  Machine learning interatomic potentials are revolutionizing large-scale,
accurate atomistic modelling in material science and chemistry. Many potentials
use atomic cluster expansion or equivariant message passing frameworks. Such
frameworks typically use spherical harmonics as angular basis functions, and
then use Clebsch-Gordan contraction to maintain rotational symmetry, which may
introduce redundancies in representations and computational overhead. We
propose an alternative: a Cartesian-coordinates-based atomic density expansion.
This approach provides a complete set of polynormially indepedent features of
atomic environments while maintaining interaction body orders. Additionally, we
integrate low-dimensional embeddings of various chemical elements and
inter-atomic message passing. The resulting potential, named Cartesian Atomic
Cluster Expansion (CACE), exhibits good accuracy, stability, and
generalizability. We validate its performance in diverse systems, including
bulk water, small molecules, and 25-element high-entropy alloys.
