Journal of Mathematical Chemistry

, Volume 53, Issue 1, pp 58–85 | Cite as

A fast algorithm for the construction of integrity bases associated to symmetry-adapted polynomial representations: application to tetrahedral \(\mathrm {XY_4}\) molecules

  • Patrick Cassam-ChenaïEmail author
  • Guillaume Dhont
  • Frédéric Patras
Original Paper


Invariant theory provides more efficient tools, such as Molien generating functions and integrity bases, than basic group theory, that relies on projector techniques, for the construction of symmetry-adapted polynomials in the symmetry coordinates of a molecular system, because it is based on a finer description of the mathematical structure of the latter polynomials. The present article extends its use to the construction of polynomial bases which span possibly, non-totally symmetric irreducible representations of a molecular symmetry group. Electric or magnetic observables can carry such irreducible representations, a common example is given by the electric dipole moment surface. The elementary generating functions and their corresponding integrity bases, where both the initial and the final representations are irreducible, are the building blocks of the algorithm presented in this article, which is faster than algorithms based on projection operators only. The generating functions for the full initial representation of interest are built recursively from the elementary generating functions. Integrity bases which can be used to generate in the most economical way symmetry-adapted polynomial bases are constructed alongside in the same fashion. The method is illustrated in detail on \(\mathrm {XY_4}\) type of molecules. Explicit integrity bases for all five possible final irreducible representations of the tetrahedral group have been calculated and are given in the supplemental material associated with this paper.


Integrity basis Covariant Tetrahedral group Methane Molien series 



Financial support for the project Application de la Théorie des Invariants à la Physique Moléculaire via a CNRS grant Projet Exploratoire Premier Soutien (PEPS) Physique Théorique et Interfaces (PTI) is acknowledged. The first and third authors also acknowledge support from the grant CARMA ANR-12-BS01-0017.

Supplementary material

10910_2014_410_MOESM1_ESM.txt (694 kb)
ESM 1 (txt 694 kb)


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Patrick Cassam-Chenaï
    • 1
    Email author
  • Guillaume Dhont
    • 2
  • Frédéric Patras
    • 1
  1. 1.CNRS, LJAD, UMR 7351University of Nice Sophia AntipolisNiceFrance
  2. 2.Université du Littoral Côte d’Opale, Laboratoire de Physico-Chimie de l’AtmosphèreDunkerqueFrance

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