### Abstract

In a previous publication [J. Chem. Phys. 118, 9911 (2003)], the derivative propagation method (DPM) was introduced as a novel numerical scheme for solving the quantum hydrodynamic equations of motion (QHEM) and computing the time evolution of quantum mechanical wave packets. These equations are a set of coupled, nonlinear partial differential equations governing the time evolution of the real-valued functions C and S in the complex action, S- =C (r,t) +iS (r,t) Ψ, where ψ (r,t) =exp (S-). Past numerical solutions to the QHEM were obtained via ensemble trajectory propagation, where the required first- and second-order spatial derivatives were evaluated using fitting techniques such as moving least squares. In the DPM, however, equations of motion are developed for the derivatives themselves, and a truncated set of these are integrated along quantum trajectories concurrently with the original QHEM equations for C and S. Using the DPM quantum effects can be included at various orders of approximation; no spatial fitting is involved; there is no basis set expansion; and single, uncoupled quantum trajectories can be propagated (in parallel) rather than in correlated ensembles. In this study, the DPM is extended from previous one-dimensional (1D) results to calculate transmission probabilities for 2D and 3D wave packet evolution on coupled Eckart barrier/harmonic oscillator surfaces. In the 2D problem, the DPM results are compared to standard numerical integration of the time-dependent Schrödinger equation. Also in this study, the practicality of implementing the DPM for systems with many more degrees of freedom is discussed.

Original language | English |
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Article number | 164104 |

Journal | Journal of Chemical Physics |

Volume | 122 |

Issue number | 16 |

DOIs | |

State | Published - Apr 22 2005 |

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## Cite this

*Journal of Chemical Physics*,

*122*(16), [164104]. https://doi.org/10.1063/1.1884606