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The Journal of Chemical Physics

Quantum fluid dynamics in the Lagrangian representation and applications to photodissociation problems

Fernando Sales Mayor, Attila Aşkar, HERSCHEL A. RABITZ

Published Aug 8, 1999• 111 citations

Abstract

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.

DOI: 10.1063/1.479520

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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.

The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.

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.

Quantum dynamics calculations scale steeply with system size, rendering calculations on a classical computer intractable for complex systems. While quantum computing offers a natural solution, its application to nuclear quantum dynamics has been scarce. Here, we present a quantum algorithm to study photodissociation dynamics on quantum computers, benchmarked on the NOCl molecule. The wave function is propagated via a split-operator method, utilizing the quantum Fourier transform and unitary transformation matrix to switch representations. To impose outgoing boundary conditions on a truncated grid, we use a nonunitary absorbing potential propagator, implemented through a dilation scheme. The photodissociation cross section is calculated from the autocorrelation function, which is extracted using the Hadamard test. Our quantum computing results agree well with benchmarks under ideal conditions, and we further demonstrate that the algorithm is robust to noise and statistical sampling errors, indicating its promise for application of noisy devices for quantum dynamics studies.