Representing spherical tensors with scalar-based machine-learning models

TL;DR

The paper tackles the challenge of encoding rotational equivariance for 3D observables by generalizing a scalar-universality result to spherical tensors, showing that irreducible tensor components can be built from maximal Clebsch–Gordan contractions of input vectors. It then proposes a practical lambda-MCoV architecture that uses three learnable vectors to form a maximal-coupled basis and a SOAP–BPNN branch to model the scalar coefficients, achieving efficient and scalable equivariant representations. The approach is demonstrated on multiple targets (dipole, polarizability, hyperpolarizability) and spectroscopic applications (water IR spectrum), with a CO2 degeneracy test highlighting the advantage over vector-only models. The work suggests that scalar-function learning, combined with maximally coupled tensor bases, offers a computationally efficient path for accurate symmetry-consistent quantum observables in atomistic modeling and related domains.

Abstract

Rotational symmetry plays a central role in physics, providing an elegant framework to describe how the properties of 3D objects -- from atoms to the macroscopic scale -- transform under the action of rigid rotations. Equivariant models of 3D point clouds are able to approximate structure-property relations in a way that is fully consistent with the structure of the rotation group, by combining intermediate representations that are themselves spherical tensors. The symmetry constraints however make this approach computationally demanding and cumbersome to implement, which motivates increasingly popular unconstrained architectures that learn approximate symmetries as part of the training process. In this work, we explore a third route to tackle this learning problem, where equivariant functions are expressed as the product of a scalar function of the point cloud coordinates and a small basis of tensors with the appropriate symmetry. We also propose approximations of the general expressions that, while lacking universal approximation properties, are fast, simple to implement, and accurate in practical settings.

Figures (9)

• Figure 1: Graphical representation of Eq. \ref{['eq:fundamental_identity']}, with vectors $\bm a = (2, -3, -1)$, $\bm b = (-2, 1, -3)$, $\bm c = (1, 2, 0)$, $\bm d = (-1, 1, 0.5)$. The polar plots are obtained by evaluating the function $f_{\bm T}(\bm{\hat{r}}) := \bm{\hat{r}}^T \bm T \bm{\hat{r}}$, where $\bm{\hat{r}}\in S^2$ is a versor of the unitary sphere, and $\bm T$ is the Cartesian representation of the $\lambda = 2$ tensors appearing in the equation. The right-hand side is magnified by a factor $\sqrt{2}$ for visual purposes.
• Figure 2: Graphical representation of the result from Prop. \ref{['prop1']}. A spherical tensor (on the left) of order $\lambda$, can be always encoded into a Cartesian tensor of rank at least $\lambda$. When the rank and the angular momentum coincide, we have the smallest Cartesian representation that can accommodate for the tensor, realized by the maximal CG coupling. In the figure, a tensor of $\lambda=2$ is encoded in the symmetric and traceless components of a Cartesian tensor of the same rank. Applying the same maximal CG coupling to the vectors in the input space (on the right) provides a representation of the same rank. In the figure, the input generates the xy-plane, and its symmetric and traceless representation of rank $2$ is depicted. The main result (converging to the center) is that, to ensure the equivariance condition of Eq. \ref{['eq:equivariance']}, we can consider only components of the tensor in the same space generated by the input, while the other must vanish.
• Figure 3: From left to right: let us consider a Cartesian tensor of arbitrary rank which is equivariant with respect to some input vectors $\{{\bm r}_i\}$. Such tensor admits a decomposition in its harmonic components by means of CG contractions: the result is a collection of irreducible representations with respect to the action of elements of the group $O(3)$. In particular, the notation $l_{\pm}$ indicates the angular momentum (dimensionality) of the representation and its behavior under inversion, proper in the case $l_+$ and pseudo for $l_-$, as per Eq. \ref{['eq:parity']}. Then, we can apply the results of \ref{['sec:methods']} on each of the irreducible components: in particular, the main idea behind the theoretical results is to encode each of these representations into the smallest Cartesian tensor that can contain them, and this leads directly to the representation in terms of the maximal coupling. For example, in the figure it is shown how a pseudovector $1_-$ can be encoded into the antisymmetric part of rank 2 Cartesian tensor, while a proper $2_+$ tensor can be encoded in the symmetric and traceless part. Since the CG contractions from the original tensors and its spherical components is invertible, after applying the results of \ref{['sec:methods']}, one can go back to a representation of the full Cartesian tensor in terms of maximal couplings only.
• Figure 4: a). An example of case for which it is not possible to uniquely associate a frame of reference to the point cloud due to the high symmetry and the permutational invariance. The point cloud is described from the center of mass of the tetrahedron, and any rotation around one of the symmetry axis will leave the points cloud unchanged while rotating any frame associated with it: to guarantee the uniqueness of the map, all the components of the frame in the perpendicular direction to the axis of rotation must vanish. Applying the same rationale to any of the symmetry axis we have that the only map compatible with the symmetry operations is the trivial one, with only null vectors. b). Polar plot representing the spherical expansion of Eq. \ref{['eq:spherical_expansion']} for $l=1$ as the point at the top of the tetrahedron traces a circle in the $xz$-plane. The $0^\circ$ position represent the fully symmetric configuration: all the components of the spherical expansion go to zero continuously (the red dot, intercepting the dashed black line) in the fully tetrahedral configuration, in accordance to a). The very same spherical expansion for $l=1$ is found also in the linear configuration, with the spherical expansion vanishing in the aligned positions. The linear case will be investigated in a real scenario for molecules of CO2 in \ref{['sec:results_CO2']}.
• Figure 5: The architecture of the $\lambda$-MCoV. From left to right: the input is the local environment centered on one atom (see right-hand side of Eq. \ref{['eq:local_atomic_contributions']}). The relative atomic positions are then used to evaluate the spherical expansion (Eq. \ref{['eq:spherical_expansion']}). The spherical expansions for $l=1$ are then fed into a linear layer that produces the three vectors $\{\bm q_\alpha\}_{\alpha=1}^3$ which are then contracted with maximal coupling according to Eqs. \ref{['eq:Q_lambda1']}-\ref{['eq:Q_lambda2']}. This produces a matrix $(2\lambda+1)$ tensors, each in $\mathbb{R}^{2\lambda+1}$. The spherical expansion for $l=\lambda$ are fed into a linear layer producing the same number of tensors, according to Eq. \ref{['eq:Q_corr']}. Finally, the scalar functions are produced by powerspectrum-SOAP contraction, according to Eq. \ref{['eq:powerspectrum']}, and then fed into a Behler-Parrinello Neural Network (BPNN) architecture behl-parr07prl which produces $2(2\lambda+1)$ scalars. These are then contracted with the matrices resulting from the other two paths, resulting in the $2\lambda+1$ harmonic components of the tensor. The weights of the BPNN and $W_{\alpha, zn}$, $W^{(\text{corr})}_{\beta, zn}$ are shared among central atoms of the same species.

Theorems & Definitions (6)

• Proposition 1
• Corollary 2
• Proposition 3
• Proposition 4
• Corollary 5
• Proposition 6