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

Guillaume Dhont (1) , Patrick Cassam-Chenaï (2) , Frédéric Patras (2)

1 LPCA - Laboratoire de Physico-Chimie de l'Atmosphère
2 JAD - Laboratoire Jean Alexandre Dieudonné

The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the SO(3) and O(3) symmetries is presented. This paper is a follow–up to our previous work that dealt with a set of two dimensional vectors under the action of the SO(2) and O(2) groups [G. Dhont and B. I. Zhilinskiı́, J. Phys. A: Math. Theor., 46, 455202 (2013)]. The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non–free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for O(3) are deduced from the SO(3) ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time.

Mots clés: orthogonal group; invariant theory; integrity basis; free module; non-free module; dipole moment; quadrupole moment

Domaines: Algèbre commutative [math.AC]; Physique mathématique [math-ph]

Guillaume Dhont, Patrick Cassam-Chenaï, Frédéric Patras. Molien generating functions and integrity bases for the action of the SO(3) and O(3) groups on a set of vectors. Journal of Mathematical Chemistry, 2021, 59, pp.2294-2326. ⟨hal-01222972v5⟩