# Transformation Properties under the Operations of the Molecular Symmetry Groups $G_{36}$ and $G_{36}\text{(EM)}$ of Ethane $\text{H}_3\text{CCH}_3$

## TL;DR

This paper details the symmetry properties of ethane using the groups G_{36} and G_{36}(EM), facilitating variational calculations of its rotation-vibration spectra through derived transformation matrices and symmetry-adapted basis functions.

## Contribution

It introduces a comprehensive derivation of transformation matrices for ethane's symmetry groups and algorithms for their numerical construction, enhancing spectral calculations for non-rigid molecules.

## Key findings

- Derived transformation matrices for G_{36} and G_{36}(EM)
- Implemented symmetry adaptation in the TROVE program
- Applied methodology to potential energy and basis functions

## 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 $\text{CH}_3$ where the internal rotation (torsion) of one $\text{CH}_3$ 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 $G_{36}\text{(EM)}$. 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 $G_{36}\text{(EM)}$ 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 $G_{36}\text{(EM)}$ 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 $G_{36}\text{(M)}$ ($G_{36}\text{(EM)}$) 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.