Exploring remnants of invariants buried in a deep potential well in chemical reactions

We revisit the concept of "remnant of invariant manifolds" originally discussed by Shirts and Reinhardt in a two degrees of freedom Henon-Heiles system [J. Chem. Phys. 77, 5204 (1982)]. This is regarded as the remnants of a destroyed invariant manifold that can dominate the transport in phase space even at high energy regions where most of all tori vanish. We present a novel technique to extract such remnants of invariants from a sea of chaos in highly nonlinear coupled molecular systems in terms of the canonical perturbation theory based on Lie transforms. As an illustrative example we demonstrate in HCN isomerization reaction that the conventional procedure based on a finite order truncation of the coordinate transformation prevent us from detecting remnants of invariants. However, our technique correctly captures the underlying remnants of invariants that shed light on the energetics of chemical reaction, that is, how the reactive mode acquires (releases) energy from (to) the other vibrational mode in order to overcome the potential barrier (to be trapped in the potential well). We also found the qualitative difference between the two potential wells, HCN and CNH, which coincides with the nearest neighbor level spacing distribution of the vibrational quantum states within the wells.