Infrared active eigenmodes (i.e. symmetry adapted coordinates, that also describe (resonant) vibrational motions) are those which change the dipole moment of the molecule. The transform all like the threefold degenerate ungerade representation called Tu in Mulliken notation.

]]>P.S.: Strange, I must have posted this about one year ago (at leat quite a few months) or so. The timeline thoroughly confuses (if not scares) me ..

]]>This a very good idea, I am sure it will help attracting more interest from people with knowledge on spectroscopy!

]]>The paper tries to push quantum computing in the direction of rotational state spaces, which have been thoroughly studied in spectroscopy. I try to include all the jargon for the groups in the text to make it easier ( and $latex $C_\infty = SO(2)=U(1)$).

]]>Thanks for the interest! The math-related part is Appx. D. I don’t think it’s at all surprising one can formulate a “grand orthogonality theorem” for coset spaces, but I’ve never seen it explicitly written either. If you’re interested in the quantum aspects, feel free to read my blog post: https://quantumfrontiers.com/2019/11/17/on-the-coattails-of-quantum-supremacy/

]]>Interesting (and long) paper! Never got in touch with quantum computing before but to me it seems to be closely related to microwave spectroscopy and theoretical chemistry, though the language is a physics/maths one ( versus , …).

]]>I’m busy grading midterms and preparing for classes but I hope someday I have time to look at your paper!

]]>• Layra Idarani, SG-invariant polynomials, 4 January 2018.

All that is a continuation of a story whose beginning is summarized here:

• John Baez, Quantum mechanics and the dodecahedron.

So, there’s a lot of serious math under the hood. But right now I just want to marvel at the fact that we’ve found a wavefunction for the hydrogen atom that not only has a well-defined energy, but is also invariant under this 7,200-element group. This group includes the usual 60 rotational symmetries of a dodecahedron, but also other much less obvious symmetries.

]]>• Layra Idarani, SG-invariant polynomials.

Namely, he takes any Coxeter group acting on its vector space and answers this puzzle:

**Puzzle.** What’s the dimension of the space homogeneous harmonic polynomials of degree on that are invariant under the even part of

**Answer.** Any Coxeter group has a list of numbers associated to it, and the answer to the puzzle is how many ways you can write as an unordered sum of s for and at most one copy of

These lists of numbers associated to Coxeter groups are explained in week186 and week187, but Layra says what these lists are. For the symmetry group of the icosahedron the list is

2, 6, 10

so the answer to the puzzle is: the number of ways we can write as a sum of 6s, 10s and at most one 15. Which is what we’d already seen!

But the list for is

2, 8, 12, 14, 18, 20, 24, 30

so we can now do this case as well. The dimension of the space of degree- harmonic polynomials on that are invariant under the even part of the Weyl group of is the number of ways you can write as an unordered sum of 2s, 8s, 12s, 14s, 18s, 20s, 24s, 30s and at most one 120.

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