# Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties

arXiv: https://arxiv.org/abs/2109.12083 · HTML: https://mlsanity.com/paper/2109.12083/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 19 references that are in the map.
- First posted on arXiv 24 Sep 2021.

## Built on by (most cited first, 19 of 19)

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- [Physics-inspired Equivariant Descriptors of Non-bonded Interactions](https://mlsanity.com/paper/2308.13208/) (2023)
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- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction](https://mlsanity.com/paper/2403.09560/) (2024)
- [Transferable E(3) equivariant parameterization for Hamiltonian of molecules and solids](https://mlsanity.com/paper/2210.16190/) (2022)
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- [Learning local equivariant representations for quantum operators](https://mlsanity.com/paper/2407.06053/) (2024)
- [Completeness of Atomic Structure Representations](https://mlsanity.com/paper/2302.14770/) (2023)
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- [Equivariant Matrix Function Neural Networks](https://mlsanity.com/paper/2310.10434/) (2023)
- [Universal Machine Learning Kohn-Sham Hamiltonian for Materials](https://mlsanity.com/paper/2402.09251/) (2024)
- [Electronic excited states from physically-constrained machine learning](https://mlsanity.com/paper/2311.00844/) (2023)
- [NeuralSCF: Neural network self-consistent fields for density functional theory](https://mlsanity.com/paper/2406.15873/) (2024)
- [How unconstrained machine-learning models learn physical symmetries](https://mlsanity.com/paper/2603.24638/) (2026)
- [Machine learning Hubbard parameters with equivariant neural networks](https://mlsanity.com/paper/2406.02457/) (2024)
- [Deep Neural Network for Phonon-Assisted Optical Spectra in Semiconductors](https://mlsanity.com/paper/2502.00798/) (2025)
- [MALOQ: Massively Accelerated Learning of Operators for Quantum Transport](https://mlsanity.com/paper/2606.28911/) (2026)

## Abstract (arXiv)

Symmetry considerations are at the core of the major frameworks used to provide an effective mathematical representation of atomic configurations that is then used in machine-learning models to predict the properties associated with each structure. In most cases, the models rely on a description of atom-centered environments, and are suitable to learn atomic properties, or global observables that can be decomposed into atomic contributions. Many quantities that are relevant for quantum mechanical calculations, however -- most notably the single-particle Hamiltonian matrix when written in an atomic-orbital basis -- are not associated with a single center, but with two (or more) atoms in the structure. We discuss a family of structural descriptors that generalize the very successful atom-centered density correlation features to the N-centers case, and show in particular how this construction can be applied to efficiently learn the matrix elements of the (effective) single-particle Hamiltonian written in an atom-centered orbital basis. These N-centers features are fully equivariant -- not only in terms of translations and rotations, but also in terms of permutations of the indices associated with the atoms -- and are suitable to construct symmetry-adapted machine-learning models of new classes of properties of molecules and materials.