Reusable Operators for Irreducible Cartesian Tensor Decomposition and Coupling

Abstract

Molecular and material properties, from the polarizability to the elastic constants, are described by tensors. Their behavior under rotations is made explicit when a tensor is decomposed into irreducible parts that transform independently. In Cartesian form these parts are the symmetric and traceless irreducible Cartesian tensors (ICTs), whose decomposition and coupling underlie selection rules, orientational averages, and the use of symmetry. The decomposition is a textbook result at rank two, but higher rank and the intrinsic symmetry of physical tensors make it nontrivial. What has been lacking, unlike in the well-established spherical formalism, is a general construction for a given intrinsic symmetry, together with reusable operators that extract the ICTs and rebuild the original tensor exactly. Here, we develop such a construction and obtain these operators explicitly. These operators depend on rank and symmetry alone, and therefore each need only be built once and then applied to any tensor of that class. Building on them, we further obtain the Cartesian harmonics of a vector and the operators that couple two ICTs into a third, both central to equivariant machine learning. The construction is demonstrated on the elastic tensor, in both its second-order form of rank four and its third-order form of rank six. The ICTs of the rank-4 tensor also define a rotation- and scale-invariant measure of anisotropy, which we evaluate across the first-principles elastic tensors of crystalline materials from the Materials Project. The construction is implemented in the open-source package natto, which produces the operators in both exact symbolic and numerical form.