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

A derivation of the mixed quantum-classical Liouville equation from the influence functional formalism

Qiang Shi, Eitan Geva

Published Aug 10, 2004• 159 citations
Publisher

Abstract

We show that the mixed quantum-classical Liouville equation is equivalent to linearizing the forward-backward action in the influence functional. Derivations are provided in terms of either the diabatic or adiabatic basis sets. An application of the mixed quantum-classical Liouville equation for calculating the memory kernel of the generalized quantum master equation is also presented. The accuracy and computational feasibility of such an approach is demonstrated in the case of a two-level system nonlinearly coupled to an anharmonic bath.

DOI
10.1063/1.1771641

Related papers

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution

Chang‐Yu Hsieh, Raymond Kapral

A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple dynamics in which a set of N coherent state coordinates evolves in forward and backward trajectories, while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward trajectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed.

arXiv:1205.1283•The Journal of Chemical Physics•Jul 18, 2012•112 citations

A new approach to calculating the memory kernel of the generalized quantum master equation for an arbitrary system–bath coupling

Qiang Shi, Eitan Geva

The Nakajima–Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system’s dynamics. The standard approach is based on using a perturbative treatment of the system–bath coupling for calculating this kernel, and is therefore restricted to systems weakly coupled to the bath. In this paper, we propose a new approach for calculating the memory kernel for an arbitrary system–bath coupling. The memory kernel is obtained by solving a set of two coupled integral equations that relate it to a new type of two-time system-dependent bath correlation functions. The feasibility of the method is demonstrated in the case of an asymetrical two-level system linearly coupled to a harmonic bath.

The Journal of Chemical Physics•Dec 4, 2003•166 citations

Nonadiabatic dynamics in open quantum-classical systems: Forward-backward trajectory solution

Chang‐Yu Hsieh, Raymond Kapral

A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple non-Hamiltonian dynamics in which a set of N coherent state coordinates evolve in forward and backward trajectories while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward tra-jectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed. 1 ar

•Aug 17, 2016•70 citations

Mixed quantum-classical dynamics

Raymond Kapral, Giovanni Ciccotti

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.

The Journal of Chemical Physics•May 8, 1999•677 citations

A semiclassical generalized quantum master equation for an arbitrary system-bath coupling

Qiang Shi, Eitan Geva

The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a, possibly anharmonic, quantum bath. In this equation, a memory kernel superoperator accounts for the influence of the bath on the dynamics of the system. In a previous paper [Q. Shi and E. Geva, J. Chem. Phys. 119, 12045 (2003)] we proposed a new approach to calculating the memory kernel, in the case of arbitrary system-bath coupling. Within this approach, the memory kernel is obtained by solving a set of two integral equations, which requires a new type of two-time system-dependent bath correlation functions as input. In the present paper, we consider the application of the linearized semiclassical (LSC) approximation for calculating those correlation functions, and subsequently the memory kernel. The new approach is tested on a benchmark spin-boson model. Application of the LSC approximation for calculating the relatively short-lived memory kernel, followed by a numerically exact solution of the GQME, is found to provide an accurate description of the relaxation dynamics. The success of the proposed LSC-GQME methodology is contrasted with the failure of both the direct application of the LSC approximation and the weak coupling treatment to provide an accurate description of the dynamics, for the same model, except at very short times. The feasibility of the new methodology to anharmonic systems is also demonstrated in the case of a two level system coupled to a chain of Lennard-Jones atoms.

The Journal of Chemical Physics•May 17, 2004•134 citations

Influence functionals with semiclassical propagators in combined forward–backward time

Keiran C. Thompson, Nancy Makri

In a recent letter [Chem. Phys. Lett. 291, 101 (1998)] we presented a semiclassical methodology for calculating influence functionals arising from many-body anharmonic environments in the path integral formulation of quantum dynamics. Taking advantage of the trace operation associated with the unobservable medium, we express the influence functional in terms of a single propagator along a combined forward–backward system path. This propagator is evaluated according to time-dependent semiclassical theory in a coherent state initial value representation. Because the action associated with propagation in combined forward and backward time is governed by the net force experienced by the environment due to its interaction with the system, the resulting propagator is generally a smooth function of coordinates and thus amenable to Monte Carlo sampling; yet, the interference between forward and reverse propagators is fully accounted for. In the present paper we present a more elaborate version of the semiclassical influence functional formalism, along with algorithms for evaluating the coherent state transform of the Boltzmann operator that enters the influence functional. This factor is evaluated by performing an imaginary time path integral, and various approximations of the resulting expression as well as sampling schemes are discussed. The feasibility of the approach is demonstrated via numerous test calculations involving a two-level system coupled to (a) a dissipative harmonic bath and (b) a ten-dimensional bath of coupled anharmonic oscillators.

The Journal of Chemical Physics•Jan 15, 1999•170 citations
We show that the mixed quantum-classical Liouville equation is equivalent to linearizing the forward-backward action in the influence functional. Derivations are provided in terms of either the diabatic or adiabatic basis sets. An application of the mixed quantum-classical Liouville equation for calculating the memory kernel of the generalized quantum master equation is also presented. The accuracy and computational feasibility of such an approach is demonstrated in the case of a two-level system nonlinearly coupled to an anharmonic bath.
A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple dynamics in which a set of N coherent state coordinates evolves in forward and backward trajectories, while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward trajectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed.
The Nakajima–Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system’s dynamics. The standard approach is based on using a perturbative treatment of the system–bath coupling for calculating this kernel, and is therefore restricted to systems weakly coupled to the bath. In this paper, we propose a new approach for calculating the memory kernel for an arbitrary system–bath coupling. The memory kernel is obtained by solving a set of two coupled integral equations that relate it to a new type of two-time system-dependent bath correlation functions. The feasibility of the method is demonstrated in the case of an asymetrical two-level system linearly coupled to a harmonic bath.
A new approximate solution to the quantum-classical Liouville equation is derived starting from the formal solution of this equation in forward-backward form. The time evolution of a mixed quantum-classical system described by this equation is obtained in a coherent state basis using the mapping representation, which expresses N quantum degrees of freedom in a 2N-dimensional phase space. The solution yields a simple non-Hamiltonian dynamics in which a set of N coherent state coordinates evolve in forward and backward trajectories while the bath coordinates evolve under the influence of the mean potential that depends on these forward and backward tra-jectories. It is shown that the solution satisfies the differential form of the quantum-classical Liouville equation exactly. Relations to other mixed quantum-classical and semi-classical schemes are discussed. 1 ar
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.
The Nakajima-Zwanzig generalized quantum master equation (GQME) provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a, possibly anharmonic, quantum bath. In this equation, a memory kernel superoperator accounts for the influence of the bath on the dynamics of the system. In a previous paper [Q. Shi and E. Geva, J. Chem. Phys. 119, 12045 (2003)] we proposed a new approach to calculating the memory kernel, in the case of arbitrary system-bath coupling. Within this approach, the memory kernel is obtained by solving a set of two integral equations, which requires a new type of two-time system-dependent bath correlation functions as input. In the present paper, we consider the application of the linearized semiclassical (LSC) approximation for calculating those correlation functions, and subsequently the memory kernel. The new approach is tested on a benchmark spin-boson model. Application of the LSC approximation for calculating the relatively short-lived memory kernel, followed by a numerically exact solution of the GQME, is found to provide an accurate description of the relaxation dynamics. The success of the proposed LSC-GQME methodology is contrasted with the failure of both the direct application of the LSC approximation and the weak coupling treatment to provide an accurate description of the dynamics, for the same model, except at very short times. The feasibility of the new methodology to anharmonic systems is also demonstrated in the case of a two level system coupled to a chain of Lennard-Jones atoms.
In a recent letter [Chem. Phys. Lett. 291, 101 (1998)] we presented a semiclassical methodology for calculating influence functionals arising from many-body anharmonic environments in the path integral formulation of quantum dynamics. Taking advantage of the trace operation associated with the unobservable medium, we express the influence functional in terms of a single propagator along a combined forward–backward system path. This propagator is evaluated according to time-dependent semiclassical theory in a coherent state initial value representation. Because the action associated with propagation in combined forward and backward time is governed by the net force experienced by the environment due to its interaction with the system, the resulting propagator is generally a smooth function of coordinates and thus amenable to Monte Carlo sampling; yet, the interference between forward and reverse propagators is fully accounted for. In the present paper we present a more elaborate version of the semiclassical influence functional formalism, along with algorithms for evaluating the coherent state transform of the Boltzmann operator that enters the influence functional. This factor is evaluated by performing an imaginary time path integral, and various approximations of the resulting expression as well as sampling schemes are discussed. The feasibility of the approach is demonstrated via numerous test calculations involving a two-level system coupled to (a) a dissipative harmonic bath and (b) a ten-dimensional bath of coupled anharmonic oscillators.