I'm going to get... So if I rotate by 90 degrees, the hydrogens on one of these corners rotates to the adjacent corner, so I'm going to get hydrogens here. The one that was here rotates to here, and the two that were in the front left and back right go to front right and back left, so the one's going to be here and one's going to be in the back here and then still a carbon in the middle. So my carbon has just been rotated by 90 degrees. That is definitely different than the one I started with. So C4 is not a symmetry element of this molecule. But if I then do a reflection through the horizontal plane that's perpendicular to the S4 axis, since my S4 axis was through the top of the cube, the perpendicular plane is going to bisect the molecule horizontally. If I reflect this molecule through that plane, this hydrogen will reflect down to the bottom. This one will reflect down to the bottom. The hydrogen in front here will reflect up to the top and the one that's obscured here in the back will reflect up to the top and the carbon will remain in the middle. So what I have there, with two diagonally opposed hydrogens on the top and two diagonally opposed hydrogens on the bottom, that's exactly the same as what I started with. So the S4 action, which consists of a C4 rotation and a perpendicular reflection, leaves the molecule unchanged. Even though the C4 by itself didn't leave the molecule unchanged, the sigma by itself didn't leave the molecule unchanged. So here's the circumstance when we do want to include an S4 symmetry element because the constituent operations, the C4 and the sigma, go up in the symmetry elements. Last thing we have to do is just to count how many such S4 axes there are. And here, because the S4 axis points out of the face of a cube at the top and the bottom of this cube, there's going to be a similar one that points out of the left and right sides of the cube and another one that points out of the front and back faces of the cube. So there's going to be three S4 axes. In fact, exactly in the same direction as the C2 axes that I've since erased. But these three C2 axes, the three S4 axes, are exactly the same axis, but I can spin halfway around with a proper rotation or a quarter of the way around with an improper rotation. And they both leave the molecule unchanged. So it can be a little bit complicated, a little bit difficult to find all of the different symmetry elements of molecule. Again, if you have access to a model kit where you can manipulate these things in 3D, that will certainly help you a lot. But it's an important skill to be able to either visualize the molecule on paper and find the symmetry elements or in your head or with a model kit to be able to identify the symmetry elements of a molecule.