Recovering hidden dynamical modes from the generalized Langevin equation

Phase space geometry of isolated to condensed chemical reactions

The complexity of gas and condensed phase chemical reactions has generally been uncovered either approximately through transition state theories or exactly through (analytic or computational) integration of trajectories. These approaches can be improved by recognizing that the dynamics and associated geometric structures exist in phase space, ensuring that the propagator is symplectic as in velocity-Verlet integrators and by extending the space of dividing surfaces to optimize the rate variationally, respectively. The dividing surface can be analytically or variationally optimized in phase space, not just over configuration space, to obtain more accurate rates. Thus, a phase space perspective is of primary importance in creating a deeper understanding of the geometric structure of chemical reactions. A key contribution from dynamical systems theory is the generalization of the transition state (TS) in terms of the normally hyperbolic invariant manifold (NHIM) whose geometric phase-space structure persists under perturbation. The NHIM can be regarded as an anchor of a dividing surface in phase space and it gives rise to an exact non-recrossing TS theory rate in reactions that are dominated by a single bottleneck. Here, we review recent advances of phase space geometrical structures of particular relevance to chemical reactions in the condensed phase. We also provide conjectures on the promise of these techniques toward the design and control of chemical reactions.