The minimum contour in the forward Lagrangian descriptor overlaps the invariant manifold (in green) dividing reactant and product regions.

We explore here the feasibility of using the recently introduced Lagrangian descriptors [A. M. Mancho et al., Commun. Nonlinear Sci. Numer. Simul. 18, 3530 (2013)] to unveil the usually rich dynamics taking place in the vibrations of molecular systems, especially if they are floppy. The principal novelty of our work is the inclusion of $p$ norms in the definition of the descriptors in this kind of system, which greatly enhances their power to discern among the different structures existing in the phase space. As an illustration we use the LiCN molecule described by realistic potentials in two and three dimensions, which exhibits chaotic motion within a mixed phase space in the isomerization between the two wells corresponding to the linear isomer stable configurations, LiNC and LiCN. In particular, we pay special attention to the manifolds emerging from the unstable fixed point between the corresponding isomer wells, and also to the marginally stable structures around a parabolic point existing near the LiNC well.

The identification of trajectories that contribute to the reaction rate is the crucial dynamical ingredient in any classical chemical reactivity calculation. This problem often requires a full scale numerical simulation of the dynamics, in particular if the reactive system is exposed to the influence of a heat bath. As an efficient alternative, we propose here to compute invariant surfaces in the phase space of the reactive system that separate reactive from nonreactive trajectories. The location of these invariant manifolds depends both on time and on the realization of the driving force exerted by the bath. These manifolds allow the identification of reactive trajectories simply from their initial conditions, without the need of any further simulation. In this paper, we show how these invariant manifolds can be calculated, and used in a formally exact reaction rate calculation based on perturbation theory for any multidimensional potential coupled to a noisy environment.

Classical transition state theory has been extended to address chemical reactions across barriers that are driven and anharmonic. This resolves a challenge to the naive theory that necessarily leads to recrossings and approximate rates because it relies on a fixed dividing surface. We develop both perturbative and numerical methods for the computation of a time-dependent recrossing-free dividing surface for a model anharmonic system in a solvated environment that interacts strongly with an oscillatory external field. We extend our previous work, which relied either on a harmonic approximation or on periodic force driving. We demonstrate that the reaction rate, expressed as the long-time flux of reactive trajectories, can be extracted directly from the stability exponents, namely, Lyapunov exponents, of the moving dividing surface. Comparison to numerical results demonstrates the accuracy and robustness of this approach for the computation of optimal (recrossing-free) dividing surfaces and reaction rates in systems with Markovian solvation forces. The resulting reaction rates are in strong agreement with those determined from the long-time flux of reactive trajectories.

We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two-dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presented.

The performance of a simple method [E. L. Sibert III, E. Vergini, R. M. Benito, and F. Borondo, New J. Phys. 10, 053016 (2008)] to efficiently compute scar functions along unstable periodic orbits with complicated trajectories in configuration space is discussed, using a classically chaotic two-dimensional quartic oscillator as an illustration. DOI: http://dx.doi.org/10.1103/PhysRevE.85.026214 ©2012 American Physical Society