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Gelin MF, Borrelli R, Chen L. Hierarchical Equations-of-Motion Method for Momentum System-Bath Coupling. J Phys Chem B 2021; 125:4863-4873. [PMID: 33929205 PMCID: PMC8279550 DOI: 10.1021/acs.jpcb.1c02431] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
For a broad class of quantum models of practical interest, we demonstrate that the Hamiltonian of the system nonlinearly coupled to a harmonic bath through the system and bath coordinates can be equivalently mapped into the Hamiltonian of the system bilinearly coupled to the bath through the system and bath momenta. We show that the Hamiltonian with bilinear system-bath momentum coupling can be treated by the hierarchical equations-of-motion (HEOM) method and present the corresponding proof-of-principle simulations. The developed methodology creates the opportunity to scrutinize a new family of nonlinear quantum systems by the numerically accurate HEOM method.
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Affiliation(s)
- Maxim F Gelin
- School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
| | | | - Lipeng Chen
- Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany
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2
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He X, Gong Z, Wu B, Liu J. Negative Zero-Point-Energy Parameter in the Meyer-Miller Mapping Model for Nonadiabatic Dynamics. J Phys Chem Lett 2021; 12:2496-2501. [PMID: 33667108 DOI: 10.1021/acs.jpclett.1c00232] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The celebrated Meyer-Miller mapping model has been a useful approach for generating practical trajectory-based nonadiabatic dynamics methods. It is generally assumed that the zero-point-energy (ZPE) parameter is positive. The constraint implied in the conventional Meyer-Miller mapping Hamiltonian for an F-electronic-state system actually requires γ∈(-1/F, ∞) for the ZPE parameter for each electronic degree of freedom. Both negative and positive values are possible for such a parameter. We first establish a rigorous formulation to construct exact mapping models in the Cartesian phase space when the constraint is applied. When nuclear dynamics is approximated by the linearized semiclassical initial value representation, a negative ZPE parameter could lead to reasonably good performance in describing dynamic behaviors in typical spin-boson models for condensed-phase two-state systems, even at challenging zero temperature.
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Affiliation(s)
- Xin He
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Zhihao Gong
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Baihua Wu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Jian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
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Tanimura Y. Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM). J Chem Phys 2021; 153:020901. [PMID: 32668942 DOI: 10.1063/5.0011599] [Citation(s) in RCA: 163] [Impact Index Per Article: 54.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022] Open
Abstract
An open quantum system refers to a system that is further coupled to a bath system consisting of surrounding radiation fields, atoms, molecules, or proteins. The bath system is typically modeled by an infinite number of harmonic oscillators. This system-bath model can describe the time-irreversible dynamics through which the system evolves toward a thermal equilibrium state at finite temperature. In nuclear magnetic resonance and atomic spectroscopy, dynamics can be studied easily by using simple quantum master equations under the assumption that the system-bath interaction is weak (perturbative approximation) and the bath fluctuations are very fast (Markovian approximation). However, such approximations cannot be applied in chemical physics and biochemical physics problems, where environmental materials are complex and strongly coupled with environments. The hierarchical equations of motion (HEOM) can describe the numerically "exact" dynamics of a reduced system under nonperturbative and non-Markovian system-bath interactions, which has been verified on the basis of exact analytical solutions (non-Markovian tests) with any desired numerical accuracy. The HEOM theory has been used to treat systems of practical interest, in particular, to account for various linear and nonlinear spectra in molecular and solid state materials, to evaluate charge and exciton transfer rates in biological systems, to simulate resonant tunneling and quantum ratchet processes in nanodevices, and to explore quantum entanglement states in quantum information theories. This article presents an overview of the HEOM theory, focusing on its theoretical background and applications, to help further the development of the study of open quantum dynamics.
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Affiliation(s)
- Yoshitaka Tanimura
- Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
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Yan Y, Xing T, Shi Q. A new method to improve the numerical stability of the hierarchical equations of motion for discrete harmonic oscillator modes. J Chem Phys 2020; 153:204109. [DOI: 10.1063/5.0027962] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yaming Yan
- 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; University of Chinese Academy of Sciences, Beijing 100049, China; and Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101407, China
| | - Tao Xing
- 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; University of Chinese Academy of Sciences, Beijing 100049, China; and Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101407, 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; University of Chinese Academy of Sciences, Beijing 100049, China; and Physical Science Laboratory, Huairou National Comprehensive Science Center, Beijing 101407, China
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Ullah A, Han L, Yan YA, Zheng X, Yan Y, Chernyak V. Stochastic equation of motion approach to fermionic dissipative dynamics. II. Numerical implementation. J Chem Phys 2020; 152:204106. [DOI: 10.1063/1.5142166] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Affiliation(s)
- Arif Ullah
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lu Han
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Yun-An Yan
- School of Physics and Optoelectronic Engineering, Ludong University, Shandong 264025, China
| | - Xiao Zheng
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - YiJing Yan
- Hefei National Laboratory for Physical Sciences at the Microscale and iChEM, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Vladimir Chernyak
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
- Department of Chemistry, Wayne State University, 5101 Cass Avenue, Detroit, Michigan 48202, USA
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Zhang HD, Cui L, Gong H, Xu RX, Zheng X, Yan Y. Hierarchical equations of motion method based on Fano spectrum decomposition for low temperature environments. J Chem Phys 2020; 152:064107. [PMID: 32061227 DOI: 10.1063/1.5136093] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The hierarchical equations of motion (HEOM) method has become one of the most popular methods for the studies of the open quantum system. However, its applicability to systems at ultra-low temperatures is largely restrained by the enormous computational cost, which is caused by the numerous exponential functions required to accurately characterize the non-Markovian memory of the reservoir environment. To overcome this problem, a Fano spectrum decomposition (FSD) scheme has been proposed recently [Cui et al., J. Chem. Phys. 151, 024110 (2019)], which expands the reservoir correlation functions using polynomial-exponential functions and hence greatly reduces the size of the memory basis set. In this work, we explicitly establish the FSD-based HEOM formalisms for both bosonic and fermionic environments. The accuracy and efficiency of the FSD-based HEOM are exemplified by the calculated low-temperature dissipative dynamics of a spin-boson model and the dynamic and static properties of a single-orbital Anderson impurity model in the Kondo regime. The encouraging numerical results highlight the practicality and usefulness of the FSD-based HEOM method for general open systems at ultra-low temperatures.
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Affiliation(s)
- Hou-Dao Zhang
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Lei Cui
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Hong Gong
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Rui-Xue Xu
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao Zheng
- Hefei National Laboratory for Physical Sciences at the Microscale and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - YiJing Yan
- Hefei National Laboratory for Physical Sciences at the Microscale, iChEM, University of Science and Technology of China, Hefei, Anhui 230026, China
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Berkelbach TC, Thoss M. Special topic on dynamics of open quantum systems. J Chem Phys 2020; 152:020401. [DOI: 10.1063/1.5142731] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Timothy C. Berkelbach
- Department of Chemistry, Columbia University, New York, New York 10027, USA
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
| | - Michael Thoss
- Institute of Physics, Albert-Ludwig University Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany
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Gong Z, Wu J. Absorption matrix of multi-site systems calculated by a hybrid quantum-classical Liouville equation. J Chem Phys 2019; 151:224109. [DOI: 10.1063/1.5138217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Zhihao Gong
- Physics Department, Zhejiang University, 38 ZheDa Road, Hangzhou, Zhejiang 310027, China
| | - Jianlan Wu
- Physics Department, Zhejiang University, 38 ZheDa Road, Hangzhou, Zhejiang 310027, China
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9
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He X, Liu J. A new perspective for nonadiabatic dynamics with phase space mapping models. J Chem Phys 2019; 151:024105. [DOI: 10.1063/1.5108736] [Citation(s) in RCA: 30] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Xin He
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Jian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
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Cui L, Zhang HD, Zheng X, Xu RX, Yan Y. Highly efficient and accurate sum-over-poles expansion of Fermi and Bose functions at near zero temperatures: Fano spectrum decomposition scheme. J Chem Phys 2019; 151:024110. [DOI: 10.1063/1.5096945] [Citation(s) in RCA: 31] [Impact Index Per Article: 6.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Lei Cui
- Hefei National Laboratory for Physical Sciences at the Microscale and Department of Chemical Physics and Synergetic Innovation Center of Quantum Information and Quantum Physics and Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Hou-Dao Zhang
- Hefei National Laboratory for Physical Sciences at the Microscale and Department of Chemical Physics and Synergetic Innovation Center of Quantum Information and Quantum Physics and Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Xiao Zheng
- Hefei National Laboratory for Physical Sciences at the Microscale and Department of Chemical Physics and Synergetic Innovation Center of Quantum Information and Quantum Physics and Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Rui-Xue Xu
- Hefei National Laboratory for Physical Sciences at the Microscale and Department of Chemical Physics and Synergetic Innovation Center of Quantum Information and Quantum Physics and Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), University of Science and Technology of China, Hefei, Anhui 230026, China
| | - YiJing Yan
- Hefei National Laboratory for Physical Sciences at the Microscale and Department of Chemical Physics and Synergetic Innovation Center of Quantum Information and Quantum Physics and Collaborative Innovation Center of Chemistry for Energy Materials (iChEM), University of Science and Technology of China, Hefei, Anhui 230026, China
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