Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Efficient quantum trajectory representation of wavefunctions evolving in imaginary time

Sophya V. Garashchuk, James William Mazzuca, Tijo Vazhappilly

Published Jul 19, 2011• 18 citations
Publisher

Abstract

The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential--computed approximately to be practical in high-dimensional systems--influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys. 132, 014112 (2010)]. For a nodeless wavefunction represented as ψ(x, t) = exp(-S(x, t)/ħ) with the trajectory momenta defined by ∇S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.

DOI
10.1063/1.3610165

Related papers

Multidimensional Quantum Trajectory Dynamics in Imaginary Time with Approximate Quantum Potential

Sophya V. Garashchuk, Tijo Vazhappilly

The quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential [ J. Chem. Phys. 2010, 132, 014112], which can be used for calculations of low-lying energy eigenstates or for Boltzmann evolution, is extended to many dimensions. Evolution of a nodeless initial wave function represented in exponential form is described in terms of a trajectory ensemble. In the Lagrangian frame of reference the trajectories evolve according to the classical equations of motion under the influence of the inverted classical potential and the quantum potential defined by the gradient of the trajectory momenta. The initial wave function is sampled by the trajectories randomly, which makes the approach scalable to high dimensions. Another practical feature is approximate computation of the quantum potential and force from the global least squares fit of the trajectory momenta in a small basis. The importance sampling and the Eulerian frame-of-reference formulation, compensating for divergent dynamics in bound potentials, are introduced to reduce the number of trajectories in multidimensional applications. The zero-point energies are computed for model and chemical (H 2 O and SO 2 ) systems.

The Journal of Physical Chemistry C•Sep 10, 2010•16 citations

Quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential

Sophya V. Garashchuk

The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodeless wave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described in the mixed coordinate space/trajectory representation at little additional computational cost. The nodeless wave function, represented by the trajectory ensemble, decays to the ground state. The mixed representation wave functions, with lower energy contributions projected out at each time step, decay to the excited energy states. The approach, illustrated by computing energy levels for anharmonic oscillators and energy level splitting for the double-well potential, can be used for the Boltzmann operator evolution.

The Journal of Chemical Physics•Jan 7, 2010•30 citations

Approximate Quantum Trajectory Dynamics in Imaginary and Real Time; Calculation of Reaction Rate Constants

In this chapter we formulate quantum trajectory dynamics in imaginary time which seamlessly connects to Bohmian dynamics in real time. Approximate determination of the quantum potential and mixed representation of wavefunctions with nodes make this trajectory approach stable and scalable to high dimensions, while including the T&F Cat # K11224, Chapter 23, Page 362, 15-9-2010 leading quantum effects into wavefunction evolution. Reaction probabilities, energy eigenstates and reaction rate constants can be obtained by propagating an ensemble of quantum trajectories in real and/or imaginary time.

CRC Press eBooks•Apr 19, 2016•0 citations

Quantum Field Dynamics from Trajectories

Peter Holland

Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.

CRC Press eBooks•Apr 19, 2016•3 citations

Theoretical methods for the study of quantum dynamics

Jonathan E. Chen

Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm's hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton's principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.

•May 19, 2010•0 citations

Quantum Dynamics with the Quantum Trajectory-Guided Adaptable Gaussian Bases

Matthew Dutra, Sachith Wickramasinghe, Sophya V. Garashchuk

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 width-to 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.

Journal of Chemical Theory and Computation•Dec 4, 2019•22 citations
The Boltzmann evolution of a wavefunction can be recast as imaginary-time dynamics of the quantum trajectory ensemble. The quantum effects arise from the momentum-dependent quantum potential--computed approximately to be practical in high-dimensional systems--influencing the trajectories in addition to the external classical potential [S. Garashchuk, J. Chem. Phys. 132, 014112 (2010)]. For a nodeless wavefunction represented as ψ(x, t) = exp(-S(x, t)/ħ) with the trajectory momenta defined by ∇S(x, t), analysis of the Lagrangian and Eulerian evolution shows that for bound potentials the former is more accurate while the latter is more practical because the Lagrangian quantum trajectories diverge with time. Introduction of stationary and time-dependent components into the wavefunction representation generates new Lagrangian-type dynamics where the trajectory spreading is controlled improving efficiency of the trajectory description. As an illustration, different types of dynamics are used to compute zero-point energy of a strongly anharmonic well and low-lying eigenstates of a high-dimensional coupled harmonic system.
The quantum trajectory dynamics in imaginary time with the momentum-dependent quantum potential [ J. Chem. Phys. 2010, 132, 014112], which can be used for calculations of low-lying energy eigenstates or for Boltzmann evolution, is extended to many dimensions. Evolution of a nodeless initial wave function represented in exponential form is described in terms of a trajectory ensemble. In the Lagrangian frame of reference the trajectories evolve according to the classical equations of motion under the influence of the inverted classical potential and the quantum potential defined by the gradient of the trajectory momenta. The initial wave function is sampled by the trajectories randomly, which makes the approach scalable to high dimensions. Another practical feature is approximate computation of the quantum potential and force from the global least squares fit of the trajectory momenta in a small basis. The importance sampling and the Eulerian frame-of-reference formulation, compensating for divergent dynamics in bound potentials, are introduced to reduce the number of trajectories in multidimensional applications. The zero-point energies are computed for model and chemical (H 2 O and SO 2 ) systems.
The quantum trajectory dynamics is extended to the wave function evolution in imaginary time. For a nodeless wave function a simple exponential form leads to the classical-like equations of motion of trajectories, representing the wave function, in the presence of the momentum-dependent quantum potential in addition to the external potential. For a Gaussian wave function this quantum potential is a time-dependent constant, generating zero quantum force yet contributing to the total energy. For anharmonic potentials the momentum-dependent quantum potential is cheaply estimated from the global Least-squares Fit to the trajectory momenta in the Taylor basis. Wave functions with nodes are described in the mixed coordinate space/trajectory representation at little additional computational cost. The nodeless wave function, represented by the trajectory ensemble, decays to the ground state. The mixed representation wave functions, with lower energy contributions projected out at each time step, decay to the excited energy states. The approach, illustrated by computing energy levels for anharmonic oscillators and energy level splitting for the double-well potential, can be used for the Boltzmann operator evolution.
In this chapter we formulate quantum trajectory dynamics in imaginary time which seamlessly connects to Bohmian dynamics in real time. Approximate determination of the quantum potential and mixed representation of wavefunctions with nodes make this trajectory approach stable and scalable to high dimensions, while including the T&F Cat # K11224, Chapter 23, Page 362, 15-9-2010 leading quantum effects into wavefunction evolution. Reaction probabilities, energy eigenstates and reaction rate constants can be obtained by propagating an ensemble of quantum trajectories in real and/or imaginary time.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.
Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm's hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton's principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.
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 width-to 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.