Multi-valued Versus Single-Valued Large-Amplitude Bending-Torsional-Rotational Coordinate Systems for Simultaneously Treating Trans-Bent and Cis-Bent Acetylene in its S1 State

Published

July 28, 2012

Author(s)

Jon T. Hougen

Abstract

There are now a large number of papers in the spectroscopic literature which make use of multiple-valued (frequently double-valued) coordinate systems and the associated multiple-groups of the permutation-inversion group to deal with the symmetry properties of large-amplitude motions in molecules of high symmetry. The use of multiple-valued coordinate systems, and the resultant appearance of more minima on the potential surface than would be found on the surface for a single-valued coordinate system, can lead to conceptual discomfort and questions of mathematical legitimacy. In the present paper we demonstrate that treatments using multiple-valued coordinate system simply represent one scheme for applying the appropriate quantum mechanical boundary conditions to Schrödinger’s partial differential equation defined in a single-valued coordinate system. The demonstration is not general, but rather focusses on the specific example of a non-linear electronic state of C2H2 and on the two-fold and eight-fold extended permutation-inversion groups recently introduced to simultaneously treat symmetry questions in trans-bent and cis-bent acetylene. Some discussion of the mathematical convenience lost by insisting on using a single-valued coordinate system is also presented.

Citation

Journal of Molecular Spectroscopy

Pub Type

Journals

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Keywords

boundary conditions, cis and trans acetylene, multiple-groups, multiple-valued coordinate systems, permutation-inversion groups, symmetry properties

Citation

Hougen, J. (2012), Multi-valued Versus Single-Valued Large-Amplitude Bending-Torsional-Rotational Coordinate Systems for Simultaneously Treating Trans-Bent and Cis-Bent Acetylene in its S1 State, Journal of Molecular Spectroscopy, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=910766 (Accessed October 15, 2025)