Irreducible Cartesian tensor decomposition: A computational approach

Cartesian tensors are widely used in physics and chemistry, especially for the formulation of linear and nonlinear spectroscopies as well as for molecular response properties. In this work, we review the problem of irreducible Cartesian tensor (ICT) decomposition of a generic Cartesian tensor of rank n into its irreducible parts, each characterized by a specific symmetry. The matrix formulation of the ICT decomposition is structurally similar to the problem of rotational averaging using isotropic Cartesian tensors. Analogously to the latter, the ICT decomposition can be considered as a problem of selecting a set of permutations of n indices that provides a linearly independent set of mappings between Cartesian tensor subspaces. This selection can be performed using a simple computational approach based on the reduced row echelon form (rref) algorithm. This protocol has been implemented in a computer code used to re-derive the already known ICT decomposition for 2 ≤ n ≤ 4. Finally, for the first time, we performed the explicit ICT decomposition of a Cartesian tensor of rank n = 5.