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Physical Review A

Classical interpretation of probability oscillations in low-energy atomic collisions

Pierre Botheron, Bernard Pons

Published Jun 15, 2011• 9 citations
Publisher

Abstract

We perform close-coupling molecular and classical trajectory Monte Carlo (CTMC) calculations of charge-exchange probabilities in low-energy ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{2+}+\mathrm{H}$ collisions. We consider head-on collisions with a zero impact parameter and study the dependence of the charge-exchange probability on the impact velocity. This probability exhibits oscillations generally understood as signatures of quantum phase effects. CTMC calculations with dimensionality reduced to one fairly reproduce the oscillations and show that these latter stem from the back and forth motion of the electron between the nuclear centers in the internuclear region. In three dimensions, the probability oscillations are generally washed out in the statistical CTMC framework because of the electron transverse degrees of freedom. However three-dimensional Bohmian trajectories ascertain the classical interpretation of the oscillations in a quantum framework.

DOI
10.1103/physreva.83.062704

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A connection between quantum mechanics and classical methods used in ion-atom collisions is developed. The connection is based on the fluid dynamical interpretation of the one-electron quantum wavefunction. The classical trajectory Monte Carlo method and the Thomson approximation are obtained as quantum models and the wavefunctions associated with these approximations are defined. A discussion is given of the procedure to be used to calculate scattering cross sections. Scaling laws derivable from the classical Liouville equation are deduced in the impact parameter formalism.
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We use an algebraic method to model the molecular collision dynamics of a collinear triatomic system. Beginning with a forced oscillator, we develop a mathematical framework upon which inelastic and reactive collisions are modeled. The model is considered algebraic because it takes advantage of the properties of a Lie algebra in the derivation of a time-evolution operator. The time-evolution operator is shown to generate both phase-space and quantum dynamics of a forced oscillator simultaneously. The model is considered semi-classical because only the molecule's internal degrees-of-freedom are quantized. The relative translation between the colliding atom and molecule in an exchange reaction (AB+C ->A+BC) contains no bound states and any possible tunneling is neglected so the relative translation is treated classically. The purpose of this dissertation is to develop a working model for the quantum dynamics of a collinear reactive collision. After a reliable model is developed we apply statistical mechanics principles by averaging collisions with molecules in a thermal bath. The initial Boltzmann distribution is of the oscillator energies. The relative velocities of the colliding particles is considered a thermal average. Results are shown of quantum transition probabilities around the transition state that are highly dynamic due to the coupling between the translational and transverse coordinate.
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Chemical dynamics, in principle, should be understood by solving the time-dependent Schrödinger equation for a molecular system, describing motion of the nuclei and electrons. However, the computational efforts to solve this partial second-order differential equation scales exponentially with the system size, which prevents us from getting exact numerical solutions for systems larger than 4-5 atoms. Thus, approximations simplifying the picture are necessary. The so-called Born-Oppenheimer approximation, separating motion of the electrons and nuclei is the central one: solution to the electronic Schrödinger equation defines the potential energy surface on which the nuclear motion unfolds, and there are standard quantum chemistry software packages for solving the electronic Schrödinger equation. For the nuclear Schrödinger equation, however, there are no widely applicable quantum-mechanical approaches, and most simulations are performed using classical Newtonian mechanics which is often adequate due to large nuclear masses. However, the nuclear quantum effects are significant for chemical processes involving light nuclei at low energies, and including these effects into simulation, even approximately, is highly desirable. In this dissertation, an approximate methodology of including quantum-mechanical effects within the quantum trajectory or the de Broglie-Bohm formulation of the Schrödinger equations is developed. Use of the trajectory framework makes the approach scalable to hundreds of degrees of freedom. The methodology is applied to study high-dimensional systems (solid He4 and others) relevant to chemistry.