Complete and Efficient Covariants for 3D Point Configurations with Application to Learning Molecular Quantum Properties

When modeling physical properties of molecules with machine learning, it is desirable to incorporate SO(3)-covariance. While such models based on low body order features are not complete, we formulate and prove general completeness properties for higher order methods, and show that 6k-5 of these features are enough for up to k atoms. We also find that the Clebsch--Gordan operations commonly used in these methods can be replaced by matrix multiplications without sacrificing completeness, lowering the scaling from O(l6) to O(l3) in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving 3D point configurations.