Back in the day, we were told that group theory was important in quantum mechanics, because it simplified the Schrodinger equation, and ultimately the interpretation of spectra. We really didn’t get into groups in the graduate QM course I took in ’61, but were told to look at Weyl’s book “The Theory of Groups and Quantum Mechanics” which didn’t get into representation theory until p. 120 after a lot of linear algebra and physics. What chemist had the time? I didn’t.
In retirement, I’ve indulged my taste for math, and audited a graduate algebra course, which had a fair amount of group theory, but no representation theory. In addition the instructor dumped all over chemistry and physics as part of his schtick of being a pure mathematician. He avoided all applications to chemistry. He did hold up for derision a book written by a chemist with several large groups explicitly written out.
So now that I’m reading chemistry again and am up to Ch. 14 of Anslyn and Dougherty on computation of orbitals, I thought it was time to give group theory another shot.
The postulates of group theory really couldn’t be simpler, and as you delve into the subject, it’s amazing how much structure can arise from them.
Here they are.
A group is just a set G, which can be finite or infinite. Chemical symmetry uses only finite sets, but in physics groups with an infinite number of elements are possible.
Just one operation is defined on G. It is a binary operation, which is to say it takes any 2 elements of the group, and produces a third element of the group. Examples include addition and multiplication, which take any two numbers and give you a third. In mathematical terms, the operation is said to be closed. Call the operation *.
* : G x G –> G ;
* : g * h |–> i, where g, h and i are elements of G not necessarily distinct.
There is only one other requirement on the operation. The operation must be associative — so that given 3 elements of the group, the order on which you subject them to the group operation doesn’t matter. e.g.
a * (b * c) = (a * b) * c ; where you do the operation inside the parentheses first
The group must have one special element (usually called e for the German einheit meaning identity), but 1 will do.
For any element of a of the group 1 * a = a * 1 = 1
Lastly, every element of the group must have an inverse (written a’ ) such that
a * a’ = 1 and a’ * a = 1, so 1 is its own inverse, and other elements can be as well.
So the positive integers are NOT a group under addition (no inverse). The integers are not a group under multiplication ( 0 has no inverse).
That’s all there is to it. Yet the classification of all finite groups took 30 years, and 10,000 pages in journals, and people aren’t really sure they’ve got it done. The classification of the quasithin groups (don’t ask) was the subject of a 1221 page paper.
Representation theory is the correspondence of the elements of a group to a set of matrices. The group operation of a representation is always matrix multiplication. You can find a gentle introduction to matrices in the 9 posts in the category — https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/.
The math books don’t have any chemistry, and while groups are worth studying for their own sweet selves, I want to see what they offer the chemist (well, more realistically- HOW they offer what they do to the chemist). So I’ve begun reading “Molecular Symmetry” by D. J. Willock. Lots of chemical structures, schematic diagrams (some not particularly intuitive), and an approach to groups through symmetry.
What is really exciting about all this, is that chemistry lets you explore the structure of groups by moving a molecule around so that it superimposes on itself. The book shows how the set of symmetry operations you can perform on a molecule form the elements of a group (showing that the elements of a group aren’t limited to just numbers). The group operation is simply doing one symmetry operation after another. So instead of screwing around with matrices, which have no visual content (although their effects do), you can play around with simple molecules like water, ammonia, benzene and watch a group in action.
Technically what you are really doing when you do this is looking at what is called a group action — the action of a group on another set (e.g. the distribution of a collection of atoms in space in this case). But, to my mind it’s really a way (and a better one) of representing a group. Water is an example of the Viergruppe (4 group), which has only 4 elements. The instructor said it’s a very strange group.
Parenthetically it is worth noting that Slater (of the Slater orbitals, and Slater determinant of quantum mechanics) hated groups — calling them the gruppenpest. Anslyn has a lot about both in pp. 817 – 822.
It’s too early to fully recommend the book, but the first 50 pages are quite fine. Stay tuned