Reconciling Semiclassical and Bohmian Mechanics: III. Scattering states for continuous potentials

In a previous paper [J. Chem. Phys. 121 4501 (2004)] a unique bipolar decomposition, Psi = Psi1 + Psi2 was presented for stationary bound states Psi of the one-dimensional Schroedinger equation, such that the components Psi1 and Psi2 approach their semiclassical WKB analogs in the large action limit. The corresponding bipolar quantum trajectories, as defined in the usual Bohmian mechanical formulation, are classical-like and well-behaved, even when Psi has many nodes, or is wildly oscillatory. A modification for discontinuous potential stationary stattering states was presented in a second paper [J. Chem. Phys. 124 034115 (2006)], whose generalization for continuous potentials is given here. The result is an exact quantum scattering methodology using classical trajectories. For additional convenience in handling the tunneling case, a constant velocity trajectory version is also developed.Comment: 16 pages and 14 figure

Investigating transition state resonances in the time domain by means of Bohmian mechanics: The F+HD reaction

In this work, we investigate the existence of transition state resonances on atom-diatom reactive collisions from a time-dependent perspective, stressing the role of quantum trajectories as a tool to analyze this phenomenon. As it is shown, when one focusses on the quantum probability current density, new dynamical information about the reactive process can be extracted. In order to detect the effects of the different rotational populations and their dynamics/coherences, we have considered a reduced two-dimensional dynamics obtained from the evolution of a full three-dimensional quantum time-dependent wave packet associated with a particular angle. This reduction procedure provides us with information about the entanglement between the radial degrees of freedom (r,R) and the angular one (\gamma), which can be considered as describing an environment. The combined approach here proposed has been applied to study the F+HD reaction, for which the FH+D product channel exhibits a resonance-mediated dynamics.Comment: 12 pages, 9 figure

Invariant Manifolds and Rate Constants in Driven Chemical Reactions

Reaction rates of chemical reactions under nonequilibrium conditions can be determined through the construction of the normally hyperbolic invariant manifold (NHIM) [and moving dividing surface (DS)] associated with the transition state trajectory. Here, we extend our recent methods by constructing points on the NHIM accurately even for multidimensional cases. We also advance the implementation of machine learning approaches to construct smooth versions of the NHIM from a known high-accuracy set of its points. That is, we expand on our earlier use of neural nets, and introduce the use of Gaussian process regression for the determination of the NHIM. Finally, we compare and contrast all of these methods for a challenging two-dimensional model barrier case so as to illustrate their accuracy and general applicability.Comment: 28 pages, 13 figures, table of contents figur

Quantum Theory of Reactive Scattering in Phase Space

We review recent results on quantum reactive scattering from a phase space perspective. The approach uses classical and quantum versions of normal form theory and the perspective of dynamical systems theory. Over the past ten years the classical normal form theory has provided a method for realizing the phase space structures that are responsible for determining reactions in high dimensional Hamiltonian systems. This has led to the understanding that a new (to reaction dynamics) type of phase space structure, a {\em normally hyperbolic invariant manifold} (or, NHIM) is the "anchor" on which the phase space structures governing reaction dynamics are built. The quantum normal form theory provides a method for quantizing these phase space structures through the use of the Weyl quantization procedure. We show that this approach provides a solution of the time-independent Schr\"odinger equation leading to a (local) S-matrix in a neighborhood of the saddle point governing the reaction. It follows easily that the quantization of the directional flux through the dividing surface with the properties noted above is a flux operator that can be expressed in a "closed form". Moreover, from the local S-matrix we easily obtain an expression for the cumulative reactio probability (CRP). Significantly, the expression for the CRP can be evaluated without the need to compute classical trajectories. The quantization of the NHIM is shown to lead to the activated complex, and the lifetimes of quantum states initialized on the NHIM correspond to the Gamov-Siegert resonances. We apply these results to the collinear nitrogen exchange reaction and a three degree-of-freedom system corresponding to an Eckart barrier coupled to two Morse oscillators.Comment: 59 pages, 13 figure