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The Journal of Chemical Physics

Dynamics of coupled Bohmian and phase-space variables: A moment approach to mixed quantum-classical dynamics

Irène Burghardt

Published Feb 22, 2005• 27 citations
Publisher

Abstract

The theoretical framework of the mixed quantum-classical description given by Burghardt and Parlant [J. Chem. Phys. 120, 3055 (2004)] is detailed. A representation in terms of partial hydrodynamic moments is developed, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. Exact equations of motion are obtained, whose quantum-classical approximants are associated with a fluid-dynamical trajectory representation which couples classical variables to quantum hydrodynamic variables. The latter evolve under a generalized hydrodynamic force which also depends upon the classical phase-space variables. The hydrodynamic moment description is shown to be closely connected to mixed quantum-classical phase-space methods.

DOI
10.1063/1.1856462

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The theoretical framework of the mixed quantum-classical description given by Burghardt and Parlant [J. Chem. Phys. 120, 3055 (2004)] is detailed. A representation in terms of partial hydrodynamic moments is developed, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. Exact equations of motion are obtained, whose quantum-classical approximants are associated with a fluid-dynamical trajectory representation which couples classical variables to quantum hydrodynamic variables. The latter evolve under a generalized hydrodynamic force which also depends upon the classical phase-space variables. The hydrodynamic moment description is shown to be closely connected to mixed quantum-classical phase-space methods.
A new approach to the coupling of quantum and classical dynamics is developed, by combining a hydrodynamic, Bohmian description for the quantum subsystem with a Liouville-space description for the classical subsystem. To this end, partial hydrodynamic moments are introduced, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. We focus on pure states (wave functions) and introduce a trajectory representation in a hybrid hydrodynamic-Liouvillian phase space. The interleaved trajectory dynamics is guided by a new type of quantum force. For illustration, we consider a pair of bilinearly coupled harmonic oscillators, for which the method is exact.
The hybrid quantum-classical approach of Burghardt and Parlant [Burghardt, I.; Parlant, G. J. Chem. Phys. 2004, 120, 3055], referred to here as the quantum-classical moment (QCM) approach, is demonstrated for the dynamics of a quantum double well coupled to a classical harmonic coordinate. The approach combines the quantum hydrodynamic and classical Liouvillian representations by the construction of a particular type of moments (that is, partial hydrodynamic moments) whose evolution is determined by a hierarchy of coupled equations. For pure states, which are at the center of the present study, this hierarchy terminates at the first order. In the Lagrangian picture, the deterministic trajectories result in dynamics which is Hamiltonian in the classical subspace, while the projection onto the quantum subspace evolves under a generalized hydrodynamic force. Importantly, this force also depends upon the classical (Q, P) variables. The present application demonstrates the tunneling dynamics in both the Eulerian and Lagrangian representations. The method is exact if the classical subspace is harmonic, as is the case for the systems studied here.
Abstract The formulation of quantum hydrodynamics for mixed states provides a complement, in a quantum‐statistical context, to its pure‐state “Bohmian mechanics” analogue. We summarize our recent work in this area, focusing on two key aspects. We first describe the formulation of a generalized hydrodynamic force, which includes the effects of thermal fluctuations and “quantum fluctuations” and thus determines the time evolution under the influence of dissipation. Second, we introduce a mixed hydrodynamic–Liouville space representation in terms of partial moments, as applied in the context of coupled electronic states and mixed quantum‐classical dynamics. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
This paper explores the hydrodynamic description of quantum densities evolving under the effects of dissipation. For illustration, we refer to analytical solutions for a harmonic system and Markovian dissipation, and derive expressions for the hydrodynamic moments and a hydrodynamic force that represents a generalization of the quantum force of Bohmian dynamics. Beyond this, we draw general conclusions on the dynamical transition, induced by dissipation, to a classical-like regime. In this context, the relation between the quantum-mechanical hydrodynamic picture and its classical-statistical analogue is investigated. This has some bearing upon mixed quantum-classical schemes emerging from the hydrodynamic description.
We propose a mixed quantum-classical hydrodynamic framework to model short-time inertial effects in the non-adiabatic evolution of a quantum solute coupled to a classical polar solvent. Drawing upon the work of Burghardt and Bagchi [Chem. Phys. 329 (2006), 343], we employ the Hamiltonian approach to incorporate consistent backreaction and preserve quantum decoherence beyond standard Ehrenfest dynamics. The solvent is treated as an ideal polar fluid and the quantum solute state is coupled to both the position and molecular orientation coordinates of the liquid. This approach retains essential solute-solvent correlations while reducing the computational complexity of previous approaches. We further incorporate dissipative terms to capture both inertial effects and polarization relaxation. After establishing the general setting for non-local dielectric continua, the Marcus local approximation is integrated into the model thereby extending traditional solvation theory to account for collective fluid sloshing on fast timescales.
Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.