Math? In Chemistry?

As you might have picked up from before, there are several kinds of symmetric molecules! That brings up a huge question-- how do we classify them? How do we differentiate one from another?

Well, that's the purpose of applying group theory in the world of molecules!

Wait a second! Isn't group theory a very advanced mathematical concept? Yes, but it can be applied here in a more simple fashion. This post will cover the basics.

We've touched upon this in one of our earliest blog posts, but it's never bad to repeat something that's so important, right?

In this case, one phrase is crucial to understanding the various types of symmetry:

"A Symmetry operation is an operation which brings an object into a new orientation which is equivalent to the old one."

So for example, in the BF3 molecule below,

we can rotate the shape around the center by 120 degrees. That would change where the atoms go, but it would still maintain the same form. So that rotation would here be a symmetry operation.

There are 5 main symmetry operations involved with molecules:

1) The identity (E)

This is what several people would call a "trivial" operation. Basically, it's doing nothing to the molecule-- that would, of course, keep it in the same form. So every molecule has at least this symmetry operation.

2) Rotation (Cn)

In this operation, we rotate the molecule around certain axes in order to get the same shape back. The axis, depending on the angle that the molecule is rotated by, is called Cn. For example,

as the picture suggests, a rotation by 360/2 degrees around the C2 axis would bring the molecule to the exact same shape! Similarly, rotation by 360/3 degrees around C3 and rotation by 360/4 degree around C4 would both preserve symmetry. 

I hope you realized the pattern: an axis used to rotate the molecule by 360/n degrees is called the Cn axis.

In addition, the operation itself is also called Cn. 

Here are some more examples:

In this famous H2O molecule, we can see that rotating it by 180 degrees around the axis in the middle would make it the same shape! Hence, this molecule has rotational symmetry.

This time, we have something more complicated! There are two different types of rotational axes in this molecule of benzene (C6H6). We can rotate it by 60 degrees around the center C6 axis or by 180 degrees around the C2 axis to get the same hexagonal shape.

Another thing to note here is this-- how many C2 axes do you think there are? I'll give you a moment to think.

Well if you answered 6, you're correct! There are three that intersect through opposite sides-- such as the one shown in the picture, and there are three more that intersect through opposite vertices! That makes 6 whole axes of just C2. This just shows you that a molecule doesn't need to just have one type of rotational symmetry.

Oh, and one more thing-- the Cn axis with the highest value of "n" is called the principal axis.

3) Reflection (σ)

Reflection is also relatively straight forward-- it's just reflecting the molecule across certain planes.

There are three types associated with this symmetry operation: 

          1) Vertical (σv): In this case, the plane of reflection is parallel to the principal axis. Here's an example:

In this ammonia molecule, the plane shown in yellow contains the principal axis. Since reflecting it across that plane would result in the same shape, this molecule is vertically reflective. (That specific reflection is also called σv.)

(If you want to see an interactive, 3-D version of this, click here.)

          2) Horizontal (σh): In this case, the plane is perpendicular to the principal axis. 

Before, we saw that benzene molecules have an C6 rotation axis, which is the principal axis in this case. So the plane shown here would count as the horizontal plane, since it's perpendicular to that axis. And clearly we can see that reflecting the molecule across this plane gets us the same shape. 

          3) Dihedral (σd): This one's a bit more complicated. In this case, the plane is parallel to the principal axis and bisects the angle between two C2 axes. It's best explained with an example:

In this picture, imagine the C2 axes connecting opposite vertices. We see that the plane bisects the angle formed by two of these axes. Since the plane is also parallel to the principal axis, this reflection counts as a dihedral reflection.

Lastly, notice how the dihedral plane was also the vertical plane! The two do not have to be distinct-- rather, all dihedral planes are actually vertical planes, since dihedral planes must also be parallel to the principal axis.

So far, we've covered 3 out of 5 symmetry operations. We'll continue on with our journey at another time.