Phase space geometry of isolated to condensed chemicalreactions

The complexity of gas and condensed phase chemical reactions has generally been uncoveredeither approximately through transition state theories or exactly through (analytic orcomputational) integration of trajectories. These approaches can be improved byrecognizing that the dynamics and associated geometric structures exist in phase space,ensuring that the propagator is symplectic as in velocity-Verlet integrators and byextending the space of dividing surfaces to optimize the rate variationally, respectively.The dividing surface can be analytically or variationally optimized in phase space, notjust over configuration space, to obtain more accurate rates. Thus, a phase spaceperspective is of primary importance in creating a deeper understanding of the geometricstructure of chemical reactions. A key contribution from dynamical systems theory is thegeneralization of the transition state (TS) in terms of the normally hyperbolic invariantmanifold (NHIM) whose geometric phase-space structure persists under perturbation. TheNHIM can be regarded as an anchor of a dividing surface in phase space and it gives riseto an exact non-recrossing TS theory rate in reactions that are dominated by a singlebottleneck. Here, we review recent advances of phase space geometrical structures ofparticular relevance to chemical reactions in the condensed phase. We also provideconjectures on the promise of these techniques toward the design and control of chemicalreactions.