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Journal of Physics Conference Series

Combination of the final element and quantum trajectory methods for numerical solution of the wave equation

V.A. Khodyrev

Published Apr 10, 2014• 0 citations
Publisher

Abstract

For description of ion-atom collisions the final elements approach is applied in regions close to the nuclei and in the quantum trajectory method is applied in the outer region. Applicability of the quasi-classics at the boundary provide a possibility to join two solutions. The method is computationally efficient and also provides detailed insight into the phenomenon.

DOI
10.1088/1742-6596/488/8/082022

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For description of ion-atom collisions the final elements approach is applied in regions close to the nuclei and in the quantum trajectory method is applied in the outer region. Applicability of the quasi-classics at the boundary provide a possibility to join two solutions. The method is computationally efficient and also provides detailed insight into the phenomenon.
We present the results of a quantum calculation designed to discuss the dynamics of an ion–atom collision at intermediate velocities. The description is based on a quantum fluid picture. The time-dependent Schrodinger equation is integrated numerically for a model proton–hydrogen collision and quantum trajectories are computed. By these means, we obtain a pictural view of the scattering process. Analysis of the resulting trajectories yields insights into the dynamics of the physical phenomena involved, with special emphasis on ionization.
The quantum trajectory method (QTM) is extended to the dynamics of electronic nonadiabiatic collisions. Equations of motion are first derived for the probability density, velocity, and action function for wave packets moving on each of the coupled electronic potential surfaces. These discretized equations are solved in the Lagrangian (moving with the fluid) picture to give the trajectory dynamics of fluid elements evolving on each potential surface. This trajectory method is fully quantum mechanical and does not involve “trajectory surface hopping.” The method is applied to nonadiabiatic collision models involving two coupled electronic states. The quantum trajectory results are in excellent agreement with solutions computed (using space-fixed grid methods) directly from the time-dependent Schrödinger equation.
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.
We consider computational methods for simulating the dynamics of molecular systems governed by the time-dependent Schrödinger equation. Solving the Schrödinger equation numerically poses a challenge due to its often highly oscillatory solutions, and to the exponential growth of work and memory with the number of particles in the system. Two different classes of problems are studied: the dynamics of the nuclei in a molecule and the dynamics of an electron in orbit around a nucleus. For the first class of problems we present new computational methods which exploit the relation between quantum and classical dynamics in order to make the computations more efficient. For the second class of problems, the lack of regularity in the solution poses a computational challenge. Using knowledge of the non-smooth features of the solution we construct a new method with two orders higher accuracy than what is achieved by direct application of a difference stencil.
This series of papers is concerned with the derivation of the equations of the classical picture of atomic collisions, $i\ensuremath{\hbar}\frac{d}{\mathrm{dt}}{d}_{i}(t)=\ensuremath{\Sigma}\stackrel{}{j}{h}_{\mathrm{ij}}(t){d}_{j}(t),$ which describe the "time" dependence of electronic-quantum-state amplitudes as the nuclei move along a classical trajectory. These equations are derived in two ways. In the first formulation, which coincides with the intuitive classical picture of the collision, the nuclear part of the wave function is treated as a superposition of narrow wave packets, each traveling along a classical trajectory. In the second formulation, a semiclassical approach is used. The validity and meaning of the two formulations are discussed and compared.