Rotational States of a Tetrahedron in a Cubic Crystal Field

Abstract

The Schrödinger equation is solved for the rotational states of a rigid cube whose center of mass is fixed at a point of symmetry in an external field. This is shown to be equivalent to the equation of motion for a regular tetrahedron in a field with symmetry Oh. The quantum states are classified under the direct product group Ō×O, where Ō is an octahedral group of rotations about body-fixed axes and O is a similar but distinct group of rotations about space-fixed axes. The wavefunctions are obtained by an expansion in a series of symmetry-adapted linear combinations of spherical-top functions. Projection operators are defined which generate the various linear combinations. Closed-form expressions are obtained for the matrix elements of the Hamiltonian and of the projection operators. Upper and lower bounds are computed for representative energy levels. The results demonstrate that using basis sets of practical size, high accuracy can be obtained. This analysis forms the basis of a more general theory of rotation of small molecules and polyatomic ions in crystals.