We report an efficient approach to accurately and efficiently compute transmission probabilities in resonant deep tunneling regime. Dynamical systems subjects to this phenomenon prove hard to simulate numerically even with exact methods, which motivates new methodological developments owing to the impact resonant phenomena have in several processes such as chemical reactions and electronic transport. Our approach is based on the original reformulation of stationnary quantum scattering as the propagation of a quantum trajectory in extended phase space. The present paper discusses in detail the node problem occurring to the time-independent quantum trajectory method in this very challenging situation, and introduces an efficient node-skipping scheme to circumvent expensive numerical integration in their vicinity. We illustrate how this numerical extension allows to treat all regimes of quantum tunneling with great versatility by comparison to existing approaches of the litterature. The quantum trajectory thus represents a very promising tool for the study of complex chemical reactions characterized by resonant tunneling effect.

Resonances are ubiquitous in a wide range of physical and chemical phenomena. Their impact on quantum scattering processes renders their study as important as it can be puzzling. In this paper, we illustrate the accuracy of a fully quantum, purely trajectory based reformulation of quantum mechanics proposed by one of the authors (Poirier) to acquire insights on shape resonances through direct and accurate computation of the diagonal elements of Smith’s lifetime matrix. This study also generalizes the relationship between the quantum trajectory propagation time and the Eisenbud- Wigner time delay—introduced in our previous publication[1] for symmetric potentials—to the general case of asymmetric potential profiles. In addition, we show how the complex amplitudes of the scattering matrix can be extracted from left- and right-incident quantum trajectories. Finally, we demonstrate that extended phase space quantum trajectories not only recover S-matrix and quantum time quantities, but they also provide their own picture of resonant phenomena, as dynamically distinct events characterised by an integer number of closed orbits in the quantum phase space.

L’effet tunnel est incontournable pour parfaire notre compréhension de la réactivité chimique. Permettant le passage de barrières de réaction cinétiquement infranchissables, son rôle primordial dans les réactions à basse température et pour l’échange d’éléments légers est bien connu. Cependant, l’assomption selon laquelle il serait sans impact hors de ces cas de figure a également été démentie ces dernières années: Ni l’échange d’éléments chimiques plus lourds que l’hélium[1], ni les réactions à température ambiante ne justifient de le négliger à priori[2]. Sa capacité à orienter la formation d’un produit de réaction l’élève au rang de troisième voie de contrôle de la réactivité, au même titre que les contrôles cinétique et thermodynamique[3]. De ce fait, il est nécessaire de disposer de méthodes numériques capables de reproduire cet effet purement quantique avec précision. Or, les méthodes approchées les plus couramment employées pour estimer la probabilité de transmission, comme WKB[4], ne garantissent pas un résultat fiable ( notamment en cas d’effet tunnel résonant, ou dans le régime deep tunneling ). L’utilisation de formules analytiques dérivées de potentiels approchés[5], une autre méthode répandue pour incorporer une estimation de l’effet tunnel, souffre également de son manque de précision. Si ces méthodes sont si largement utilisées, c’est avant tout pour leur simplicité et leur faible coût calculatoire. L’objet de cette contribution est de présenter une méthode efficace pour déterminer la probabilité de transmission le long d’un chemin de réaction de manière exacte[6,7]. Issue d’un formalisme rigoureusement équivalent à l’équation de Schrödinger stationnaire, cette approche remplace la détermination des états de diffusion par la propagation d’une trajectoire dans un espace des phases étendu[8,9]. La méthode et son efficacité seront illustrées sur des systèmes modèles ainsi qu’une réaction d’intérêt pour l’astrochimie[10].

Owing to its reactivity enhancing properties, quantum tunneling represents one of the most crucial effects to account for in order to achieve accurate prediction of rate constants for numerous chemical processes[1], even at ambient temperature[2]. Over the years, efficient methods emerged to accurately reproduce quantum tunneling in approximate atomistic simulations, with much progress being made on assessing the multidimensional character of the optimal tunneling path[3]. However resonant tunneling still proves to be a difficult phenomenon to characterize in the aforementioned methodological framework. In this talk, we present a purely trajectory based[4] approach of great accuracy and efficiency[5] applied to potential energy profiles subject to resonant tunneling. The working equations are a set of first order ODEs for a Hamiltonian in an extended phase space with respect to its classical analog. Trajectory propagation time enjoys a close relationship with collision lifetime, allowing to directly recover Smith's quantal time delay[6] at the energy of interest and thus giving further insight into resonant phenomena[7]. Trajectories describing scattering states with a reflection probability of nearly unity manifest strong destructive interference patterns, resulting in a very arduous numerical integration. This is reminiscent of pathologic numerical behavior encountered by the log-derivative approach in the deep tunneling regime[8] and constitutes a specific form of the well-known « node problem » encountered in Bohmian Dynamics[9]. To cope with the node problem, we propose an efficient semi-analytic scheme allowing trajectories to bypass nodes without significant loss of accuracy. As a result the method is a robust tool to analyse resonant reactive scattering.

In this work, we present the efficient combination of Smolyak representations with time independent quantum mechanical approach using absorbing boundary conditions for the cumulative reaction probability calculations of a multidimensional reactive scattering problem. Our approach uses both kinds of Smolyak representations (finite basis and grid) which drastically reduces the size of the basis representation for the cumulative reaction operator. The cumulative reaction probability is thus obtained by solving the eigenvalue problem within the context of reaction path Hamiltonian using the compact Smolyak basis combined with an iterative Lanczos algorithm. Benchmark calculations are presented for reactive scattering models with a linear reaction coordinate and applied to a 25D model highlighting the efficiency of the present approach for multidimensional reactive processes.

We present a quantum dynamics method based on the propagation of interacting quan- tum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys. 370 4–14 (2010)] and Schiff and Poirier [J. Chem. Phys. 136 031102 (2012)] on quantum dynamics with- out wavefunctions. It consists in determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quan- tum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

The accurate description of quantum effects, particularly quantum tunneling[1], in bimolecular chemical reactions constitutes one of the major goals of theoretical chemistry. Due to an unfavorable scaling law, exact quantum scattering methods lead to expensive and often unfeasible simulations when considering chemical systems with a large number of degrees of freedom. Moreover, at low temperature quantum effects often do not represent a small correction to classical calculations, but contrariwise are essential to obtain quantitative results. In this talk, I will show the feasibility of accurate quantum stationary scattering simulations for high dimensional systems using hybrid quantum-classical trajectories[2]. The configuration space is partitioned in one reaction coordinate going from reactants to products along which trajectories reproduce quantum effects, and bath coordinates with small amplitude motion which are treated classically[3]. Quantization of bath modes by a semi-classical phase-space sampling approximation based on Wigner-Weyl mapping allows to accurately reproduce quantum structure effects such as reaction thresholds. As was acknowledged many years ago [4], when the motion along the reaction coordinate is sufficiently slow compared to perpendicular motion, both subsystems undergo adiabatic evolution. By taking advantage from the local character of trajectory dynamics, I will illustrate how on-the-fly conversion from position-momenta to action-angle representation for bath coordinates in specific regions of vibrational adiabaticity leads to a computationally efficient scheme[5].