Moving Boundary Truncated Grid Method for Wave Packet Dynamics

Abstract

The moving boundary truncated grid method is developed to significantly reduce the number of grid points required for wave packet propagation. The time-dependent Schrödinger equation (TDSE) and the imaginary time Schrödinger equation (ITSE) are integrated using an adaptive algorithm which economizes the number of grid points. This method employs a variable number of grid points in the Eulerian frame (grid points fixed in space) and adaptively defines the boundaries of the truncated grid. The truncated grid method is first applied to the time integration of the TDSE for the photodissociation dynamics of NOCl and a three-dimensional quantum barrier scattering problem. The time-dependent truncated grid precisely captures the wave packet evolution for the photodissociation of NOCl and finely adjusts according to the process of the wave packet bifurcation into reflected and transmitted components for the barrier scattering problem. The truncated grid method is also applied to the time integration of the ITSE for the eigenstates of quantum systems. Compared to the full grid calculations, the truncated grid method requires fewer grid points to achieve high accuracy for the stationary state energies and wave functions for a two-dimensional double well potential and the Ar trimer. Therefore, the truncated grid method demonstrates a significant reduction in the number of grid points needed to perform accurate wave packet propagation governed by the TDSE or the ITSE.

Moving boundary truncated grid method: Multidimensional quantum dynamics

“…We first briefly review the moving boundary TG method for quantum wave packet propagation . As an example, the 2D TDSE governing the time evolution of a wave packet ψ ( x , y , t ) is given by …”

“…In a recent study by Lee and Chou, the moving boundary truncated grid (TG) method was developed to greatly decrease the total number of grid points needed for propagating the complex‐valued wave function. This approach employs a variable number of Eulerian grid points and adaptively defines the boundaries of the TG.…”

“…With the success of computational results from the previous investigation, the focus of this study concerns applications of the moving boundary TG method to the Henon‐Heiles potential and wave packet barrier scattering in two, three, and four dimensions. For the Henon‐Heiles potential, the TG method can capture the complicated dynamics of a time‐evolving wave packet.…”

A time-reversible integrator for the time-dependent Schrödinger equation on an adaptive grid

“…The system of differential and algebraic Eqs. (25), (26), (27) for ψ t , q t , p t is equivalent to and, hence, can be replaced with the system of differential Eqs. (25), (40), (41).…”

“…20 Another way of reducing the required computational resources is by appropriately truncating a lattice of Gaussian basis functions [21][22][23][24] or a set of grid points. 25,26 In particular, sparse-grid methods 10,[27][28][29] reduce the number of required grid points, e.g., by employing the Smolyak quadrature. 30 Making the grid adaptive 31 is another coma) Electronic mail: jiri.vanicek@epfl.ch mon approach to reduce the required number of grid points.…”