Abstract
Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical ab initio solution of the time-dependent Schr\"odinger equation, thereby motivating the development of very efficient time propagators. These usually involve the solution of very large systems of first-order differential equations that are characterized by a high degree of stiffness. In this contribution, we analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy, however, differs significantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. In this contribution, we consider two physical processes. The first one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time-independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding differential equation shows a striking stiffness which makes the time integration extremely difficult. In the case of the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiffness of the problem.
Highlights
The numerical integration of the time-dependent Schrodinger equation (TDSE) has become the main theoretical approach for the quantitative study of a vast amount of phenomena, including strong field processes in atoms and molecules, quantum collisions and chemical reactions
We examine the performances of these explicit schemes in a completely different context namely the calculation of a fidelity function that measures the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are confined in an anharmonic potential
The electron wavepacket is developed in a basis of 200 B-splines and we use the time scaled coordinate method during the propagation [38]
Summary
The numerical integration of the time-dependent Schrodinger equation (TDSE) has become the main theoretical approach for the quantitative study of a vast amount of phenomena, including strong field processes in atoms and molecules, quantum collisions and chemical reactions. In the high frequency regime where the photon energy is of the order or larger than the ionization potential, very intense coherent X-ray sources are under development They are based on the collective electronic response of a plasma to ultra intense laser fields [6] as well as the generation free electron lasers (FEL) such as the European XFEL project. The latter is expected to boost the average photon flux by about two orders of magnitude in comparison with already existing FELs. The interaction of atoms or molecules with intense X-ray pulses with a duration in the femtosecond or subfemtosecond regime is expected to lead to highly non-linear processes which can no longer be described within perturbation theory as is currently the case
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