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

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 th...

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Bibliographic Details
Published in:arXiv.org 2019-06
Main Authors: Mellor, Thomas M, Yurchenko, Sergey N, Mant, Barry P, Jensen, Per
Format: Article
Language:English
Subjects:
Algorithms
Basis functions
Ethane
Mathematical analysis
Matrix representation
Potential energy
Rotation
Symmetry
Torsion
Transformations (mathematics)
Vibration
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Summary: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.
ISSN:2331-8422