Completeness of finite-order atom-centered descriptors

Can atom-centered correlation descriptors distinguish every pair of environments up to rigid congruence? The question includes complete correlations through a fixed neighbor order and descriptors with both finite correlation order and angular cutoff.

Why it matters and what counts as progress

Why it matters: Descriptor collisions limit how much atomistic geometry can be recovered from finitely many correlation orders and angular components.

What counts as progress: Construct noncongruent environments with identical descriptors at higher orders, or prove injectivity under explicit restrictions on environments and cutoffs.

How the problem developed

2007 · Descriptor introduction [1]
Jörg Behler and Michele Parrinello Introduced atom-centred symmetry functions for neural-network potentials, encoding local atomic environments through radial and angular terms.
2020 · Counterexample [2]
Gábor Pozdnyakov and colleagues Showed that widely used three-body atomistic representations, including SOAP-type descriptors, can identify distinct atomic environments; higher-order bispectrum examples were also found.
29 Jul 2026 · AI-assisted construction [3]
Michelangelo Domina and Michele Ceriotti Constructed noncongruent environments indistinguishable by finite-order atom-centred correlations, including examples in three dimensions.

AI contribution

Using large language models to probe the limits of atom-centered structural descriptors

Michelangelo Domina, Michele Ceriotti

Domina and Ceriotti give noncongruent atom-centered environments with identical complete correlations through seven-neighbor order, including three-dimensional lifts. For every finite correlation order and angular cutoff, they also construct indistinguishable noncongruent environments.

AI involvementAI-centered

ImportanceNotable

Technical verificationPartial

Human understandingFull