Publication

A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

Résumé

One of the most accurate methods for solving the time-dependent Schrödinger equation uses a combination of the dynamic Fourier method with the split-operator algorithm on a tensor-product grid. To reduce the number of required grid points, we let the grid move together with the wavepacket but find that the naïve algorithm based on an alternate evolution of the wavefunction and grid destroys the time reversibility of the exact evolution. Yet, we show that the time reversibility is recovered if the wavefunction and grid are evolved simultaneously during each kinetic or potential step; this is achieved by using the Ehrenfest theorem together with the splitting method. The proposed algorithm is conditionally stable, symmetric, and time-reversible and conserves the norm of the wavefunction. The preservation of these geometric properties is shown analytically and demonstrated numerically on a three-dimensional harmonic model and collinear model of He–H2 scattering. We also show that the proposed algorithm can be symmetrically composed to obtain time-reversible integrators of an arbitrary even order. We observed 10 000-fold speedup by using the tenth-order instead of the second-order method to obtain a solution with a time discretization error below 10−9. Moreover, using the adaptive grid instead of the fixed grid resulted in a 64-fold reduction in the required number of grid points in the harmonic system and made it possible to simulate the He–H2 scattering for six times longer while maintaining reasonable accuracy. Applicability of the algorithm to high-dimensional quantum dynamics is demonstrated using the strongly anharmonic eight-dimensional Hénon–Heiles model.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Concepts associés (38)
Réseau électrique intelligent
Un réseau électrique intelligent, ou smart grid en anglais, est un réseau de distribution d'électricité qui favorise la circulation d’information entre les fournisseurs et les consommateurs afin d’ajuster le flux d’électricité en temps réel et d'en permettre une gestion plus efficace. Ce type de réseaux intelligents utilise des techniques informatiques pour optimiser la production, la distribution, la consommation et éventuellement le stockage de l'énergie afin de mieux coordonner l'ensemble des mailles du réseau électrique, du producteur au consommateur final.
Numerical methods for ordinary differential equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
Intégrateur symplectique
Un intégrateur symplectique est une méthode numérique de résolution approchée des équations de la mécanique hamiltonienne, valable pour des faibles variations de temps. Les hypothèses de la mécanique hamiltonienne sont souvent appliquées à la mécanique céleste. Le système à étudier peut s'écrire sous la forme d'une action I et d'un angle φ, de manière que le système différentiel se réduise à : x := (I, φ) et : où l'on a noté : le crochet de Poisson de et . On voudrait connaître la solution formelle au système intégrable .
Afficher plus
Publications associées (42)

High-order geometric integrators for the variational Gaussian wavepacket dynamics and application to vibronic spectra at finite temperature

Roya Moghaddasi Fereidani

Molecular quantum dynamics simulations are essential for understanding many fundamental phenomena in physics and chemistry. They often require solving the time-dependent Schrödinger equation for molecular nuclei, which is challenging even for medium-sized ...
EPFL2024

High-order geometric integrators for the variational Gaussian approximation

Jiri Vanicek, Roya Moghaddasi Fereidani

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller’s original thawed Gaussian approximation, it is symplectic, conse ...
2023

Efficient geometric integrators for the linear and nonlinear time-dependent Schrödinger equation

Julien Roulet

Many physical and chemical reactions are driven by nonadiabatic processes, which imply the breakdown of the celebrated Born-Oppenheimer approximation. To understand these processes, experimentalists employ spectroscopic techniques. However, the obtained re ...
EPFL2022
Afficher plus
MOOCs associés (4)
SES Swiss-Energyscope
La transition énergique suisse / Energiewende in der Schweiz
Warm-up for EPFL
Warmup EPFL est destiné aux nouvelles étudiantes et étudiants de l'EPFL.
Thermodynamique II
Ce cours complète le MOOC « Thermodynamique : fondements » qui vous permettra de mettre en application les concepts fondamentaux de la thermodynamique. Pour atteindre cet objectif, le Professeur J.-P
Afficher plus