Hybrid adaptive grid algorithm for wave packet propagation

Lucas R. Pettey, Róbert E. Wyatt

Róbert E. Wyatt

A new hybrid moving grid approach to wave packet dynamics is described. Exterior points within a nonrigid frame follow Lagrangian trajectories obtained by solving the hydrodynamic equations of motion. Internal grid points within one or more open windows follow non-Lagrangian adaptive paths. Within these windows, problems encountered with quantum trajectories near wave function nodes are circumvented by directly solving the moving path transform of the Schrödinger equation. Excellent results are obtained for evolution of the density in a double well potential even though multiple ripples develop in the density.

Róbert E. Wyatt, Eric R. Bittner

The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.

Hans‐Dieter Meyer, Uwe Manthe, Jasna Kučar +1 more

David J. Tannor, Anna Besprozvannaya, Carl J. Williams

Two significant developments in the formulation of the equations of motion in the interaction representation (IR) are introduced in this paper. These developments have direct bearing on the efficient propagation in the interaction representation. The first of these developments involves the concept of sequential, or ‘‘nested’’ interaction transformations on the Hamiltonian and the wave function. Two such nested IRs are described. The first is a momentum- or ‘‘P-adapted’’ IR which, in addition to removing wave packet spreading, systematically maintains the average momentum 〈P〉 at zero. This facilitates a grid which not only has a minimal range, but whose points are maximally sparse. Evaluating the Hamiltonian operator in the P-adapted interaction representation involves only one simple algorithmic modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈P〉t/m) eiH0t/ℏ, i.e., using the potential energy function which is downstream from the wave function by the classical propagation distance. A PR-adapted IR is also defined, which systematically maintains both the average position 〈R〉 and momentum 〈P〉 at zero. The PR-adapted IR also involves only one modification to the ordinary IR: replacing eiH0t/ℏV(R)e−iH0t /ℏ→eiH0t/ℏV(R+〈R〉 +〈P〉t/m)eiH0t/ℏ. The second formal development consists of the application of the operator equation eiH0t/ℏV(R)e−iH0t /ℏ=V(R+Pt/m) in conjunction with the nested IRs. This formal identity, which is essentially the Heisenberg representation for V(R), provides a convenient route to propagation in the interaction representation without reconstructing the Schrödinger wave function as an intermediate. The time propagation in all representations is performed using an iterative Lanczos reduction scheme combined with a second- order Magnus expansion. Numerical results for the exponential and the 1/R potential are presented, and illustrate that the new representations can easily lead to a savings of an order of magnitude in the size of the grid required for the propagation.