Integrity bases for covariants of tetrahedral XY4 molecules. Application to the electric dipole moment surface
Abstract
Techniques of invariant theory such as Molien generating functions and integrity bases offer mathematical tools for the efficient construction of symmetry--adapted polynomials in the symmetrized coordinates of a molecular system. The present article is the prolongation of our previous work [P.~Cassam-Chenaï and F.~Patras, J. Math. Chem., 44(4), 938--966 (2008).] to the case of polynomials that transform as a non--totally symmetric irreducible representation of the symmetry group $G$ of the molecule. Such a covariant representation occurs with electric or magnetic properties, for example with the electric dipole moment surface. The symmetrized coordinates span an initial reducible representation from which polynomials transforming as an irreducible representation are built. The number of linearly independent polynomials of degree $k$ within this final representation is given by the coefficient of degree $k$ in the Taylor expansion of the associated Molien function. This generating function is built from combination of elementary generating functions where both the initial and final representations are irreducible. In parallel, Clebsch--Gordan coefficients of the symmetry group $G$ recursively couples the corresponding elementary integrity bases in order to build the integrity bases for the initial representation associated to symmetrized coordinates. The method is illustrated in detail on XY4 type of molecules for which the explicit integrity bases for the five final irreducible representations are given.
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covariant1.pdf (201.68 Ko)
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files-description.pdf (124.68 Ko)
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symmetry_A1.txt (50.69 Ko)
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symmetry_A2.txt (50.32 Ko)
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symmetry_E.txt (137.91 Ko)
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symmetry_F1.txt (225.71 Ko)
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symmetry_F2.txt (225.19 Ko)
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