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

Autor: Frédéric Patras, Patrick Cassam-Chenaï, Guillaume Dhont
Rok vydání: 2021
Předmět:
Zdroj: Journal of Mathematical Chemistry. 59:2294-2326
ISSN: 1572-8897
0259-9791
Popis: The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the $${{\mathrm {SO(3)}}}$$ and $${{\mathrm {O(3)}}}$$ 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 $${{\mathrm {SO(2)}}}$$ and $${{\mathrm {O(2)}}}$$ groups (Dhont and Zhilinskii in 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 $${\mathrm {O(3)}}$$ are deduced from the $${\mathrm {SO(3)}}$$ 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.
Databáze: OpenAIRE
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