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Reservoir computing is a machine learning algorithm that excels at predicting the evolution of time series, in particular, dynamical systems. Moreover, it has also shown superb performance at solving partial differential equations. In this work, we adapt this methodology to integrate the time-dependent Schrödinger equation, propagating an initial wavefunction in time. Since such wavefunctions are complex-valued high-dimensional arrays, the reservoir computing formalism needs to be extended to cope with complex-valued data. Furthermore, we propose a multi-step learning strategy that avoids overfitting the training data. We illustrate the performance of our adapted reservoir computing method by application to four standard problems in molecular vibrational dynamics. [ABSTRACT FROM AUTHOR]

A classical-quantum correspondence study of a saddle-node bifurcation in a realistic molecular system is presented. The relevant classical structures (periodic orbits and manifolds) and its origin are examined in detail. The most important conclusion of this study is that, below the bifurcation point, there exists an infinite sequence of precursor orbits, which mimic for a significant period of time the (future) saddle-node orbits. These structures have a profound influence in the quantum mechanics of the molecule and several vibrational wave functions of the system present a strong localization along the saddle-node periodic orbits. A striking result is that this scarring effect also takes place well below the bifurcation energy, which constitutes a manifestation of the so-called ‘‘ghost’’ orbits in configuration and phase space. This localization effect has been further investigated using wave packet dynamics. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this Communication we describe how the transition from regularity to classical chaos in molecular Hamiltonian systems shows up at the quantum level in the structure of the corresponding wave functions and spectra. By changing the value of h we show how the scars result from combinations of regular wave functions. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The behavior of semiclassical wave functions during adiabatic deformations of the Hamiltonian is studied. A practical technique based on Gaussian wave packet dynamics is presented which allows a semiclassical eigenstate for one Hamiltonian to be adiabatically switched into an eigenstate for a different Hamiltonian. The validity of this technique is established by studying the properties of the Hamilton–Jacobi equation during adiabatic processes. The method is shown to yield good results for several one- and two-dimensional model problems. It is shown that converged semiclassical eigenstates can be obtained even if the trajectories have become chaotic by the end of the switching process. [ABSTRACT FROM AUTHOR]

By constructing continuation/bifurcation diagrams of families of periodic orbits of LiNC/LiCN system the spectroscopy and dynamics for this species are deduced and compared with accurate quantum mechanical calculations up to 13 000 cm-1. The interesting phenomenon of the appearance of gaps in the continuation diagram of the principal family that corresponds to the bend motion is shown to occur in both isomers, LiNC and LiCN. Through semiclassical quantization a one to one correspondence of specific periodic orbits to certain eigenstates is demonstrated. One interesting example is the case of periodic orbits that are generated from a saddle-node bifurcation and describe rotations of the Li+ ion around the CN- fragment. The correspondence of these periodic orbits to regular rotating type eigenfunctions is shown, thus, demonstrating in a clear way the importance of the saddle-node bifurcations in locating localized wavefunctions in highly energized molecules. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The feasibility of using the zeros of the Husimi function to characterize, i.e. ascribe quantum numbers to, dynamical systems with mixed dynamics is explored using a realistic model for the vibrations of the HCP molecule as an example. Further implications are also discussed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]