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Journal of Chemical Theory and Computation

Adiabatic and Nonadiabatic Dynamics with Interacting Quantum Trajectories

Lucien Dupuy, Francesco Talotta, Federica Agostini, David Lauvergnat, Bill Poirier, Yohann Scribano

Published Oct 17, 2022• 19 citations
Publisher

Abstract

We present a quantum dynamics method based on the propagation of interacting quantum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [ Chem. Phys. 2010, 370, 4–14] and Schiff and Poirier [ J. Chem. Phys. 2012, 136, 031102] on quantum dynamics without wavefunctions. It consists of determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quantum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

DOI
10.1021/acs.jctc.2c00744

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We present a quantum dynamics method based on the propagation of interacting quantum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [ Chem. Phys. 2010, 370, 4–14] and Schiff and Poirier [ J. Chem. Phys. 2012, 136, 031102] on quantum dynamics without wavefunctions. It consists of determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quantum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.
We present a quantum dynamics method based on the propagation of interacting quan- tum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys. 370 4–14 (2010)] and Schiff and Poirier [J. Chem. Phys. 136 031102 (2012)] on quantum dynamics with- out wavefunctions. It consists in determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quan- tum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.
We present a quantum dynamics method based on the propagation of interacting quan- tum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys. 370 4–14 (2010)] and Schiff and Poirier [J. Chem. Phys. 136 031102 (2012)] on quantum dynamics with- out wavefunctions. It consists in determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quan- tum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.
In this work we present a numerical analysis of the so-called ‘interacting quantum trajectory method’ (IQTM), which is based on a formal theory developed by Poirier [Bohmian mechanics without pilot waves, Chem. Phys. 370 (2010), pp. 4–14.], and later extended by Schiff and Poirier [Communication: Quantum mechanics without wavefunctions, J. Chem. Phys. 136 (2012), p. 031102.], for representing the quantum mechanical state of a particle in terms of an ensemble of Bohmian quantum trajectories. This has led to a number of numerical algorithms over the years, mostly for time-independent applications. A numerically robust, exact time-dependent wavepacket propagation scheme would have to await the refinements of Dupuy et al. [L. Dupuy, F. Talotta, F. Agostini, D. Lauvergnat, B. Poirier, and Y. Scribano, Adiabatic and nonadiabatic dynamics with interacting quantum trajectories, J. Chem. Theory Comput. 18 (2022), pp. 6447–6462.], which serves as the starting point for our continued development in the present study. Here we will discuss the numerical protocols comprising IQTM, and present detailed error analysis for several 1D model problems involving simple free particles and oscillators. These studies reveal some key advantages of IQTM; however, they also highlight problems for which the algorithms need further work.
The method, described in the paper, incorporates ideas of the Bohmian formulation of quantum mechanics into a trajectory approach designed for systems comprised of light (quantum) and heavy (nearly classical) particles. The goal is to treat all degrees of freedom on the same footing but, for practical reasons, to include the quantum-mechanical effects into dynamics for just the light particles. The total wavefunction, taken as a product of slow and fast components representing the heavy and light particles respectively is represented in terms of trajectories. The fast component depends on the trajectory-guided configurations of the heavy particles. The initial conditions for the trajectories are defined by the full-dimensional initial wavefunction according to Bohm's prescription of associating the trajectory momentum with the gradient of the wavefunction phase. The time-dependent Schrödinger equation is solved for the quantum degrees of freedom using an ensemble of the quantum trajectories (or other time-propagation method) for each guiding trajectory of the classical subspace. Neglect of the quantum force acting on the heavy particles allows decoupling of the quantum calculations associated with different guiding trajectories. The approach allows reconstruction of the full-dimensional wavefunction and conserves the total wavefunction energy as illustrated for the vibrationally non-adiabatic model.
A novel time-dependent hybrid quantum/classical propagation scheme based on Bohmian quantum trajectories is presented. The quantum subsystem is described by a wave packet depending on the quantum variables x and, via the total potential energy of the system, parametrically on the classical trajectories X(t). The wave packet is used to calculate de Broglie–Bohm quantum trajectories x(t) which are used to calculate the force acting on the classical variables. Quantum corrections of the classical equation of motion are also included. The method is applied to a simple case of two coupled oscillators. Comparison between exact quantum and approximate results demonstrates that these MQCB (Mixed Quantum/Classical Bohmian) trajectories provide a good description of the energy exchange between the two oscillators.
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.