Schrödinger–Langevin equation with quantum trajectories for photodissociation dynamics

2018

J. Phys. Chem. A

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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.

Section: Computational Resultsmentioning

confidence: 99%

Moving Boundary Truncated Grid Method for Wave Packet Dynamics

Lee

2018

J. Phys. Chem. A

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“…Because Bohmian trajectories describe the flow of probability density in quantum mechanics, − they can serve as an optimal adaptive moving grid for integrating the TDSE . The real-valued and complex-valued quantum trajectory methods have been developed to study a diverse spectrum of physical processes. − Electronic nonadiabatic dynamics has been analyzed using the semiclassical quantum trajectory method. − Quantum trajectories have been propagated under the influence of an approximate quantum potential. − The bipolar expansion of the total wave function has been employed to circumvent the node problem during the evolution of quantum trajectories. − The complex quantum trajectory method has been applied to a diverse range of phenomena, including barrier scattering problems and nonadiabatic molecular dynamics. − Based upon ideas from computational fluid dynamics, arbitrary Lagrangian–Eulerian methods have been developed to efficiently solve the TDSE. − Moving boundary conditions determined from the quantum trajectory method have been applied to a fixed grid for integration of the TDSE. − …”

Section: Introductionmentioning

confidence: 99%

Moving Boundary Truncated Grid Method: Application to the Time Evolution of Distribution Functions in Phase Space

Lee

2020

J. Phys. Chem. A

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The moving boundary truncated grid (TG) method, previously developed to integrate the time-dependent Schrödinger equation and the imaginary time Schrödinger equation, is extended to the time evolution of distribution functions in phase space. A variable number of phase space grid points in the Eulerian representation are used to integrate the equation of motion for the distribution function, and the boundaries of the TG are adaptively determined as the distribution function evolves in time. Appropriate grid points are activated and deactivated for propagation of the distribution function, and no advance information concerning the dynamics in phase space is required. The TG method is used to integrate the equations of motion for phase space distribution functions, including the Klein–Kramers, Wigner–Moyal, and modified Caldeira–Leggett equations. Even though the initial distribution function is nonnegative, the solutions to the Wigner–Moyal and modified Caldeira–Leggett equations may develop negative basins in phase space originating from interference effects. Trajectory-based methods for propagation of the distribution function do not permit the formation of negative regions. However, the TG method can correctly capture the negative basins. Comparisons between the computational results obtained from the full grid and TG calculations demonstrate that the TG method not only significantly reduces the computational effort but also permits accurate propagation of various distribution functions in phase space.

Quantum‐classical transition of the dissipative wave packet dynamics for barrier scattering

Int J of Quantum Chemistry

2018

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A nonlinear equation based on the Schrödinger–Langevin equation is employed to analyze the quantum‐classical transition of dissipative systems for wave packet barrier scattering. The transition equation is obtained by subtracting a modified quantum potential term depending on a degree of quantumness in the Schrödinger–Langevin equation, and it provides a continuous description for the transition process of dissipative systems from purely quantum to purely classical regimes. Important properties possessed by the equation are derived. The energy dissipation rate of the dissipative system does not depend on the degree of quantumness. It is shown that the equations of motion for the expectation values of the position and momentum operators for dissipative systems hold for all the degree of quantumness. This feature establishes the correspondence between classical and quantum dissipative systems. Computational results for the transition process of dissipative wave packet dynamics and transition trajectories are presented and analyzed for model systems involving the propagation of a free Gaussian wave packet and the wave packet scattering from an Eckart barrier. Computational results demonstrate that the agreement between the classical‐limit transition trajectories and classical trajectories is excellent, except in some regions. The single‐valued transition wave function implies that transition trajectories cannot pass through the same configuration space point at the same time.