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