Equivalent rotations associated with the permutation inversion group revisited: symmetry projection of the rovibrational functions of methane

Abstract

In this work the analysis of the equivalent rotations from the permutation inversion group formalism is revisited. We emphasize that explicit knowledge of changes in the Euler angles are not required in order to determine the transformation that a given symmetry operation causes to the rotational functions when dealing with the permutation inversion group formalism. Indeed, matrix elements of the equivalent rotations are provided by a single Wigner's D ^(j)(R) function. Taking advantage of this, we propose a symmetry projection approach to build the rovibrational functions of methane. This approach focuses on the relevance of the isomorphism between permutations and equivalent rotations. In our method, symmetry adapted functions are obtained by simultaneous diagonalization of a set of commuting operators, whose representation is given in terms of direct products of Wigner's D functions and vibrational matrix representations provided by a local scheme. The proposed approach is general and permits us to obtain in a systematic fashion an orthonormal set of symmetry-projected functions, with good total angular momentum, and carrying the irreducible representations of the molecular symmetry group.

Keywords:

• permutation-inversion group
• equivalent rotations
• ro-vibrational functions
• symmetry projection
• methane

Acknowledgements

This work has partially been supported by CONACyT, México. This work has also been partially supported by the Spanish MEC and by European regional development funds (FEDER) under project CPAN-Ingenio (CSD2007-00042), and by the Junta de Andalucía (projects P07-FQM-02962 and P07-FQM-03014).