Barriers to Linearity in Triatomic Molecules. I. Development of a Curvilinear Quantum Mechanical Model Appropriate for Large Amplitude Bending Motion in Triatomic Molecules of C2ν Symmetry

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Abstract
A classical kinetic energy expression is developed for a nonlinear triatomic molecule of C2ν symmetry with two degrees of freedom. These are changes in the valence angle plus K‐type rotation about the molecular z axis whose moment vanishes as the molecule becomes linear. The bonds are considered rigid, and the z axis is assumed fixed in space, but no other assumptions are made; in particular, neither small amplitude vibrations nor a nearly linear equilibrium configuration is assumed. The quantum mechanical operators are obtained and vibration‐rotation eigenvalues are obtained for two cases: (a) rectilinear motion applicable only to small deviations from a nearly linear equilibrium geometry and (b) curvilinear motion applicable to any bending amplitude or any equilibrium geometry. The former case follows as a special condition of the latter case. For each kinematic model the wave equation is solved using potential functions identical in form, and the results illustrate the fallacy of placing the entire burden of unevenly spaced vibrational levels upon the potential energy expression. Furthermore, it has been demonstrated that eigenvalues can be obtained from a five‐term recursion relation which arises in the solution for case (b).

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