 So, we understand now that the symmetry elements of a molecule form a group, a mathematical group under composition. So for example, we saw that the water molecule with these four symmetry elements, two vertical reflection planes, a rotation, C2 rotation and identity forms a group. That group is an example of something mathematicians and chemists call a point group. And in fact, we can give names to these individual groups. So the group that's formed by identity, a C2 rotation and two different reflection planes, we call that group C2V. And really that's just a name that we give it to remind ourselves that there's a C2 rotation and there's some vertical reflection planes in that group. Some other molecule with some different collection of symmetry elements will have a different group, a different name for that group. But what's remarkable about and useful in fact about point groups is that any molecule with this collection of symmetry elements will belong to the same group. And there are in fact many molecules that have this collection of symmetry elements, there are many molecules that are in this C2V point group. So for example, water as we've seen repeatedly is in the C2V point group because it has a C2 rotation, it has a reflection plane that bisects it in this direction and a reflection plane that bisects it in this direction. So those are the elements, the symmetry elements of the molecule. Any other molecule with the same geometry will certainly have the same point group and it doesn't matter if the bond angles and bond lengths are the same. Any other bent molecule, for example sulfur dioxide, which I'll draw, I could draw it with a double bond and a single bond but to emphasize the fact that it's symmetric, I'll draw the resonance structures like that. So there's sulfur dioxide, that's a molecule with the bent geometry. Any other molecule with the bent geometry is going to have a C2V symmetry group. But it doesn't have to be a triatomic molecule with a bent geometry. For maldehyde, for example, you can see perhaps after I've drawn the three dimensional structure, also is in the C2V point group. It has a C2 rotation along the carbonyl bond axis. It has the same two vertical reflection planes, one in the plane of the board, one bisecting the molecule in half in this direction. So it has the same symmetry elements, it will be in the same point group. And even more surprisingly, there's some other molecules that are in the C2V point group. So sulfur hexafluoride, for example, which I'll draw with its three dimensional geometry in this saw horse or T shaped, sorry, saw horse geometry. This is also C2V. Again, there's a two-fold rotation. Here's the same C2 axis of rotation. There's a symmetry, a reflection plane in the plane of the board. There's a reflection plane that bisects the molecule this way. And those are the only symmetry elements we can find for this molecule. So all of these four molecules and many others are in the C2V point group, even though they don't look like they have very much in comment at all. But because of the symmetry of the molecules, because they have very similar identical symmetries, many of the properties of the molecules are related. For example, the shapes and symmetries of their molecular orbitals are similar in many ways. The different vibrational modes that they have or can undergo have some connections to each other. The polarity of the molecule has some similarity because of what point group they're in. The spectroscopy of the molecule, what frequencies of light they can or can't absorb to change which quantum mechanical states. There's similarities in all those properties that we can predict knowing the point group of the molecule, even though the geometry of the molecule at first glance doesn't look terribly similar. So what that means is it's very important, very useful, to be able to predict the point group of the molecule by finding this symmetry elements and then identifying what that point group is. So to give you an idea of how that's done, I'll pull up the table here of several different point groups. In fact, the full list of all the point groups that molecules can have. And we see here there's a number of different point groups. So this column lists the point groups of a molecule, and then next to that is listed the different symmetry elements. So we know the one example we've seen so far. Water is in the C2V point group with an E, a C2, and two reflection planes. That is this example right here. So a molecule with, if I say, N equals 2, a molecule is in the CNV or the C2V point group if it has an identity. If it has a CN or a C2 rotation axis, if it has N or two different sigma V reflection planes. So if we've determined the symmetry elements of water to be E, C2, and two sigma Vs, then we've discovered that it's in the C2V point group. If we write down another molecule and we find that it has some different list of symmetry elements, then we can name the point group of that molecule. So you can treat this like a list of definitions. If you've listed the symmetry elements, you can determine the point group of the molecule. But what's truly remarkable about this list is even though we can synthesize thousands, maybe millions of different molecules, this is pretty much the complete list of point groups that molecules can have. There's only a handful of different point groups that molecules can have. And again, those point groups determine many of the properties of, or at least restrict many of the properties of those molecules. So it's kind of remarkable that out of all the molecules that we can imagine synthesizing, they fall into this finite list of different point groups. So the next thing we'll do is there's a few features of this list that deserve a little bit of comment or explanation. So that's the next thing we'll do is explore the details of this list in a little more detail.