Trying to grasp molecular symmetry

Not all molecules are created equal. Some look neat and tidy when you draw them out in 2D, whereas others can't feasibly be drawn out that way. It would be easy to assume that therefore, only some molecules possess symmetry, but then you realise molecules exist in 3D space. Therefore, in reality, all molecules, in their own way, possess a certain type of symmetry. It's just less obvious at times.

Symmetry refers to how an object looks unchanged after you do something to it. In maths, you may have heard about various different operations, such as reflecting, rotating, or scaling a shape, and the same can apply to molecules. It just so happens the maths behind all this can get very confusing, but which is sadly necessary if you want to handily characterise all your molecules.

That maths is known as group theory, and I think I'll be having a series of lectures on it. If anything, today, I'll start simple, and go over symmetry. There may be future group theory posts to come...

Symmetry operations

There are a few straightforward symmetry operations I'd like to mention which are relevant to my module:

• A molecule is inherently symmetrical. Don't believe me? Keep the molecule the same and note whether it looks the same - it does (obviously). This is the identity operation, or E. Every molecule has it, it's how you'd denote an exceptionally non-symmetric molecule like CHFClBr.
• It's very elementary, so let's not bother with it.
• It's also technically equal to one full 360° rotation, come to think of it. That's going to prove useful in a few paragraphs.

CHFClBr

• You can rotate a molecule around an axis. You've got three axes to pick from - the x, y, and z axes - and so you have three different operations to play around with. These are denoted as C_n, where n is the number of rotations performed within 360°.

I can rotate water around the z axis, and it will look identical every 180°. That means water is C₂.

• You can also reflect a molecule along a mirror plane. These are denoted as σ, and you've got three to choose from: 
• Horizontal mirror planes (σ_h) are perpendicular to the main rotation axis.
• Vertical mirror planes (σ_v) include the main rotation axis.
• Dihedral planes (σ_d) are a type of vertical plane which also happen to bisect bond angles.
• Another operation involves taking a molecule and moving each point to its inverted coordinate. For example, a point (x, y, z) is adjusted to (-x, -y, -z). This is known as inversion, or i.
• The final operation is known as improper rotation, or S_n. Here, you perform a C_n rotation followed by a reflection in the horizontal mirror plane. 
• If you're wondering what makes this operation so special, considering it's literally just combining two aforementioned ones, it's because some molecules don't have a rotation or horizontal reflection as symmetry elements on their own, but they do possess an improper rotation. A good example of a molecule like this is methane.

Point groups

Some molecules have loads of different symmetry operations, which can get tedious after a while. Wouldn't it be easier to classify molecules based on how similar they are symmetrically to each other, rather than listing them out line by line?

That's where point groups come in. They're basically just groups that denote various symmetry operations. For example, all molecules that only have an identity operation are classified as C₁. Alternatively, a molecule with only a rotational operation are classified as C_n, and molecules with only mirror planes are known as C_s. That's pretty straightforward stuff, but point groups get far more confusing as soon as they represent a combination of operations.

Say your molecule has rotational and reflectional symmetry, like water. From the diagram above, it should be somewhat clear that placing a mirror plane along the z axis produces an identical reflection in two different planes. That gives water a C₂ axis as well as two σ_v planes, meaning it belongs to the C_2v point group. Other axes and mirror plane orientations are available.

If you were to place a mirror along the x and y axes, you'd get reflectional symmetry - σ_v. However, if you did the same along the z axis - so in the plane - you lack reflectional symmetry, your water would look upside down. Therefore, you have no σ_h.

There are two other point group types I want to bring up:

• Some molecules have loads of different rotational axes. BF₃, for example, has four different rotational axes: one going straight through the boron atom (the principle axis), as well as three perpendicular axes along the B-F bonds. Alongside this, there are three mirror planes along the principle axis, as well as one perpendicular axis along the plane of the molecule. The respective point group of BF₃ is therefore D_3h. 
• Why D? That refers to the dihedral point group, which includes molecules with different rotational axes. 
• Why 3h and not 3v? Well, horizontal mirror planes take priority over vertical planes.

You can imagine rotating around each of the axes in BF₃; along the x and y axes, you can complete two 180° rotations, hence C₂, but around the z axis, you can complete three 120° rotations - C₃. Two differenct axes makes this a D point group, and it's D₃ not D₂ because the larger number wins out.

• Some molecules have really high amounts of symmetry. Take a centrosymmetric octahedral molecule like SF₆: three different rotational axes, a centre of inversion, even some improper rotation and a fair few mirror planes. In fact, its symmetry resembles that of an octahedron. Similarly, CH₄'s symmetry somewhat resembles that of a cube.

Methane inside a cube. Image not to scale; all the bond lengths are equal. Of course, methane is tetrahedral, which you can visualise by drawing bonds between the hydrogens.

• For these kinds of molecules, you have cubic, such as T (tetrahedral) and O (octahedral) point groups. These refer to the symmetry of the molecule, not their shape. 
• CH₄ is T_d and SF₆ is O_h, to be specific. In other words, they're perfect tetrahedrons and octahedrons, respectively.
• At least compared to, say, CHCl₃ and SClF₅, where you have various different atoms around the centre. You lose a few symmetry operations here, most notably the loss of a reflective plane of symmetry. 
• Specifically comparing CH₄ and CHCl₃, you lose a horizontal mirror plane with CHCl₃. You can visualise this by looking at my methane cube and replacing three hydrogens with chlorines, then imagine placing a mirror across one of the Cl-C-Cl axes.

Chirality

To finish off this post, I thought it would be worth discussing chirality, specifically how it relates to symmetry. Chirality is really just a property of symmetry: specifically, a molecule is chiral if it lacks an improper rotation axis. So that's why it was worth its own category earlier...

Going back to basics, a molecule is chiral if it doesn't have a superimposable mirror image. A straightforward example would be your hands: they're chiral to each other, because they're mirror images but you can't superimpose them. If your hands were molecules, they'd be enantiomers of each other, to be specific. 

That said, you can also have diastereomers, where you have two stereoisomers that aren't mirror images, but also aren't superimposable. A nice example is glucose. All I need to do is invert one of the stereocentres (bring an alcohol group to the back) and they're no longer the same molecule.

Two diastereomers of glucose

As an aside, you may have heard recently that they've announced the Chemistry Nobel Prize winners this year. Henri B. Kagan and Kenso Soai won the prize “for the discovery of non-linear effects and autocatalysis in asymmetric organic synthesis”, and part of that related to the notion of homochirality - the tendency for some molecules to opt for one specific stereoisomer in nature. I might write a blogpost about this in the future, but if anything, I just found this to be somewhat cool and relevant to mention.

Epilogue 

I can't really explain all this in a blogpost, and I suspect those of you who read this blog for the occasional trip to a random town are feeling a bit bewildered now. That's okay, this is just a revision post for myself after all. Either way, I suppose this is just a skill that will take lots of practice, the kind that gets much easier when you can visualise all the molecules in 3D space. 

Time to bring out the modelling kits...

The main source for this blogpost was the book Group Theory for Chemists by Kieran C. Molloy.