Identification of the invariant manifolds of the LiCN molecule using Lagrangian descriptors.

In this paper, we apply Lagrangian descriptors to study the invariant manifolds that emerge from the top of two barriers existing in the LiCN⇌LiNC isomerization reaction. We demonstrate that the integration times must be large enough compared with the characteristic stability exponents of the periodic orbit under study. The invariant manifolds manifest as singularities in the Lagrangian descriptors. Furthermore, we develop an equivalent potential energy surface with 2 degrees of freedom, which reproduces with a great accuracy previous results [F. Revuelta, R. M. Benito, and F. Borondo, Phys. Rev. E 99, 032221 (2019)2470-004510.1103/PhysRevE.99.032221]. This surface allows the use of an adiabatic approximation to develop a more simplified potential energy with solely 1 degree of freedom. The reduced dimensional model is still able to qualitatively describe the results observed with the original 2-degrees-of-freedom potential energy landscape. Likewise, it is also used to study in a more simple manner the influence on the Lagrangian descriptors of a bifurcation, where some of the previous invariant manifolds emerge, even before it takes place.

Correspondence between classical and quantum resonances.

Bifurcations take place in molecular Hamiltonian nonlinear systems as the excitation energy increases, leading to the appearance of different classical resonances. In this paper, we study the quantum manifestations of these classical resonances in the isomerizing system CN-Li⇆Li-CN. By using a correlation diagram of eigenenergies versus Planck constant, we show the existence of different series of avoided crossings, leading to the corresponding series of quantum resonances, which represent the quantum manifestations of the classical resonances. Moreover, the extrapolation of these series to ℏ=0 unveils the correspondence between the bifurcation energy of classical resonances and the energy of the series of quantum resonances in the semiclassical limit ℏ→0. Additionally, in order to obtain analytical expressions for our results, a semiclassical theory is developed.

Short-periodic-orbit method for excited chaotic eigenfunctions.

An alternative method for the calculation of excited chaotic eigenfunctions in arbitrary energy windows is presented. We demonstrate the feasibility of using wave functions localized on unstable periodic orbits as efficient basis sets for this task in classically chaotic systems. The number of required localized wave functions is only of the order of the ratio t_{H}/t_{E}, with t_{H} the Heisenberg time and t_{E} the Ehrenfest time. As an illustration, we present convincing results for a coupled two-dimensional quartic oscillator with chaotic dynamics.

Using correlation diagrams to study the vibrational spectrum of highly nonlinear floppy molecules: The K-CN case.

The correlation diagrams of vibrational energy levels considering the Planck constant as a variable parameter have proven as a very useful tool to study vibrational molecular states, and more specifically in relation to the quantum manifestations of chaos in such dynamical systems. In this paper, we consider the highly nonlinear K-CN molecule, showing how the regular classical structures, i.e., Kolmogorov-Arnold-Moser tori, existing in the mixed classical phase space appear in the quantum levels correlation diagram as emerging diabatic states, something that remains hidden when only the actual value of the Planck constant is considered. Additionally, a quantum transition from order to chaos is unveiled with the aid of these correlation diagrams, where it appears as a frontier of scarred functions.

Identifying reaction pathways in phase space via asymptotic trajectories.

In this paper, we revisit the concepts of the reactivity map and the reactivity bands as an alternative to the use of perturbation theory for the determination of the phase space geometry of chemical reactions. We introduce a reformulated metric, called the asymptotic trajectory indicator, and an efficient algorithm to obtain reactivity boundaries. We demonstrate that this method has sufficient accuracy to reproduce phase space structures such as turnstiles for a 1D model of the isomerization of ketene in an external field. The asymptotic trajectory indicator can be applied to higher dimensional systems coupled to Langevin baths as we demonstrate for a 3D model of the isomerization of ketene.

Shannon entropy at avoided crossings in the quantum transition from order to chaos.

Shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics. In order to be able to study jointly this entropy for discrete and continuous probability, calculations have been performed on a quantized map, the kicked Harper map, resulting in a different behavior, as order-chaos transition takes place, for the discrete (position representation) and continuous (coherent state representation) cases. This different behavior is analyzed in terms of the distribution of zeros of the Husimi function.

Finite-barrier corrections for multidimensional barriers in colored noise.

The usual identification of reactive trajectories for the calculation of reaction rates requires very time-consuming simulations, particularly if the environment presents memory effects. In this paper, we develop a method that permits the identification of reactive trajectories in a system under the action of a stochastic colored driving. This method is based on the perturbative computation of the invariant structures that act as separatrices for reactivity. Furthermore, using this perturbative scheme, we have obtained a formally exact expression for the reaction rate in multidimensional systems coupled to colored noisy environments.

Unveiling the chaotic structure in phase space of molecular systems using Lagrangian descriptors.

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)1007-570410.1016/j.cnsns.2013.05.002] 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.

Water phase transitions from the perspective of hydrogen-bond network analysis.

We analyze the water vapour-liquid and solid-liquid phase transitions from the perspective of hydrogen bond networks. Using molecular dynamics simulation data for the TIP4P/2005 and TIP4P/ice water models, we built hydrogen bond networks in the neighbourhood of the transitions. We studied the behaviour of some topological network properties: the average degree, clustering coefficient, and average path length. We found that these properties exhibit a discontinuity while approaching a phase transition region, similar to those that appear for some thermodynamic properties in the same region. This approach can be extended to characterize other water phase transitions. Besides, it can also be applied to study the phase transitions of other hydrogen-bonded substances or to other scenarios whose relevant "interaction" could be identified together with a "proper criterion" defined in an analogous way as in the case of hydrogen bonded systems.

Role of short periodic orbits in quantum maps with continuous openings.

We apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map. In this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions. This is relevant in many situations that include optical microresonators and more complicated boundary conditions. In a perturbative regime, the shortest periodic orbits belonging to the classical repeller of the open map-a cantor set given by a region of exactly zero reflectivity-prove to be extremely robust in supporting a set of long-lived resonances of the continuously open quantum maps. Moreover, for steplike functions a significant reduction in the number needed is obtained, similarly to the completely open situation. This happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case. In order to give a more realistic interpretation of these results we compare with a Fresnel-type reflectivity function.

Above Saddle-Point Regions of Order in a Sea of Chaos in the Vibrational Dynamics of KCN.

The dynamical characteristics of a region of regular vibrational motion in the sea of chaos above the saddle point corresponding to the linear C-N-K configuration is examined in detail. To explain the origin of this regularity, the associated phase space structures were characterized using suitably defined Poincaré surfaces of section, identifying the different resonances between the stretching and bending modes, as a function of excitation energy. The corresponding topology is elucidated by means of periodic orbit analysis.

Transition state theory for activated systems with driven anharmonic barriers.

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.

Semiclassical basis sets for the computation of molecular vibrational states.

In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN molecule.

Theory of short periodic orbits for partially open quantum maps.

We extend the semiclassical theory of short periodic orbits [M. Novaes et al., Phys. Rev. E 80, 035202(R) (2009)PLEEE81539-375510.1103/PhysRevE.80.035202] to partially open quantum maps, which correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity R. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of this kind of system. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belongs to the classical repeller of the open map (R=0) is able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of R values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behavior, providing an explanation in terms of short periodic orbits.

Agricultural activity shapes the communication and migration patterns in Senegal.

The communication and migration patterns of a country are shaped by its socioeconomic processes. The economy of Senegal is predominantly rural, as agriculture employs over 70% of the labor force. In this paper, we use mobile phone records to explore the impact of agricultural activity on the communication and mobility patterns of the inhabitants of Senegal. We find two peaks of phone calls activity emerging during the growing season. Moreover, during the harvest period, we detect an increase in the migration flows throughout the country. However, religious holidays also shape the mobility patterns of the Senegalese people. Hence, in the light of our results, agricultural activity and religious holidays are the primary drivers of mobility inside the country.

Transition state theory for solvated reactions beyond recrossing-free dividing surfaces.

The accuracy of rate constants calculated using transition state theory depends crucially on the correct identification of a recrossing-free dividing surface. We show here that it is possible to define such optimal dividing surface in systems with non-Markovian friction. However, a more direct approach to rate calculation is based on invariant manifolds and avoids the use of a dividing surface altogether, Using that method we obtain an explicit expression for the rate of crossing an anharmonic potential barrier. The excellent performance of our method is illustrated with an application to a realistic model for LiNC⇌LiCN isomerization.

Transition state geometry of driven chemical reactions on time-dependent double-well potentials.

Reaction rates across time-dependent barriers are difficult to define and difficult to obtain using standard transition state theory approaches because of the complexity of the geometry of the dividing surface separating reactants and products. Using perturbation theory (PT) or Lagrangian descriptors (LDs), we can obtain the transition state trajectory and the associated recrossing-free dividing surface. With the latter, we are able to determine the exact reactant population decay and the corresponding rates to benchmark the PT and LD approaches. Specifically, accurate rates are obtained from a local description regarding only direct barrier crossings and to those obtained from a stability analysis of the transition state trajectory. We find that these benchmarks agree with the PT and LD approaches for obtaining recrossing-free dividing surfaces. This result holds not only for the local dynamics in the vicinity of the barrier top, but also for the global dynamics of particles that are quenched at the reactant or product wells after their sojourn over the barrier region. The double-well structure of the potential allows for long-time dynamics related to collisions with the outside walls that lead to long-time returns in the low-friction regime. This additional global dynamics introduces slow-decay pathways that do not result from the local transition across the recrossing-free dividing surface associated with the transition state trajectory, but can be addressed if that structure is augmented by the population transfer of the long-time returns.

Scar Functions, Barriers for Chemical Reactivity, and Vibrational Basis Sets.

The performance of a recently proposed method to efficiently calculate scar functions is analyzed in problems of chemical interest. An application to the computation of wave functions associated with barriers relevant for the LiNC ⇄ LiCN isomerization reaction is presented as an illustration. These scar functions also constitute excellent elements for basis sets suitable for quantum calculation of vibrational energy levels. To illustrate their efficiency, a calculation of the LiNC/LiCN eigenfunctions is also presented.

Solvated molecular dynamics of LiCN isomerization: All-atom argon solvent versus a generalized Langevin bath.

The reaction rate rises and falls with increasing density or friction when a molecule is activated by collisions with the solvent particles. This so-called Kramers turnover has recently been observed in the isomerization reaction of LiCN in an argon bath. In this paper, we demonstrate by direct comparison with those results that a reduced-dimensional (generalized) Langevin description gives rise to similar reaction dynamics as the corresponding (computationally expensive) full molecular dynamics calculations. We show that the density distributions within the Langevin description are in direct agreement with the full molecular dynamics results and that the turnover in the reaction rates is reproduced qualitatively and quantitatively at different temperatures.

Frequency analysis of the laser driven nonlinear dynamics of HCN.

We study the vibrational dynamics of a model for the HCN molecule in the presence of a monochromatic laser field. The variation of the structural behavior of the system as a function of the laser frequency is analyzed in detail using the smaller alignment index, frequency maps, and diffusion coefficients. It is observed that the ergodicity of the system depends on the frequency of the excitation field, especially in its transitions from and into chaos. This provides a roadmap for the possibility of bond excitation and dissociation in this molecule.