 So, we can determine the point group of a molecule if we know it's symmetry elements by looking it up on this chart. So it turns out the only collections of symmetry elements molecules will ever have will be one of the lines listed on this chart. So a few features of this chart deserve some extra comment, however, one of them we've already seen is that this chart involves some shorthand. We've seen the C2V point group. There's also a C3V and a C4V point group, so this N in many of these lines stands for an integer. You're never going to have N equal one, but you could have N equal two or three or four, any integer two or larger. So we've seen water is in the C2V point group because it has identity, a C2 rotation and two vertical reflection planes. But a molecule like ammonia, if I draw the 3D structure of ammonia, ammonia has a C3 rotational axis and it's trigonal pyramidal structure. So I can rotate it a third of the way around and not change the molecule. It's also going to have three different vertical reflection planes that I won't attempt to draw. But if I bisect the molecule along this NH bond or this NH bond or this NH bond, each of those is a different vertical reflection plane. So that would be a C3V point group, molecule in the C3V point group because it has an identity, a C3 rotation and three different vertical planes. So N in this chart could stand for any integer two or larger. So it's a little bit of an exaggeration to say there's only a dozen point groups. If I count C2V and C3V and C4V and so on, there's more than are just listed here. I've just collapsed many of them into one line on the chart. Let's see. Another feature of the chart that I should explain is, in fact, what these sigma V as opposed to sigma H as opposed to a new one, a sigma D planes look like. So we've seen, we've already been using this terminology of sigma V and sigma H for vertical and horizontal planes. If there's a rotational axis and the plane is aligned with the axis, contains that rotational axis, some CN axis and a sigma V plane contains it, that would be a vertical reflection plane. On the other hand, if there's a horizontal reflection plane, we call that a sigma H. There's, so that would be a vertical and here vertical and horizontal mean with respect to this CN axis. If I've drawn the molecule in some odd orientation where the C2 axis points in a weird direction, I have to make the C2 axis point straight up and then, if the plane is vertical, I call it a vertical reflection plane. If the plane is horizontal, then I call it a horizontal reflection plane. A sigma D is new, that's a dihedral reflection plane and this one is a little less common and harder to remember because you see it less commonly. Dihedral, if you remember that like a polyhedron is a surface made up of many, many surfaces or many planes, dihedral is two different planes, two different surfaces. A dihedral reflection plane, if I have a rotation axis, a CN rotation axis, a dihedral plane is also a vertical reflection plane, meaning it contains the C2 axis. But it also happens to bisect two different C2 axes. So if a molecule has a C2 or some other rotational axis and it has, let's say, a C2 axis pointing off in this direction, so here's the C2 axis and also a C2 axis pointing off in that direction. So I've got two crossed C2 axes in addition to the vertical CN axis. If there's a vertical plane that bisects these two C2 axes, cuts them in half, splits them down the middle, then we would call this a dihedral reflection plane rather than a vertical reflection plane. If it doesn't bisect two C2 axes, if either these C2 axes aren't there or if it lies along one of the C2 axes rather than bisecting it, then we would just call it an ordinary sigma V vertical reflection plane. So other features to explain in this list, we have some interesting features like this C infinity. That looks a little strange. We understand what a C2 rotational axis is, what a C3 rotational axis is, what's a C infinity rotational axis? It's really just the same thing just in the limit of very large values for the subscript. A C2 axis means rotate halfway around and you don't change the molecule. A C3 axis means rotate a third of the way around and you don't change the molecule. A C infinity axis, all that means is if you rotate the molecule by an infinitesimally small, an infinitely tiny fraction of a circle, you don't change the molecule. So that would be, so a C infinity axis would be for molecules like say HCl, a linear molecule like HCl has a C infinity axis because certainly if I rotate that molecule by halfway around that axis, the H doesn't change, the chlorine doesn't change, but I could also rotate it by one degree or half a degree or a tenth of a degree. Whatever amount I rotate the molecule, the spherical, the cylindrically symmetrical molecule or if you want to think about it as the spherical H and the spherical chlorine don't look any different after that rotation. So it doesn't matter how many degrees I rotate it, the molecule is unchanged. So the only difference between these C infinity V and D infinity H, point groups then they both have this C infinity axis, but molecules that are symmetric and in addition to the C infinity, so the H2 also has a C infinity plane, but it also has some additional features like a horizontal reflection plane. So because it has a horizontal reflection plane and an inversion, then it falls in this D infinity H point group. So we have several different types of planes, we have the C infinity rotational axis, what else should I explain? One feature of this chart that is in fact a slight fib as it's written now, some of these C2, C2H, C3H and so on are a little bit different for even values of N than they are for odd values of N. So the CNH in particular, sometimes in addition has an inversion operation. So it might have identity, a C2, a sigma H and an I. So if the N value is an even number then there will be an inversion center. A C3H on the other hand just has identity, a C3 axis and a horizontal reflection plane but does not have the inversion center. So the ones that require this little addition, CNH has an inversion if N is even, DNH also has an inversion if N is even, but DND is the opposite, it has an inversion center if N happens to be an odd number. So D3D for example would have an inversion center but D4D would not. Alright so that explains that additional feature of this chart and the last thing to mention about the chart is that this list of symmetry elements actually doesn't include one particular type of symmetry element that we assume is always present for a molecule like NH3 with its C3 rotational axis. So that would be C3V. For NH3 falls under this category, it's got its identity, C3 rotation, 3 sigma V planes but there's one that's not listed. If I rotate this molecule by 120 degrees, sure I've not changed the molecule but what if I rotate the molecule by 240 degrees or if you prefer by negative 120 degrees. That operation, that symmetry element is necessary, it needs to be there as the inverse of the C3 rotation so that we have a C3 rotation, we also have a C3 rotation that happens twice or if you prefer a C3 operation that happens in the opposite direction. That symmetry element you'll notice is not listed on this table. In fact any time we have a CN symmetry element in this table, we also assume that it's in there as CN squared and CN cubed and as many different copies of that, let me write that up here. If we have a CN, we also have a CN squared and a CN cubed and so on. All those incremental amounts of, additional amounts of that rotation also are symmetry elements, we just don't bother to list them in the table because we know if there's a C3 there's also going to be two C3s, if there's a C5 there's going to be two C5s or three C5s or four C5s. Those are all the little asterisks and caveats and extra details required to understand the details of this table but with all those extra details, now we're definitely able to, if we're given a molecule, find the symmetry elements, find them on this table and identify what the point group is but it turns out there's an even easier way to identify the point group. And that's what we'll explore next.