Quantum theory of rotational isomerism and Hill equation

Abstract

The process of rotational isomerism of linear triatomic molecules is described by the potential with two different-depth minima and one barrier between them. The corresponding quantum-mechanical equation is represented in the form that is a special case of the Hill equation. It is shown that the Hill-Schrödinger equation has a Klein's quadratic group symmetry which, in its turn, contains three invariant subgroups. The presence of these subgroups makes it possible to create a picture of energy spectrum which depends on a parameter and has many merging and branch points. The parameter-dependent energy spectrum of the Hill-Schrödinger equation, like Mathieu-characteristics, contains branch points from the left and from the right of the demarcation line. However, compared to the Mathieu-characteristics, in the Hill-Schrödinger equation spectrum the “right” points are moved away even further for some distance that is the bigger, the bigger is the less deep well. The asymptotic wave functions of the Hill-Schrödinger equation for the energy values near the potential minimum contain two isolated sharp peaks indicating a possibility of the presence of two stable isomers. At high energy values near the potential maximum, the height of two peaks decreases, and between them there appear chaotic oscillations. This form of the wave functions corresponds to the process of isomerization.