The irreducible tensor approach in the separation of collective angles in the quantum N-body problem

The problem of the separation of collective rotational degrees of freedom in a quantum N-body problem is considered from the viewpoint of the irreducible tensor theory. The vector differentiation technique is used as a universal tool for the derivation of equations for internal wave functions. As a tensor basis for the elimination of collective angles both Wigner D-functions and minimal bipolar harmonics (inspired by the problem of the helium atom) are considered. For both these choices consideration of the collective angular motion necessarily leads to singular terms in the Schrödinger equation (“gauge singularities”). The general question of the possibility of elimination of such singularities is answered positively by describing procedures for the separation of rotations based on the use of an overcomplete basis set of minimal (N−1)-polar harmonics or on a set of translated spherical harmonics. As a result, the internal motion of the system is described by a set of coupled equations for the radial wave functions dependent on (3N−6) internal variables (“shape coordinates”). At collinear configurations the dimension of the system reduces, being equal to unity for the configuration when all particles lie on a straight line, thus demonstrating the physical importance of collinear configurations. Generalized radial equations for the P- and D- states of an N-body system are written explicitly and, in the particular case of three- and four-body systems, choices of internal coordinates are discussed.