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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Solution of the quantum fluid dynamical equations with radial basis function interpolation

Xu-Guang Hu, Tak‐San Ho, HERSCHEL A. RABITZ, Attila Aşkar

Published May 1, 2000• 30 citations
Publisher

Abstract

The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation psi = exp(R + iS)/[symbol: see text]. The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density rho = magnitude of psi 2 and on the structure of R = [symbol: see text]/2 ln rho generally being simpler and smoother than rho. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool.

DOI
10.1103/physreve.61.5967

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Transactions of Beijing Institute of Technology•Jan 1, 2010•0 citations
The paper proposes a numerical technique within the Lagrangian description for propagating the quantum fluid dynamical (QFD) equations in terms of the Madelung field variables R and S, which are connected to the wave function via the transformation psi = exp(R + iS)/[symbol: see text]. The technique rests on the QFD equations depending only on the form, not the magnitude, of the probability density rho = magnitude of psi 2 and on the structure of R = [symbol: see text]/2 ln rho generally being simpler and smoother than rho. The spatially smooth functions R and S are especially suitable for multivariate radial basis function interpolation to enable the implementation of a robust numerical scheme. Examples of two-dimensional model systems show that the method rivals, in both efficiency and accuracy, the split-operator and Chebychev expansion methods. The results on a three-dimensional model system indicates that the present method is superior to the existing ones, especially, for its low storage requirement and its uniform accuracy. The advantage of the new algorithm is expected to increase for higher dimensional systems to provide a practical computational tool.
This paper considers the practical utility of quantum fluid dynamics (QFD) whereby the time-dependent Schrödinger’s equation is transformed to observing the dynamics of an equivalent “gas continuum.” The density and velocity of this equivalent gas continuum are respectively the probability density and the gradient of the phase of the wave function. The numerical implementation of the QFD equations is carried out within the Lagrangian approach, which transforms the solution of Schrödinger’s equation into following the trajectories of a set of mass points, i.e., subparticles, obtained by discretization of the continuum equations. The quantum dynamics of the subparticles which arise in the present formalism through numerical discretization are coupled by the density and the quantum potential. Numerical illustrations are performed for photodissociation of NOCl and NO2 treated as two-dimensional models. The dissociation cross sections σ(ω) are evaluated in the dramatically short CPU times of 33 s for NOCl and 40 s for NO2 on a Pentium-200 MHz PC machine. The computational efficiency comes from a combination of (a) the QFD representation dealing with the near monotonic amplitude and phase as dependent variables, (b) the Lagrangian description concentrating the computation effort at all times into regions of highest probability as an optimal adaptive grid, and (c) the use of an explicit time integrator whereby the computational effort grows only linearly with the number of discrete points.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
A new method is proposed for computing the time evolution of quantum mechanical wave packets. Equations of motion for the real-valued functions C and S in the complex action S=C(r,t)+iS(r,t)/ℏ, with ψ(r,t)=exp(S), involve gradients and curvatures of C and S. In previous implementations of the hydrodynamic formulation, various time-consuming fitting techniques of limited accuracy were used to evaluate these derivatives around each fluid element in an evolving ensemble. In this study, equations of motion are developed for the spatial derivatives themselves and a small set of these are integrated along quantum trajectories concurrently with the equations for C and S. Significantly, quantum effects can be included at various orders of approximation, no spatial fitting is involved, there are no basis set expansions, and single quantum trajectories (rather than correlated ensembles) may be propagated, one at a time. Excellent results are obtained when the derivative propagation method is applied to anharmonic potentials involving barrier transmission.
In this paper we report upon our computational methodology for numerically integrating the quantum Hamilton–Jacobi equations using hydrodynamic trajectories. Our method builds upon the moving least squares method developed by Lopreore and Wyatt [Phys. Rev. Lett. 82, 5190 (1999)] in which Lagrangian fluid elements representing probability volume elements of the wave function evolve under Newtonian equations of motion which include a nonlocal quantum force. This quantum force, which depends upon the third derivative of the quantum density, ρ, can vary rapidly in x and become singular in the presence of nodal points. Here, we present a new approach for performing quantum trajectory calculations which does not involve calculating the quantum force directly, but uses the wavefront to calculate the velocity field using mv=∇S, where S/ℏ is the argument of the wave function ψ. Additional numerical stability is gained by performing local gauge transformations to remove oscillatory components of the wave function. Finally, we use a dynamical Rayleigh–Ritz approach to derive ancillary equations-of-motion for the spatial derivatives of ρ, S, and v. The methodologies described herein dramatically improve the long time stability and accuracy of the quantum trajectory approach even in the presence of nodes. The method is applied to both barrier crossing and tunneling systems. We also compare our results to semiclassical based descriptions of barrier tunneling.
We explore the usefulness of a quantum fluid dynamics (QFD) approach for quantitative electronic structure calculations of many-electron systems. By combining QFD and density functional theory, a single time-dependent nonlinear QFD equation can be derived. The equation is further transformed into a diffusion-type form by an imaginary-time evolution method, whose asymptotic solution reaches a global minimum and the many-body ground-state wavefunction. The time-dependent generalized pseudospectral method is extended to solve the diffusion equation in spherical coordinates, allowing optimal and nonuniform spatial discretization and accurate and efficient solution of the diffusion function in space and time. The procedure is applied to the study of electronic energies, densities and other ground-state properties of noble gas atoms (He, Ne, Ar, Kr, Xe). The results are in good agreement with other best available values. The method offers a conceptually appealing and computationally practical procedure for the treatment of many-electron systems beyond the Hartree-Fock level.
The quantum hydrodynamics dynamics (QHD) is introduced to study the probability of electron through the tunneling junction. The complex form of the wave function was transformed into polar form, the time evolution equations of C and S were given in the Lagrange framework, and the explicit central differencing scheme was adopted for time integration of the QHD equations. In the Lagrangian framework, the grid points of the system will inevitable become scattered and unstructured during the evolution, the spatially smooth functions C and S are especially suitable for rational function approximation method to enable the implementation of a robust numerical scheme. As an example, the dynamics of a Gaussian wave packet is calculated and results are presented in the figures. In these figures, wave packet penetration of the barrier can be clearly observed, showing the quantum effect obviously.