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Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths. Symmetry (Basel) 2018. [DOI: 10.3390/sym10100518] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.
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Xie W, Xu M, Bai S, Shi Q. Mixed Quantum-Classical Study of Nonadiabatic Curve Crossing in Condensed Phases. J Phys Chem A 2016; 120:3225-32. [DOI: 10.1021/acs.jpca.5b11695] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Weiwei Xie
- Beijing
National Laboratory
for Molecular Sciences, State Key Laboratory for Structural Chemistry
of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
| | - Meng Xu
- Beijing
National Laboratory
for Molecular Sciences, State Key Laboratory for Structural Chemistry
of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
| | - Shuming Bai
- Beijing
National Laboratory
for Molecular Sciences, State Key Laboratory for Structural Chemistry
of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
| | - Qiang Shi
- Beijing
National Laboratory
for Molecular Sciences, State Key Laboratory for Structural Chemistry
of Unstable and Stable Species, Institute of Chemistry, Chinese Academy of Sciences, Zhongguancun, Beijing 100190, China
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Kapral R. Quantum dynamics in open quantum-classical systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:073201. [PMID: 25634784 DOI: 10.1088/0953-8984/27/7/073201] [Citation(s) in RCA: 49] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Often quantum systems are not isolated and interactions with their environments must be taken into account. In such open quantum systems these environmental interactions can lead to decoherence and dissipation, which have a marked influence on the properties of the quantum system. In many instances the environment is well-approximated by classical mechanics, so that one is led to consider the dynamics of open quantum-classical systems. Since a full quantum dynamical description of large many-body systems is not currently feasible, mixed quantum-classical methods can provide accurate and computationally tractable ways to follow the dynamics of both the system and its environment. This review focuses on quantum-classical Liouville dynamics, one of several quantum-classical descriptions, and discusses the problems that arise when one attempts to combine quantum and classical mechanics, coherence and decoherence in quantum-classical systems, nonadiabatic dynamics, surface-hopping and mean-field theories and their relation to quantum-classical Liouville dynamics, as well as methods for simulating the dynamics.
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Affiliation(s)
- Raymond Kapral
- Department of Chemistry, Chemical Physics Theory Group, University of Toronto, Toronto, ON, M5S 3H6 Canada
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Martinez F, Hanna G. Assessment of approximate solutions of the quantum–classical Liouville equation for dynamics simulations of quantum subsystems embedded in classical environments. MOLECULAR SIMULATION 2014. [DOI: 10.1080/08927022.2014.923573] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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Nassimi A, Bonella S, Kapral R. Analysis of the quantum-classical Liouville equation in the mapping basis. J Chem Phys 2010; 133:134115. [DOI: 10.1063/1.3480018] [Citation(s) in RCA: 54] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Rank JP, Kapral R. Decoherence and quantum-classical dynamics in a dissipative bath. J Chem Phys 2010; 132:074106. [DOI: 10.1063/1.3310811] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022] Open
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Abstract
The calculation of quantum canonical time correlation functions is considered in this paper. Transport properties, such as diffusion and reaction rate coefficients, can be determined from time integrals of these correlation functions. Approximate quantum-classical expressions for correlation functions, which are amenable to simulation, are derived. These expressions incorporate the full quantum equilibrium structure of the system but approximate the dynamics by quantum-classical evolution where a quantum subsystem is coupled to a classical environment. The main feature of the formulation is the use of a mapping basis where the subsystem quantum states are represented by fictitious harmonic oscillator states. This leads to a full phase space representation of the dynamics that can be simulated without appeal to surface-hopping methods. The results in this paper form the basis for new simulation algorithms for the computation of quantum transport properties of large many-body systems.
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Affiliation(s)
- Ali Nassimi
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronton ON M5S 3H6, Canada
| | - Raymond Kapral
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronton ON M5S 3H6, Canada
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Kim H, Nassimi A, Kapral R. Quantum-classical Liouville dynamics in the mapping basis. J Chem Phys 2008; 129:084102. [DOI: 10.1063/1.2971041] [Citation(s) in RCA: 110] [Impact Index Per Article: 6.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Gelman D, Schwartz SD. Tunneling dynamics with a mixed quantum-classical method: Quantum corrected propagator combined with frozen Gaussian wave packets. J Chem Phys 2008; 129:024504. [DOI: 10.1063/1.2949818] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Hanna G, Kapral R. Quantum-classical Liouville dynamics of proton and deuteron transfer rates in a solvated hydrogen-bonded complex. J Chem Phys 2008; 128:164520. [DOI: 10.1063/1.2907847] [Citation(s) in RCA: 29] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Grunwald R, Kim H, Kapral R. Surface-hopping dynamics and decoherence with quantum equilibrium structure. J Chem Phys 2008; 128:164110. [DOI: 10.1063/1.2906485] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Kim H, Kapral R. Proton and Deuteron Transfer Reactions in Molecular Nanoclusters. Chemphyschem 2008; 9:470-4. [DOI: 10.1002/cphc.200700709] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
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Grunwald R, Kapral R. Decoherence and quantum-classical master equation dynamics. J Chem Phys 2007; 126:114109. [PMID: 17381198 DOI: 10.1063/1.2567164] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/28/2022] Open
Abstract
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
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Affiliation(s)
- Robbie Grunwald
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
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Kim H, Kapral R. Solvation and proton transfer in polar molecule nanoclusters. J Chem Phys 2006; 125:234309. [PMID: 17190560 DOI: 10.1063/1.2404956] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Proton transfer in a phenol-amine complex dissolved in polar molecule nanoclusters is investigated. The proton transfer rates and mechanisms, as well as the solvation of the complex in the cluster, are studied using both adiabatic and nonadiabatic dynamics. The phenol-amine complex exists in ionic and covalent forms and as the size of the cluster increases the ionic form gains stability at the expense of the covalent form. Both the adiabatic and nonadiabatic transfer reaction rates increase with cluster size. Given a fixed cluster size, the stability of the covalent state increases with increasing temperature. The proton transfer rates do not change monotonously with an increase in temperature. A strong correlation between the solvent polarization reaction coordinate and the location of the phenol-amine complex in the cluster is found. The ionic form of the complex strongly prefers the interior of the cluster while the covalent form prefers to lie on the cluster surface.
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Affiliation(s)
- Hyojoon Kim
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada.
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