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The rate of a chemical reaction can often be determined by the properties of a rank-1 saddle and the associated transition state separating reactants and products. We have found evidence that such rates can be controlled and even enhanced by external driving in at least one such system. Specifically, we analyze a reactive model in two degrees of freedom that has been used earlier to describe driven chemical reactions. Therein, changes in the external driving can lead to a local maximum of the decay rate constant or even to bifurcations of periodic trajectories on the normally hyperbolic invariant manifold (NHIM) corresponding to the transition state. Inspired by these bifurcations, we show that in this case, the dynamics on the NHIM can be connected to the geometry of reactive trajectories and to reaction probabilities of Maxwell-Boltzmann distributed reactant ensembles., Comment: Main article has 8 pages, 6 figures

The framework of transition state theory (TST) provides a powerful way for analyzing the dynamics of physical and chemical reactions. While TST has already been successfully used to obtain reaction rates for systems with a single time-dependent saddle point, multiple driven saddles have proven challenging because of their fractal-like phase space structure. This paper presents the construction of an approximately recrossing-free dividing surface based on the normally hyperbolic invariant manifold in a time-dependent two-saddle model system. Based on this, multiple methods for obtaining instantaneous (time-resolved) decay rates of the underlying activated complex are presented and their results discussed., Comment: 13 pages and 9 figures

Recent work has shown that in a non-thermal, multidimensional system, the trajectories in the activated complex possess different instantaneous and time-averaged reactant decay rates. Under dissipative dynamics, it is known that these trajectories, which are bound on the normally hyperbolic invariant manifold (NHIM), converge to a single trajectory over time. By subjecting these dissipative systems to thermal noise, we find fluctuations in the saddle-bound trajectories and their instantaneous decay rates. Averaging over these instantaneous rates results in the decay rate of the activated complex in a thermal system. We find, that the temperature dependence of the activated complex decay in a thermal system can be linked to the distribution of the phase space resolved decay rates on the NHIM in the non-dissipative case. By adjusting the external driving of the reaction, we show that it is possible to influence how the decay rate of the activated complex changes with rising temperature., Comment: Main article has 9 pages, 7 figures. Supplemental material has 2 pages, 1 figure

The framework of transition state theory relies on the determination of a geometric structure identifying reactivity. It replaces the laborious exercise of following many trajectories for a long time to provide chemical reaction rates and pathways. In this paper, recent advances in constructing this geometry even in time-dependent systems are applied to the LiCN $\rightleftharpoons$ LiNC isomerization reaction, driven by an external field. We obtain decay rates of the reactant population close to the transition state by exploiting local properties of the dynamics of trajectories in and close to it. We find that the external driving has a large influence on these decay rates when compared to the non-driven isomerization reaction. This, in turn, provides renewed evidence for the possibility of controlling chemical reactions, like this one, through external time-dependent fields., Comment: Main article has 11 pages, 6 figures. Supplemental material has 4 pages, 1 figure

Chemical reactions in multidimensional systems are often described by a rank-1 saddle, whose stable and unstable manifolds intersect in the normally hyperbolic invariant manifold (NHIM). Trajectories started on the NHIM in principle never leave this manifold when propagated forward or backward in time. However, the numerical investigation of the dynamics on the NHIM is difficult because of the instability of the motion. We apply a neural network to describe time-dependent NHIMs and use this network to stabilize the motion on the NHIM for a periodically driven model system with two degrees of freedom. The method allows us to analyze the dynamics on the NHIM via Poincar\'e surfaces of section (PSOS) and to determine the transition state (TS) trajectory as a periodic orbit with the same periodicity as the driving saddle, viz. a fixed point of the PSOS surrounded by near-integrable tori. Based on Transition State Theory and a Floquet analysis of a periodic TS trajectory we compute the rate constant of the reaction with significantly reduced numerical effort compared to the propagation of a large trajectory ensemble., Comment: 9 pages, 4 figures

Chemical reactions in multidimensional driven systems are typically described by a time-dependent rank-1 saddle associated with one reaction and several orthogonal coordinates (including the solvent bath). To investigate reactions in such systems, we develop a fast and robust method ---viz., local manifold analysis (LMA)--- for computing the instantaneous decay rate of reactants. Specifically, it computes the instantaneous decay rates along saddle-bound trajectories near the activated complex by exploiting local properties of the stable and unstable manifold associated with the normally hyperbolic invariant manifold (NHIM). The LMA method offers substantial reduction of numerical effort and increased reliability in comparison to direct ensemble integration. It provides an instantaneous flux that can be assigned to every point on the NHIM and which is associated with a trajectory ---regardless of whether it is periodic, quasi-periodic, or chaotic--- that is bound on the NHIM. The time average of these fluxes in the driven system corresponds to the average rate through a given local section containing the corresponding point on the NHIM. We find good agreement between the results of the LMA and direct ensemble integration obtained using numerically constructed, recrossing-free dividing surfaces., Comment: 9 pages, 7 figures

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 figure

Transition state theory formally provides a simplifying approach for determining chemical reaction rates and pathways. Given an underlying potential energy surface for a reactive system, one can determine the dividing surface in phase space which separates reactant and product regions, and thereby also these regions. This is often a difficult task, and it is especially demanding for high-dimensional time-dependent systems or when a non-local dividing surface is required. Recently, approaches relying on Lagrangian descriptors have been successful at resolving the dividing surface in some of these challenging cases, but this method can also be computationally expensive due to the necessity of integrating the corresponding phase space function. In this paper, we present an alternative method by which time-dependent, locally recrossing-free dividing surfaces can be constructed without the calculation of any auxiliary phase space function, but only from simple dynamical properties close to the energy barrier., Comment: 8 pages, 6 figures

In a dynamical system, the transition between reactants and products is typically mediated by an energy barrier whose properties determine the corresponding pathways and rates. The latter is the flux through a dividing surface (DS) between the two corresponding regions and it is exact only if it is free of recrossings. For time-independent barriers, the DS can be attached to the top of the corresponding saddle point of the potential energy surface, and in time-dependent systems, the DS is a moving object. The precise determination of reaction rates, eg using transition state theory, requires the actual construction of a DS for a given saddle geometry which is in general a demanding methodical and computational task, especially in high-dimensional systems. In this paper, we demonstrate how such time-dependent, global, and recrossing-free DSs can be constructed using neural networks. In our approach, the neural network uses the bath coordinates and time as input and it is trained in a way that its output provides the position of the DS along the reaction coordinate. An advantage of this procedure is that, once the neural network is trained, the complete information about the dynamical phase space separation is stored in the network's parameters, and a precise distinction between reactants and products can be made for all possible system configurations, all times, and with little computational effort. We demonstrate this general method for two- and three-dimensional systems, and explain its straightforward extension to even more degrees of freedom., Comment: 8 pages, 7 figures

Dynamics between reactants and products are often mediated by a rate-determining barrier and an associated dividing surface leading to the transition state theory rate. This framework is challenged when the barrier is time-dependent because its motion can give rise to recrossings across the fixed dividing surface. A non-recrossing time-dependent dividing surface can nevertheless be attached to the TS trajectory resulting in recrossing-free dynamics. We extend the formalism---constructed using Lagrangian Descriptors---to systems with additional bath degrees of freedom. The propagation of reactant ensembles provides a numerical demonstration that our dividing surface is recrossing-free and leads to exact TST rates., Comment: 55 pages, 4 figures, 66 references

In chemical or physical reaction dynamics, it is essential to distinguish precisely between reactants and products for all time. This task is especially demanding in time-dependent or driven systems because therein the dividing surface (DS) between these states often exhibits a nontrivial time-dependence. The so-called transition state (TS) trajectory has been seen to define a DS which is free of recrossings in a large number of one-dimensional reactions across time-dependent barriers, and, thus, allows one to determine exact reaction rates. A fundamental challenge to applying this method is the construction of the TS trajectory itself. The minimization of Lagrangian descriptors (LDs) provides a general and powerful scheme to obtain that trajectory even when perturbation theory fails. Both approaches encounter possible breakdowns when the overall potential is bounded, admitting the possibility of returns to the barrier long after trajectories have reached the product or reactant wells. Such global dynamics cannot be captured by perturbation theory. Meanwhile, in the LD-DS approach, it leads to the emergence of additional local minima which make it difficult to extract the optimal branch associated with the desired TS trajectory. In this work, we illustrate this behavior for a time-dependent double-well potential revealing a self-similar structure of the LD, and we demonstrate how the reflections and side-minima can be addressed by an appropriate modification of the LD associated with the direct rate across the barrier., Comment: 7 pages, 6 figures, 50 references

We present a method to calculate the exciton spectra of all direct semiconductors with a complex valence band structure. The Schr\"odinger equation is solved using a complete basis set with Coulomb Sturmian functions. This method also allows for the computation of oscillator strengths. Here we apply this method to investigate the impact of the valence band structure of cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ on the yellow exciton spectrum. Results differ from those in [Phys. Rev. Lett. 115, 027402 (2015)]; the differences are discussed and explained. The difference between the second and third Luttinger parameter can be determined by comparisons with experiments, however, the evaluation of all three Luttinger parameters is not uniquely possible. Our results are consistent with band structure calculations. Considering also a finite momentum $\hbar K$ of the center of mass, we show that the large $K$-dependent line splitting observed for the $1S$ exciton state [Phys. Rev. Lett. 91, 107401 (2003)] is not related to an exchange interaction but rather to the complex valence band structure of $\mathrm{Cu_{2}O}$., Comment: Some corrections and improvements. Accepted for publication in Phys. Rev. B

Exceptional points are special parameter points in spectra of open quantum systems, at which resonance energies degenerate and the associated eigenvectors coalesce. Typical examples are Rydberg systems in parallel electric and magnetic fields, for which we solve the Schr\"odinger equation in a complete basis to calculate the resonances and eigenvectors. Starting from an avoided crossing within the parameter-dependent spectra and using a two-dimensional matrix model, we develop an iterative algorithm to calculate the field strengths and resonance energies of exceptional points and to verify their basic properties. Additionally, we are able to visualise the wave functions of the degenerate states. We report the existence of various exceptional points. For the hydrogen atom these points are in an experimentally inaccessible regime of field strengths. However, excitons in cuprous oxide in parallel electric and magnetic fields, i. e., the corresponding hydrogen analogue in a solid state body, provide a suitable system, where the high-field regime can be reached at much smaller external fields and for which we propose an experiment to detect exceptional points., Comment: 22 pages, 7 figures. Some improvements and one additional figure. Accepted for publication in J. Phys. B