Electronic Structure Reading Group: Clebsch-Gordan Coefficients: Making Sense of Rotations in ML and Beyond

When trying to extract the rotational behavior of composite systems, the result is often a reducible mess - but Clebsch-Gordan coefficients are the elegant tools we use to restore order. They tell us how to decompose tensor products into irreducible representations and how to keep track of what’s actually transforming nicely under rotations.

In the seminar, we’ll take a practical, example-driven tour through the world of Clebsch-Gordan coefficients and their role in making sense of transformations, particularly in Machine Learning (ML), where we often target quantities with well-defined rotational behavior.

We’ll follow a hands-on approach to rotations and their decomposition into irreducible components, all while building intuition for how this machinery really works, and what it means. With just enough math to keep everything honest.

While one goal is to understand why this formalism is so widespread in the ML community, the broader aim is to equip you with tools that are just as useful in other fields where angular momentum decomposition plays a key role - from quantum field theory to spin-½ representations of electrons, to the Dirac belt trick (sometimes good for parties).

My favorite references that use the Clebsch-Gordan contractions in ML:
Drautz, Atomic cluster expansion for accurate and transferable interatomic potentials, Phys. Rev. B 100, 249901 (2019).
Batatia et al., MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields, NeurIPS (2022).
Nigam et al., Recursive evaluation and iterative contraction of N-body equivariant features, J. Chem. Phys. 153, 121101 (2020).
Bartók et al., On representing chemical environments, Phys. Rev. B 87, (2017).
Batzner et al., E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials, Nat. Commun. 13, 2453 (2022).

My favourite reference book for this:
Varshalovich, et. al. Quantum Theory of Angular Momentum (1988).

This event is jointly organized with the COSMO seminar.
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The electronic structure reading group brings together researchers and students interested in mathematical aspects of electronic structure problems and adjacent topics, including:

• Density Functional Theory
• Many-body Schrödinger equation for electrons
• Born-Oppenheimer Molecular Dynamics
• Numerical analysis and error control