Complete and Efficient Covariants for Three-Dimensional Point Configurations with Application to Learning Molecular Quantum Properties

When physical properties of molecules are being modeled with machine learning, it is desirable to incorporate <i>SO</i>(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 6<i>k</i> - 5 of these features are enough for up to <i>k</i> 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 <i>O</i>(<i>l</i>⁶) to <i>O</i>(<i>l</i>³) in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving three-dimensional point configurations.