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

Abstract

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. @S1050-2947~98!07211-4# PACS number~s!: 34.50.2s, 31.15.2p, 02.40.2k This paper is the second in a series concerning body frames and their singularities in the quantum dynamics of n-particle systems. The first of these papers @1#, concerning body frames in the three-body problem, is necessary background for the present paper, which focuses on the four-body problem. The meaning of frame singularities, their effect on internal wave functions, their topological inevitability, and the latitude one has in moving them around in the internal space are all exemplified by the three-body case and are discussed in Ref. @1#. Many of the facts presented in Ref. @1# on body frames in the three-body problem are familiar to practitioners in the field, although our geometrical perspective is almost completely different. In the four-body problem, how