Transition states near rank-two saddles: Correlated electron dynamics of helium

The phase space structure in the vicinity of a rank-two saddle is explored both analytically and numerically. In particular, the geometry of electron dynamics in the neighborhood of the rank-two saddle associated with the nonsequentual double ionization of helium is analyzed. As in the rank-one saddle case, codimension-one normally hyperbolic invariant manifolds turn out to control the rates of the correlated electron dynamics within each energy surface. The construction of these manifolds, however, is more involved here and requires the use of pseudo-hyperbolic invariant manifold theory. Two distinct correlated motions occur, the nonsequential double ionization and the nonsequential exchange of electrons. The relative rates of crossing the barrier are related to the Lyapunov exponents. The dynamics associated with the larger of the two Lyapunov exponents predominates.