The n-Category Café

This paper gives a link to Racah’s work on $\mathrm{G}_2$ and the f shell:

• R. Bruce King, The group-theoretical structure of the atomic g shell: connection with the alternating group $\mathrm{A}_6$ as $\mathrm{L}_2(9)$, Journal of Mathematical Chemistry 44(1) (2008), 5–19. (paywalled link.)

King writes:

The detailed study of the atomic d shell was initiated by Condon and Shortley [1] in 1935 following earlier work by Slater [2] in 1929. In 1949 Racah [3] developed group-theoretical methods for study of both the atomic d and f shells…”

Reference [3] is

• G. Racah, Theory of complex spectra IV, Physical Review 76(9) (1949), 1352–1365. (paywalled link.)

and it indeed does quite a bit with $\mathrm{G}_2$!

But King’s paper also has its own interesting stuff on f orbitals, and it refers to this for more:

• R. Bruce King, Group-theoretical structure of the atomic f shell: connection with the non-Euclidean heptakisoctahedral (didodecahedral) group, Molecular Physics 104(20-21) (2007), 3261–3268. (paywalled link.)

The ‘heptakisoctahedral group’ is our friend the 168-element group $PSL(2,\mathbb{F}_7)$.

Now that you’ve mentioned $A_6$, I’m suddenly hoping that there’s some amazing chemistry that comes from the existence of its weird outer automorphism.

I still don’t know King is embedding $A_6 \cong PSL(2,\mathbb{F}_9$ in $SO(9)$ or $PSL(2,\mathbb{F}_7)$ in $SO(7)$ — he doesn’t say, and I haven’t succeeded in guessing!

But in pondering this, I ran into something which seems to say the outer automorphism of $A_6$ can be understood as the outer automorphism of $PSL(2,\mathbb{F}_9)$ that arises from the nontrivial automorphism of the field $\mathbb{F}_9$, so that’s nice. The field $\mathbb{F}_9$ can be thought of as $\mathbb{F}_3$ with a square root of $-1$ thrown in, say “$i$”, and the automorphism sends $i$ to $-i$. So it’s just like complex conjugation.

I still don’t understand the isomomorphism $A_6 \cong PSL(2,\mathbb{F}_9)$, so if anyone knows how that works, I’d be interested.

@Jack Morava : the book is available on the Internet Archive !

And it has been reviewed (the 1931 german edition) by none other than Bartel van der Waerden for the Zentralblatt.

Jack wrote:

I believe you [JB] have had a shot at explaining shell theory elsewhere and I have been waiting a long time to understand this story.

Yes, I summarized the key results of shell theory here:

• The Madelung rules.

The biggest thing I left out is the quite difficult calculation starting from the hydrogen atom Hamiltonian and concluding with the decomposition of its Hilbert space of bound states (which I called $\mathbf{H}$) into eigenspaces of the Hamiltonian (which I called $\mathbf{H}_n$, and which are called ‘shells’) and then the further decomposition of each eigenspace into irreducible representations of $SO(3)$ (which I called $\mathbf{H}_{n,\ell}$, and which are called ‘subshells’.) When you understand this stuff, you hold the magic key to understanding the periodic table!

I also have a ‘pop’ version of this story, which should be very easy and fun for you:

• Mathematical mysteries of the periodic table.

Keep hitting the “next” button at upper left. (It’s hard to see on the very first page.)