Localized chaos and partial assignability of dynamical constants of motion in the transition to molecular chaos

The implications of approximate dynamical constants of motion for statistical analysis of highly excited vibrational spectra are investigated. The existence of approximate dynamical constants is related to localized chaos and partial assignability of a "chaotic spectrum." Approximate dynamical constants are discussed in a dynamical symmetry breaking formulation of the transition from periodic to quasiperiodic motion, and from quasiperiodic to chaotic motion. Level repulsion, leading to a Wigner distribution in the case of a strongly chaotic system, is shown to originate in dynamical symmetry breaking via the noncrossing rule that states of the same symmetry do not cross. It is argued that quantum numbers for dynamical constants must be correctly assigned to detect localized chaos in statistical spectroscopy. Two possible kinds of approximate constants, for a "total polyad number" and a bend normal mode, are discussed in relation to two coupling schemes that could govern the transition to chaos in H2 O.