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In the present work, we report a detailed description of the symmetry properties of the eight-atomic molecule ethane, with the aim of facilitating the variational calculations of rotation-vibration spectra of ethane and related molecules. Ethane consists of two methyl groups CH 3 where the internal rotation (torsion) of one CH 3 group relative to the other is of large amplitude and involves tunnelling between multiple minima of the potential energy function. The molecular symmetry group of ethane is the 36-element group G 36 , but the construction of symmetrised basis functions is most conveniently done in terms of the 72-element extended molecular symmetry group G 36 (EM). This group can subsequently be used in the construction of block-diagonal matrix representations of the ro-vibrational Hamiltonian for ethane. The derived transformation matrices associated with G 36 (EM) have been implemented in the variational nuclear motion program TROVE (Theoretical ROVibrational Energies). TROVE variational calculations are used as a practical example of a G 36 (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 G 36 (M) (G 36 (EM)) and also describe algorithms for a numerical construction of these matrices based on a set of four (five) generators. The methodology presented is illustrated on the construction of the symmetry-adapted representations both of the potential energy function of ethane and of the rotation, torsion and vibration basis set functions.

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