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Invariant theory provides more efficient tools, such as Molien generatingfunctions and integrity bases, than basic group theory, that relies onprojector techniques for the construction of symmetry--adapted polynomials inthe symmetry coordinates of a molecular system, because it is based on a finerdescription of the mathematical structure of the latter. The present articleextends its use to the construction of polynomial bases which span possibly,non--totally symmetric irreducible representations of a molecular symmetrygroup. Electric or magnetic observables can carry such irreduciblerepresentations, a common example is given by the electric dipole momentsurface. The elementary generating functions and their corresponding integritybases, where both the initial and the final representations are irreducible,are the building blocks of the algorithm presented in this article, which isfaster than algorithms based on projection operators only. The generatingfunctions for the full initial representation of interest are built recursivelyfrom the elementary generating functions. Integrity bases which can be used togenerate in the most economical way symmetry--adapted polynomial bases areconstructed alongside in the same fashion. The method is illustrated in detailon XY4 type of molecules. Explicit integrity bases for all five possible finalirreducible representations of the tetrahedral group have been calculated andare given in the supplemental material associated with this paper.

A Fast Algorithm for the Construction of Integrity Bases Associated to  Symmetry-Adapted Polynomial Representations. Application to Tetrahedral XY4  Molecules