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

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 confusion and questions of mathematical legitimacy. In the present paper we demonstrate that treatments using multiple-valued coordinate systems 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 focuses on the specific example of the S1 electronic state of C2H2, which has local minima only for nonlinear configurations, and on the twofold and eightfold 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 using a single-valued coordinate system is also presented. [Copyright &y& Elsevier]