Modified quantum trajectory dynamics using a mixed wave function representation (original) (raw)
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Energy conserving approximations to the quantum potential: Dynamics with linearized quantum force
The Journal of Chemical Physics, 2004
Solution of the Schrödinger equation within the de Broglie-Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives exact time-evolution of a correlated Gaussian wave function in a locally quadratic potential. The method is computationally cheap in many dimensions, conserves total energy and satisfies the criterion on the average quantum force. Expectation values are readily found by summing over trajectory weights. Efficient extraction of the phase-dependent quantities is discussed. We illustrate the efficiency and accuracy of the linear quantum force approximation by examining a one-dimensional scattering problem and by computing the wavepacket reaction probability for the hydrogen exchange reaction and the photodissociation spectrum of ICN in two dimensions.
The Journal of Chemical Physics, 2003
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential ͑AQP͒. One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman-Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
Approximate quantum trajectory dynamics for reactive processes in condensed phase
Molecular Simulation, 2014
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Semiclassical dynamics based on quantum trajectories
Chemical Physics Letters, 2002
We present a trajectory-based method that incorporates quantum effects in the context of Hamiltonian dynamics. It is based on propagation of trajectories in the presence of quantum potential within the hydrodynamic formulation of the Schr€ o odinger equation. The quantum potential is derived from the density approximated as a linear combination of gaussian functions. One-gaussian fit gives exact result for parabolic potentials, as do successful semiclassical methods. The limit of the large number of fitting gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum, as demonstrated on a transmission through the Eckart barrier.
An introduction to the problem of bridging quantum and classical dynamics
The European Physical Journal Special Topics, 2015
Simulating the exact quantum dynamics of realistic interacting systems is presently a task beyond reach but for the smallest of them, as the numerical cost for solving the time-dependent Schrödinger equation scales exponentially with the number of degrees of freedom. Mixed quantum-classical methods attempt to solve this problem by starting from a full quantum description of the system and subsequently partitioning the degrees of freedom in two subsets: the quantum subsystem and the bath. A classical limit is then taken for the bath while preserving, at least approximately, the quantum evolution of the subsystem. A key, as yet not fully resolved, theoretical question is how to do so by constructing a consistent description of the overall dynamics. An exhaustive review of this class of methods is beyond the scope of this paper and we shall limit ourselves to present, as an example, a specific approach, known as the LANDM-Map method. The method stems from an attempt at taking a rigorous limit for the classical degrees of freedom starting from a path integral formulation of the full quantum problem. The results that we discuss are not new, but our intent here is to present them as an introduction to the problem of mixed quantum classical dynamics. We shall also indicate a broad classification of the available approaches, their limitations, and some open questions in this field.
Stabilization of Quantum Energy Flows within the Approximate Quantum Trajectory Approach †
The Journal of Physical Chemistry A, 2007
The hydrodynamic, or the de Broglie-Bohm, formulation provides an alternative to the conventional time-dependent Schrödinger equation based on quantum trajectories. The trajectory dynamics scales favorably with the system size, but it is, generally, unstable due to singularities in the exact quantum potential. The approximate quantum potential based on the fitting of the nonclassical component of the momentum operator in terms of a small basis is numerically stable but can lead to inaccurate large net forces in bound systems. We propose to compensate errors in the approximate quantum potential by applying a semiempirical friction-like force. This significantly improves the description of zero-point energy in bound systems. Examples are given for one-dimensional models relevant to nuclear dynamics.
Quantum Dynamics with the Quantum Trajectory-Guided Adaptable Gaussian Bases
Journal of Chemical Theory and Computation, 2019
The computational cost of describing a general quantum system fully coupled by anharmonic interactions scales exponentially with the system size. Thus, an efficient basis representation of wave functions is essential, and when it comes to the large-amplitude motion of high-dimensional systems, the dynamic bases of Gaussian functions are often employed. The time dependence of such bases is determined from the variational principle or from classical dynamics; the former is challenging in implementation due to singular matrices, while the latter may not cover the configuration space relevant to quantum dynamics. Here we describe a method using Quantum Trajectory-guided Adaptable Gaussian (QTAG) bases "tuned"including the basis position, phase, and widthto the wave function evolution, thanks to the continuity of the probability density in the course of the quantum trajectory dynamics. Thus, an efficient basis in configuration space is generated, bypassing the variational equations on the parameters of the Gaussians. We also propose a time propagator with basis transformation by projections which lends efficiency and stability to the QTAG dynamics, as demonstrated on standard tests and the ammonia inversion model.
Semiclassical nonadiabatic dynamics using a mixed wave-function representation
The Journal of Chemical Physics, 2005
Nonadiabatic effects in quantum dynamics are described using a mixed polar/coordinate space representation of the wave function. The polar part evolves on dynamically determined potential surfaces that have diabatic and adiabatic potentials as limiting cases of weak localized and strong extended diabatic couplings. The coordinate space part, generalized to a matrix form, describes transitions between the surfaces. Choice of the effective potentials for the polar part and partitioning of the wave function enables one to represent the total wave function in terms of smooth components that can be accurately propagated semiclassically using the approximate quantum potential and small basis sets. Examples are given for two-state one-dimensional problems that model chemical reactions that demonstrate the capabilities of the method for various regimes of nonadiabatic dynamics.
Trajectory Based Simulations of Quantum-Classical Systems
2009
In this Chapter we review the core ingredients of a class of mixed quantum-classical methods that can naturally account for quantum coherence effects. In general, quantum-classical schemes partition degrees of freedom between a quantum subsystem and an environment. The various approaches are based on different approximations to the full quantum dynamics, in particular in the way they treat the environment. Here we compare and contrast two such methods: the Quantum Classical Liouville (QCL) approach, and the Iterative Linearized Density Matrix (ILDM) propagation scheme. These methods are based on evolving ensembles of surface-hopping trajectories in which the ensemble members carry weights and phases and their contributions to time-dependent quantities must be added coherently to approximate interference effects. The side by side comparison we offer highlights similarities and differences between the two approaches and serves as a starting point to explore more fundamental connections between such methods. The methods are applied to compute the evolution of the density matrix of a challenging condensed phase model system in which coherent dynamics plays a critical role: the asymmetric spin-boson. Various implementation questions are addressed.
Progress in the Theory of Mixed Quantum-Classical Dynamics
Annual Review of Physical Chemistry, 2006
▪ Quantum-classical Liouville dynamics can be used to study the properties of open quantum systems that are coupled to bath or environmental degrees of freedom whose dynamics can be approximated by classical equations of motion. In contrast to many open quantum system approaches, quantum-classical dynamics provides a detailed description of the bath molecules. Such a description is especially appropriate for the study of quantum rate processes, such as proton and electron transport, where the detailed dynamics of the bath has a strong influence on the quantum rate. The quantum-classical Liouville equation can also serve as a starting point for the derivation of reduced descriptions where all or some of the bath degrees of freedom are projected out. Quantum-classical Liouville dynamics can be simulated in terms of an ensemble of surface-hopping trajectories whose character differs from that in other surface-hopping schemes. The results of studies of proton transfer in condensed pha...