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The Journal of Chemical Physics

Quantum-classical description of rotational diffractive scattering using Bohmian trajectories: Comparison with full quantum wave packet results

Etienne Gindensperger, C. Meier, Juan Alberto Beswick, M-C. Heitz

Published May 30, 2002• 37 citations
Publisher

Abstract

We apply the mixed quantum/classical method based on the Bohmian formulation of quantum mechanics [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case of rotational diffractive surface scattering of a diatomic molecule. The rotation as well as the normal translational degree of freedom are treated classically while the two parallel degrees of freedom that account for the diffraction are treated quantum mechanically. The effects of treating some degrees of freedom classically are discussed in detail by comparing our novel approximate method to quantum wave packet results obtained by the multiconfiguration time-dependent Hartree method.

DOI
10.1063/1.1471904

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We apply the mixed quantum/classical method based on the Bohmian formulation of quantum mechanics [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case of rotational diffractive surface scattering of a diatomic molecule. The rotation as well as the normal translational degree of freedom are treated classically while the two parallel degrees of freedom that account for the diffraction are treated quantum mechanically. The effects of treating some degrees of freedom classically are discussed in detail by comparing our novel approximate method to quantum wave packet results obtained by the multiconfiguration time-dependent Hartree method.
We apply the MQCB (mixed quantum/classical mechanics based on Bohmian trajectories) [E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, 9369 (2000)] to the case where the quantum degree of freedom contains both bound and continuum states. The quantum degree of freedom is treated by a wave packet that is propagated on a spatial grid with a Hamiltonian that depends parametrically on the classical degrees of freedom, while the classical degrees of freedom themselves are coupled to the wave function via quantum trajectories. The method is applied to a simple model of a light particle colliding with a heavy (classical) particle absorbed on a surface, which has been used by other authors to develop methods to combine quantum degrees of freedom including continuum states with classical mechanics. The results are compared to the Ehrenfest mean-field method as well as to full quantum results.
A novel time-dependent hybrid quantum/classical propagation scheme based on Bohmian quantum trajectories is presented. The quantum subsystem is described by a wave packet depending on the quantum variables x and, via the total potential energy of the system, parametrically on the classical trajectories X(t). The wave packet is used to calculate de Broglie–Bohm quantum trajectories x(t) which are used to calculate the force acting on the classical variables. Quantum corrections of the classical equation of motion are also included. The method is applied to a simple case of two coupled oscillators. Comparison between exact quantum and approximate results demonstrates that these MQCB (Mixed Quantum/Classical Bohmian) trajectories provide a good description of the energy exchange between the two oscillators.
The method, described in the paper, incorporates ideas of the Bohmian formulation of quantum mechanics into a trajectory approach designed for systems comprised of light (quantum) and heavy (nearly classical) particles. The goal is to treat all degrees of freedom on the same footing but, for practical reasons, to include the quantum-mechanical effects into dynamics for just the light particles. The total wavefunction, taken as a product of slow and fast components representing the heavy and light particles respectively is represented in terms of trajectories. The fast component depends on the trajectory-guided configurations of the heavy particles. The initial conditions for the trajectories are defined by the full-dimensional initial wavefunction according to Bohm's prescription of associating the trajectory momentum with the gradient of the wavefunction phase. The time-dependent Schrödinger equation is solved for the quantum degrees of freedom using an ensemble of the quantum trajectories (or other time-propagation method) for each guiding trajectory of the classical subspace. Neglect of the quantum force acting on the heavy particles allows decoupling of the quantum calculations associated with different guiding trajectories. The approach allows reconstruction of the full-dimensional wavefunction and conserves the total wavefunction energy as illustrated for the vibrationally non-adiabatic model.
Diffraction and interference of matter waves are key phenomena in quantum mechanics. Here we present some results on particle diffraction in a wide variety of situations, ranging from simple slit experiments to more complicated cases such as atom scattering by corrugated metal surfaces and metal surfaces with simple and isolated adsorbates. The principal novelty of our study is the use of the so-called Bohmian formalism of quantum trajectories. These trajectories are able to satisfactorily reproduce the main features of the experimental results and, more importantly, they provide a causal intuitive interpretation of the underlying dynamics. In particular, we will focus our attention on: (a) a revision of the concepts of near and far field in undulatory optics; (b) the transition to the classical limit, where it is found that although the quantum and classical diffraction patterns tend to be quite similar, some quantum features are maintained even when the quantum potential goes to zero; and (c) a qualitative description of the scattering of atoms by metal surfaces in the presence of a single adsorbate.
The numerical analysis of the diffraction features rendered by transmission electron microscopy typically relies either on classical approximations (Monte Carlo simulations) or quantum paraxial tomography (the multislice method and any of its variants). Although numerically advantageous (relatively simple implementations and low computational costs), they involve important approximations and thus their range of applicability is limited. To overcome such limitations, an alternative, more general approach is proposed, based on an optimal combination of wave-packet propagation with the on-the-fly computation of associated Bohmian trajectories. For the sake of clarity, but without a loss of generality, the approach is used to analyze the diffraction of an electron beam by a thin aluminum slab as a function of three different incidence (working) conditions which are of interest in electron microscopy: the probe width, the tilting angle, and the beam energy. Specifically, it is shown that, because there is a dependence on particular thresholds of the beam energy, this approach provides a clear description of the diffraction process at any energy, revealing at the same time any diversion of the beam inside the material towards directions that cannot be accounted for by other conventional methods, which is of much interest when dealing with relatively low energies and/or relatively large tilting angles.
This paper proposes a new mixed quantum mechanics (QM)-molecular mechanics (MM) approach, where MM is replaced by quantum Hamilton mechanics (QHM), which inherits the modeling capability of MM, while preserving the state-dependent nature of QM. QHM, a single mechanics playing the roles of QM and MM simultaneously, will be employed here to derive the three-dimensional quantum dynamics of diatomic molecules. The resulting state-dependent molecular dynamics including vibration, rotation and spin are shown to completely agree with the QM description and well match the experimental vibration-rotation spectrum. QHM can be incorporated into the framework of a mixed quantum-classical Bohmian method to enable a trajectory interpretation of orbital-spin interaction and spin entanglement in molecular dynamics.