Knowledge Bank

In molecular spectroscopy problems, the use of symmetrized wavefunctions, i.e. basis vectors transforming according to irreducible representations of the molecular point group, helps to classify the quantum states according to their symmetries. This is very useful when considering the selection rules for transitions and it also facilitates the calculations of eigenvalues. Thus, the symmetrization can be used as a powerful tool in the analysis of high-resolution molecular spectra. While this process is easy to perform in the case of symmetric- and asymmetric-top molecules thanks to projection methods, it is much harder in the case of spherical-tops (tetrahedral or octahedral molecules). After briefly reviewing the general principles of this topic, we propose a detailed method for the symmetrization of the standard basis $|j,m⟩$ into the $Oh$ or $Td$ point $groupa$. This is realized by means of an orientation matrix called G determined following the ideas developed formerly by Moret-$Baillyb$. It allows matrix element calculations for rovibronic spectroscopic problems concerning octahedral or tetrahedral molecules. A numerical calculation of all the G matrix elements for both integer and half-integer j values up to 399/2 has been performed. Such high angular momentum values are necessary for the case of heavy molecules with high rotational excitation. The complete list of these coefficients can be obtained freely at the URL: http://www.u-bourgogne.fr/LPUB/group.html. As an illustration, we also present briefly an application to a case where both high and half angular momentum values are to be considered: the $\nu 3$ band of $ReF6$ (an open-shell molecule with an odd number of electrons and a fourfold degenerate electronic ground state).