TY - JOUR

T1 - Multidimensional quantum trajectories

T2 - Applications of the derivative propagation method

AU - Trahan, Corey J.

AU - Wyatt, Robert E.

AU - Poirier, Bill

N1 - Funding Information:
This work was supported in part by both the Welch and National Science Foundations. Special thanks are given to L. Pettey and D. Babyuk for many enlightening conversations.

PY - 2005/4/22

Y1 - 2005/4/22

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21244432373&partnerID=8YFLogxK

U2 - 10.1063/1.1884606

DO - 10.1063/1.1884606

M3 - Article

AN - SCOPUS:21244432373

SN - 0021-9606

VL - 122

JO - The Journal of Chemical Physics

JF - The Journal of Chemical Physics

IS - 16

M1 - 164104

ER -