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The Journal of Physical Chemistry A

A Hybrid Hydrodynamic−Liouvillian Approach to Mixed Quantum−Classical Dynamics: Application to Tunneling in a Double Well

Keith H. Hughes, Steven M. Parry, Gérard Parlant, Irène Burghardt

Published Aug 9, 2007• 14 citations
Publisher

Abstract

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.

DOI
10.1021/jp072642f

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physics.chem-phCross-listedarXiv:2605.05658•May 8, 2026
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.
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.
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
The hydrodynamic analogy for quantum mechanics is developed in a quantum phase-space representation. The key technical step in this development is the formulation of a phase-space flux (or current) density expression. Lagrangian fluid trajectories in phase space are explicitly found as solutions to a set of first-order ordinary differential equations which are formulated without making semiclassical approximation to the quantum dynamics. These fluid trajectories may be used to construct sharp structures in quantum phase space in the same way that one uses classical trajectories in classical mechanics. While these sharp structures do not themselves represent states, their properties ``control'' the dynamics of states in a manner familiar from the theory of first-order flows. The formalism reduces to Liouville dynamics in the classical limit, \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. In addition to several analytic applications to the pure- and mixed-state dynamics, the approach is numerically implemented for tunneling dynamics in a double-well problem. A new picture for quantum tunneling emerges where the tunneling transport occurs when density spirals outward from one well along real-valued fluid trajectories and then into the other well. The tunneling is found to occur through localized portals, or ``flux gates'' in the phase space. We advocate the phase-space hydrodynamic model as a general tool to understand classical quantum correspondence.
In this paper we compute quantum trajectories arising from Bohm’s causal description of quantum mechanics. Our computational methodology is based upon a finite-element moving least-squares method (MWLS) presented recently by Wyatt and co-workers [Lopreore and Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. This method treats the “particles” in the quantum Hamilton–Jacobi equation as Lagrangian fluid elements that carry the phase, S, and density, ρ, required to reconstruct the quantum wave function. Here, we compare results obtained via the MWLS procedure to exact results obtained either analytically or by numerical solution of the time-dependent Schrödinger equation. Two systems are considered: first, dynamics in a harmonic well and second, tunneling dynamics in a double well potential. In the case of tunneling in the double well potential, the quantum potential acts to lower the barrier, separating the right- and left-hand sides of the well, permitting trajectories to pass from one side to another. However, as probability density passes from one side to the other, the effective barrier begins to rise and eventually will segregate trajectories in one side from the other. We note that the MWLS trajectories exhibited long time stability in the purely harmonic cases. However, this stability was not evident in the barrier crossing dynamics. Comparisons to exact trajectories obtained via wave packet calculations indicate that the MWLS trajectories tend to underestimate the effects of constructive and destructive interference effects.
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.