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 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 tunneling 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 symmetrized 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 will be 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.

A numerical-tensorial “hybrid” nuclear motion Hamiltonian and dipole moment operator for spectra calculation of polyatomic nonrigid molecules

“…Indeed, many large groups can be expressed as a group product, making easier the determination of the symmetry species and generators. For example, for ethanelike molecules we have the direct product structure 72…”

“…With the spectacular advances in modern experimental techniques, even small splittings can be now resolved, making necessary the development of sophisticated theoretical models to analyse spectra. Though it does not give any information about the magnitude of the splittings, group theory plays a central role [66][67][68][69][70][71][72] and is inseparable from the theory of the nonrigid molecules.…”

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

“…The introduced structure is similar to that of Extended molecular symmetry groups G(EM), which are used in the ro-vibrational problems of non-rigid molecules, such as hydrogen peroxide (H 2 O 2 ) [34] and ethane (C 2 H 6 ) [53].…”

“…For example, the rotational or vibrational wavefunctions of G 36 (EM), the MS group of ethane C 2 H 6 , can transform as odd or even under E , labelled as s and d, respectively. Likewise, the full rotation-vibrational wavefunction can only be of s type and the artificial d types must be eliminated [53].…”