Internal spaces, kinematic rotations, and body frames for four-atom systems

Four-atom systems may soon be subject to state-to-state reactive scattering calculations and understanding body frames and their singularities will be an important part of this effort. This paper examines body frames in four-atom systems, building on a geometrical analysis of the nine-dimensional configuration space and the six-dimensional internal space. Kinematic rotations are an important tool in this analysis. A central role is played by the ``kinetic cube,'' the space of all asymmetric top shapes related by kinematic rotations. The singularities, multiple branches, and connectivity of the principal axis frame are examined in detail and related to the topology of the kinetic cube. The principal axis frame has singularities on all symmetric top shapes, both oblate and prolate, of both chiralities. A version of the Eckart frame, however, has singularities only on prolate symmetric top shapes of one chirality. Frame singularities are inevitable in the four-body problem and no other frame has a smaller singular set than the Eckart frame.