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Mapping an atomistic configuration to an $N$-point correlation of a field associated with the atomic positions (e.g.
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O ( 3 ) {O(3)} equivariants can be obtained symmetrizing over both R ^ \hat{R} and i ^ \hat{i} , a field built as the tensor product of ν \nu atom-centred densities, a spherical harmonic, and a parity field | σ ⟩ \ket{\sigma} that is invariant under rotations and behaves as a scalar/tensor ( σ = 1 \sigma=1 ) or a pseudoscalar/pseudotensor ( σ = − 1 \sigma=-1 ) under inversion
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