It’s apparent from getting through 127 Pages of “Molecular Symmetry” by David J. Willock, that he put a lot of effort into the book, and that group theory is quite important to chemistry. But, it appears that this book is of the ‘plug and chug’ variety. You are given formulas and codes for irreducible representations of point groups (which the book explained quite well) and then asked to apply them. Just how the formulas are obtained hasn’t been given. So you are expected to learn the concepts by example, which I find irritating for something basically mathematical, where concepts are clearly defined and results deduced. I was hoping to understand characters, but I really don’t at this point. The book made it clear how useful they are to group theory — another way to slice and dice group structure (along with subgroups, and conjugate classes). So I’ll put the book aside, and hopefully come back to it, once I read some more group theory.
Plug and chug can even get you through a math course, but understanding is what’s crucial. Even you haven’t looked at something for a long time (in my case decades), if you once understood something, it comes back very quickly. Plug and chug does not, just like the combination of key strokes you need for a program you last used years ago.
Students are being cheated when plug and chug is all they get. I did an alumni interview last night with a very bright young woman taking AP calculus (2200 on SAT’s etc. etc.). She knew how to calculate a derivative, but had literally no idea what a derivative actually is. She couldn’t give a geometric example of a limiting process. It’s not because she’s stupid, because she’s far from that, it’s because she isn’t being well taught.
For those of you plowing through the book along with me, here are my comments (up to page 127 when I gave up). Comments on earlier pages can be found in the category “Willock: “Molecular Symmetry”
pp. 105 — 127
p. 105 — If you think about a vibration, you are thinking about a molecule in motion, so how can it be the subject of a symmetry element? Because at any instant in (some) vibrations the position of the atoms are fixed, and these positions can (italics) possess varying degrees of symmetry as the example of the guitar shows. Nice.
p. 109 — “contravenes the restrictions of the irreducible representation” — probably should add “(e.g. A1)” to avoid the idea the restrictions of all (italics) irreducible representations have somehow been given.
p. 109 — “The molecule, as a whole, never actually moves during a vibration” — otherwise you’d violate the first law of thermodynamics, getting translational energy from nothing.
p. 110 – Further understanding of what a basis can actually be. You can put 3 basis vectors on each atom of water, giving a total of 9 basis vectors.
p. 115 – “Property 3: The number of irreducible represntations in a point group is equal to the number of classes”. But class was defined very incompletely on p. 93 as ‘an equivalent set of operations’ — bu how two distinct symmetry operations are equivalent is never (italics) defined, just illustrated.
What a culture clash — “Property 3 can be confirmed” — confirmation isn’t proof. Hopefully I’ll learn something about representations and their utility in chemistry by continuing to plow forward.
My guess is that the notion of equivalence for classes is conjugate equivalence (from my knowledge of group theory) and that the classes are conjugate classes (we’ll see).
p. 116 — I never got to character theory when I studied groups (which is one reason I’m reading this book), but property 4 tells why they are important — they give another way of slicing and dicing groups (like subgroups and conjugate classes). Again property 4 is confirmed for a variety of examples rather than proven.
p. 117 — Property 5 — Yet another way characters give you a way to slice and dice a group — orthogonality.
p. 118 — What is a fundamental vibration of a molecule? A vibration having some symmetry (not all vibrations do)?
p. 119 — you can simplify a reducible representation using the data for the set of irreducible representations in the standard character tables — how are these irreducible representations obtained? Willock isn’t telling.