Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors

TL;DR

This paper develops methods to construct integrity bases for invariant and covariant polynomials under SO(3) and O(3) actions on vectors, with applications in quantum chemistry for molecular property descriptions.

Contribution

It introduces generalized integrity bases for O(3) derived from SO(3) and explores their structure using Molien generating functions, extending previous work on lower-dimensional groups.

Findings

Derived explicit integrity bases for O(3) from SO(3) bases.

Expressed Molien functions as sums of rational functions.

First-time construction of bases for quadrupole moments in tetratomic molecules.

Abstract

The construction of integrity bases for invariant and covariant polynomials built from aset of three dimensional vectors under the SO(3) and O(3) symmetries is presented. Thispaper is a follow--up to our previous work that dealt with a set of two dimensional vectorsunder the action of the SO(2) and O(2) groups [G. Dhont and B. I. Zhilinski{\i}, J. Phys. A:Math. Theor., 46, 455202 (2013)]. The expressions of the Molien generating functions asone rational function are a useful guide to build integrity bases for the rings of invariantsand the free modules of covariants. The structure of the non--free modules of covariants ismore complex. In this case, we write the Molien generating function as a sum of rationalfunctions and show that its symbolic interpretation leads to the concept of generalizedintegrity basis. The integrity bases and generalized integrity bases for O(3) are deducedfrom the SO(3) ones. The results are useful in quantum chemistry to describe the potentialenergy or multipole moment hypersurfaces of molecules. In particular, the generalizedintegrity bases that are required for the description of the electric and magnetic quadrupolemoment hypersurfaces of tetratomic molecules are given for the first time.