The symmetric spherical oscillator, and the rotational motion of homopolar molecules in crystals

Abstract

In a recent paper, L. Pauling has discussed the motion of molecules in crystals. By the use of approximate methods, criteria are deduced in that paper for determining when the motion of molecules in crystals is “rotational,” and when it is “oscillational” about positions of equilibrium. While it served Pauling’s purpose, the investigation did not go deep enough to satisfy the needs of the present author, who desired information about the statistical weights of the lowest quantum states of molecules in crystals, more certain and definite than that provided by Pauling’s analysis. The present more rigorous investigation was accordingly undertaken. The results go somewhat beyond the author’s needs, but they are perhaps interesting in themselves. The investigation involves a problem in quantum mechanics which is capable of exact treatment, without the use of perturbation methods. We imagine a homopolar molecule of type X 2 free to rotate in a field of force of axial symmetry. Symmetry considerations show that if a certain orientation is one of equilibrium, then the orientation obtained by reversing the molecule end for end will also be one of equilibrium; and we see also, if the orientation of the molecule is specified by the co-ordinates θ, ϕ (the axis of θ being the axis of equilibrium), that the potential function V(θ) of the homopolar molecule must be symmetric about the equatorial circle θ = π/2. We suppose that our molecule is tree from axial spin. If V(θ) can be expanded in a Fourier series, then it can be expanded in a Fourier series made up of cosines of even multiples of θ. A good first approximation will be obtained by taking only the first two terms of this series; higher terms can be taken account of later, if desired, by a perturbation method. We therefore consider, with Pauling, the potential function (1) V = V 0 (1 - cos 2θ).