^{1}and Randall S. Dumont

^{1,a)}

### Abstract

This article presents a new approach to long time wave packet propagation. The methodology relies on energy domain calculations and an on-the-surface straightforward energy to time transformation to provide wave packet time evolution. The adaptive bisection fast Fourier transform method employs selective bisection to create a multiresolution energy grid, dense near resonances. To implement fast Fourier transforms on the nonuniform grid, the uniform grid corresponding to the finest resolution is reconstructed using an iterative interpolation process. By proper choice of the energy grid points, we are able to produce results equivalent to grids of the finest resolution, with far fewer grid points. We have seen savings 20-fold in the number of eigenfunction calculations. Since the method requires computation of energy eigenfunctions, it is best suited for situations where many wave packet propagations are of interest at a fixed small set of points—as in time dependent flux computations. The fast Fourier transform (FFT) algorithm used is an adaptation of the Danielson-Lanczos FFT algorithm to sparse input data. A specific advantage of the adaptive bisection FFT is the possibility of long time wave packet propagations showing slow resonant decay. A method is discussed for obtaining resonance parameters by least squares fitting of energy domain data. The key innovation presented is the means of separating out the smooth background from the sharp resonance structure.

The authors thank Paul W. Ayers for helpful discussions on data smoothing, Stephen Lam for help in preparation of the plots, and the Natural Sciences and Engineering Research Council of Canada for financial support.

I. INTRODUCTION

II. METHODOLOGY

A. Energy representation of initial wave packet

B. Wave packet time evolution

C. Adaptive bisection fast Fourier transformation

III. NUMERICAL RESULTS

A. Convergence studies

B. Long time propagation

C. Characterization of resonances

IV. SUMMARY

### Key Topics

- Fourier transforms
- 20.0
- Interpolation
- 10.0
- Eigenvalues
- 4.0
- Molecular dynamics
- 3.0
- Time series analysis
- 3.0

## Figures

Adiabatic potentials for the two level avoided crossing system of Eq. (12).

Adiabatic potentials for the two level avoided crossing system of Eq. (12).

Log plot of the transmitted flux, for a wave packet with and . The solid and dashed lines are for ABFFT (coarse grid of energies plus five stages of bisection, bisection ) and simple FFT (uniform grid of energies), respectively. The results from ABFFT overlap the results from a uniform grid of energies, with difference less than the thickness of the line. The dotted line is wave packet propagation data from Ref. 14.

Log plot of the transmitted flux, for a wave packet with and . The solid and dashed lines are for ABFFT (coarse grid of energies plus five stages of bisection, bisection ) and simple FFT (uniform grid of energies), respectively. The results from ABFFT overlap the results from a uniform grid of energies, with difference less than the thickness of the line. The dotted line is wave packet propagation data from Ref. 14.

Log plot of the reflected flux for a wave packet of and . Results obtained using ABFFT (same grid as in Fig. 2) and wave packet propagation data from Ref. 14—solid and dotted lines, respectively. Again, the difference between these ABFFT results and those for a uniform grid with energies is less than the thickness of the line. In fact, in this case a uniform grid of energies is sufficient to reproduce these results.

Log plot of the reflected flux for a wave packet of and . Results obtained using ABFFT (same grid as in Fig. 2) and wave packet propagation data from Ref. 14—solid and dotted lines, respectively. Again, the difference between these ABFFT results and those for a uniform grid with energies is less than the thickness of the line. In fact, in this case a uniform grid of energies is sufficient to reproduce these results.

As in Fig. 3, except , both reflected and transmitted fluxes are shown—panels (a) and (b), respectively. Note that the uniform grid of energies does not reproduce the higher resolution data.

As in Fig. 3, except , both reflected and transmitted fluxes are shown—panels (a) and (b), respectively. Note that the uniform grid of energies does not reproduce the higher resolution data.

As in Fig. 4, except and the dotted line is for a uniform grid of energies, i.e., there is now discernible but small difference between these data. Note that wave packet propagation data match the uniform grid data almost perfectly in this case.

As in Fig. 4, except and the dotted line is for a uniform grid of energies, i.e., there is now discernible but small difference between these data. Note that wave packet propagation data match the uniform grid data almost perfectly in this case.

Log plot of the long time reflected flux up to , for a wave packet with and , obtained using ABFFT (same grid as in Fig. 2) and a uniform grid of energies—solid and dotted lines, respectively.

Log plot of the long time reflected flux up to , for a wave packet with and , obtained using ABFFT (same grid as in Fig. 2) and a uniform grid of energies—solid and dotted lines, respectively.

As in Fig. 6, except for transmitted flux, and only the ABFFT results (same grid as in Fig. 2) are shown.

As in Fig. 6, except for transmitted flux, and only the ABFFT results (same grid as in Fig. 2) are shown.

(solid line) and the base line (dotted line) as a function of energy, evaluated at . 13 resonances with mixed Lorentzian and dispersion profiles are evident (just barely). The inset blows up the first resonance and reveals the associated base line, not visible in larger energy range plot.

(solid line) and the base line (dotted line) as a function of energy, evaluated at . 13 resonances with mixed Lorentzian and dispersion profiles are evident (just barely). The inset blows up the first resonance and reveals the associated base line, not visible in larger energy range plot.

## Tables

Mean relative error in transmitted flux: (A) from ABFFT, using a uniform coarse grid of 8192 energies (stage 1) and extra points from bisections (i.e., higher stages) using and (B) from simple FFT using the corresponding uniform grid.

Mean relative error in transmitted flux: (A) from ABFFT, using a uniform coarse grid of 8192 energies (stage 1) and extra points from bisections (i.e., higher stages) using and (B) from simple FFT using the corresponding uniform grid.

Mean relative error in transmitted flux for different coarse grids and higher bisection stages, with resolution up to the resolution of a uniform fine grid of energies. Bisection .

Mean relative error in transmitted flux for different coarse grids and higher bisection stages, with resolution up to the resolution of a uniform fine grid of energies. Bisection .

Mean relative error in transmitted flux for different bisection thresholds using a fixed uniform coarse grid of energies plus five stages of bisection.

Mean relative error in transmitted flux for different bisection thresholds using a fixed uniform coarse grid of energies plus five stages of bisection.

Mean relative error in reflected and transmitted flux for wave packets with and 9, for a coarse grid of energies, a bisection threshold of 5% and 5 stages of bisections. Total number of .

Mean relative error in reflected and transmitted flux for wave packets with and 9, for a coarse grid of energies, a bisection threshold of 5% and 5 stages of bisections. Total number of .

The resonant and nonresonant integrated transmitted flux densities for wave packets with , 7, and 9.

The resonant and nonresonant integrated transmitted flux densities for wave packets with , 7, and 9.

Resonance parameters. (Only the intrinsic parameters are reported. The amplitude depends on the wavepacket.)

Resonance parameters. (Only the intrinsic parameters are reported. The amplitude depends on the wavepacket.)

Article metrics loading...

Full text loading...

Commenting has been disabled for this content