English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The construction of integrity bases for invariant and covariant polynomialsbuilt 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 thatdealt with a set of two dimensional vectorsunder the action of the SO(2) andO(2) groups [G. Dhont and B. I. Zhilinski{\i}, J. Phys. A:Math. Theor., 46,455202 (2013)]. The expressions of the Molien generating functions asonerational function are a useful guide to build integrity bases for the rings ofinvariantsand the free modules of covariants. The structure of the non--freemodules of covariants ismore complex. In this case, we write the Moliengenerating function as a sum of rationalfunctions and show that its symbolicinterpretation leads to the concept of generalizedintegrity basis. Theintegrity bases and generalized integrity bases for O(3) are deducedfrom theSO(3) ones. The results are useful in quantum chemistry to describe thepotentialenergy or multipole moment hypersurfaces of molecules. In particular,the generalizedintegrity bases that are required for the description of theelectric and magnetic quadrupolemoment hypersurfaces of tetratomic moleculesare given for the first time.

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