Transformation Properties under the Operations of the Molecular Symmetry Groups G36 and G36(EM) of Ethane H3CCH3

Abstract

In the present work, we report a detailed description of the symmetry propertiesof the eight-atomic molecule ethane, with the aim of facilitating the variational calculations ofrotation-vibration spectra of ethane and related molecules. Ethane consists of two methyl groupsCH3 where the internal rotation (torsion) of one CH3 group relative to the other is of large amplitudeand involves tunnelling between multiple minima of the potential energy function. The molecularsymmetry group of ethane is the 36-element group G36, but the construction of symmetrised basisfunctions is most conveniently done in terms of the 72-element extended molecular symmetrygroup G36(EM). This group can subsequently be used in the construction of block-diagonal matrixrepresentations of the ro-vibrational Hamiltonian for ethane. The derived transformation matricesassociated with G36(EM) have been implemented in the variational nuclear motion program TROVE(Theoretical ROVibrational Energies). TROVE variational calculations are used as a practical exampleof a G36(EM) symmetry adaptation for large systems with a non-rigid, torsional degree of freedom.We present the derivation of irreducible transformation matrices for all 36 (72) operations of G36(M)(G36(EM)) and also describe algorithms for a numerical construction of these matrices based on aset of four (five) generators. The methodology presented is illustrated on the construction of thesymmetry-adapted representations both of the potential energy function of ethane and of the rotation,torsion and vibration basis set functions.