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Bose A. Quantum correlation functions through tensor network path integral. J Chem Phys 2023; 159:214110. [PMID: 38051096 DOI: 10.1063/5.0174338] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2023] [Accepted: 11/07/2023] [Indexed: 12/07/2023] Open
Abstract
Tensor networks have historically proven to be of great utility in providing compressed representations of wave functions that can be used for the calculation of eigenstates. Recently, it has been shown that a variety of these networks can be leveraged to make real time non-equilibrium simulations of dynamics involving the Feynman-Vernon influence functional more efficient. In this work, a tensor network is developed for non-perturbatively calculating the equilibrium correlation function for open quantum systems using the path integral methodology. These correlation functions are of fundamental importance in calculations of rates of reactions, simulations of response functions and susceptibilities, spectra of systems, etc. The influence of the solvent on the quantum system is incorporated through an influence functional, whose unconventional structure motivates the design of a new optimal matrix product-like operator that can be applied to the so-called path amplitude matrix product state. This complex-time tensor network path integral approach provides an exceptionally efficient representation of the path integral, enabling simulations for larger systems strongly interacting with baths and at lower temperatures out to longer time. The derivation, design, and implementation of this method are discussed along with a wide range of illustrations ranging from rate theory and symmetrized spin correlation functions to simulation of response of the Fenna-Matthews-Olson complex to light.
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Affiliation(s)
- Amartya Bose
- Department of Chemical Sciences, Tata Institute of Fundamental Research, Mumbai 400005, India
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2
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Tao X, Shushkov P, Miller TF. Microcanonical rates from ring-polymer molecular dynamics: Direct-shooting, stationary-phase, and maximum-entropy approaches. J Chem Phys 2020; 152:124117. [DOI: 10.1063/1.5144307] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Affiliation(s)
- Xuecheng Tao
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
| | - Philip Shushkov
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
| | - Thomas F. Miller
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
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3
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Ke Y, Zhao Y. An extension of stochastic hierarchy equations of motion for the equilibrium correlation functions. J Chem Phys 2017; 146:214105. [PMID: 28576086 PMCID: PMC5453806 DOI: 10.1063/1.4984260] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2017] [Accepted: 05/16/2017] [Indexed: 11/14/2022] Open
Abstract
A traditional stochastic hierarchy equations of motion method is extended into the correlated real-time and imaginary-time propagations, in this paper, for its applications in calculating the equilibrium correlation functions. The central idea is based on a combined employment of stochastic unravelling and hierarchical techniques for the temperature-dependent and temperature-free parts of the influence functional, respectively, in the path integral formalism of the open quantum systems coupled to a harmonic bath. The feasibility and validity of the proposed method are justified in the emission spectra of homodimer compared to those obtained through the deterministic hierarchy equations of motion. Besides, it is interesting to find that the complex noises generated from a small portion of real-time and imaginary-time cross terms can be safely dropped to produce the stable and accurate position and flux correlation functions in a broad parameter regime.
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Affiliation(s)
- Yaling Ke
- State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| | - Yi Zhao
- State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, and Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, People's Republic of China
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4
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Song L, Shi Q. A new approach to calculate charge carrier transport mobility in organic molecular crystals from imaginary time path integral simulations. J Chem Phys 2015; 142:174103. [DOI: 10.1063/1.4919061] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Affiliation(s)
- Linze Song
- 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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5
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Shushkov P, Li R, Tully JC. Ring polymer molecular dynamics with surface hopping. J Chem Phys 2012; 137:22A549. [DOI: 10.1063/1.4766449] [Citation(s) in RCA: 90] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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6
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Wilner EY, Levy TJ, Rabani E. Analytical continuation approaches to electronic transport: The resonant level model. J Chem Phys 2012; 137:214107. [DOI: 10.1063/1.4768674] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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7
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Nan G, Li Z. Stationary phase approximation of electron transfer rate constants for the spin-boson model with Debye spectral density. COMPUT THEOR CHEM 2011. [DOI: 10.1016/j.comptc.2011.04.024] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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8
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Kletenik-Edelman O, Rabani E, Reichman DR. Analytic continuation average spectrum method for transport in quantum liquids. Chem Phys 2010. [DOI: 10.1016/j.chemphys.2010.01.012] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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9
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Zhang R, Lev B, Cuervo JE, Noskov SY, Salahub DR. A Guide to QM/MM Methodology and Applications. ADVANCES IN QUANTUM CHEMISTRY 2010. [DOI: 10.1016/s0065-3276(10)59010-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/16/2023]
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10
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Collepardo-Guevara R, Suleimanov YV, Manolopoulos DE. Bimolecular reaction rates from ring polymer molecular dynamics. J Chem Phys 2009; 130:174713. [DOI: 10.1063/1.3127145] [Citation(s) in RCA: 111] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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11
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Reichman DR, Rabani E. Analytic continuation average spectrum method for quantum liquids. J Chem Phys 2009; 131:054502. [DOI: 10.1063/1.3185728] [Citation(s) in RCA: 13] [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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12
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Pérez A, Tuckerman ME, Müser MH. A comparative study of the centroid and ring-polymer molecular dynamics methods for approximating quantum time correlation functions from path integrals. J Chem Phys 2009; 130:184105. [DOI: 10.1063/1.3126950] [Citation(s) in RCA: 86] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
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13
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Liu J, Miller WH. Linearized semiclassical initial value time correlation functions with maximum entropy analytic continuation. J Chem Phys 2008; 129:124111. [PMID: 19045010 DOI: 10.1063/1.2981065] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jian Liu
- Department of Chemistry and K. S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720-1460, USA
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14
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Collepardo-Guevara R, Craig IR, Manolopoulos DE. Proton transfer in a polar solvent from ring polymer reaction rate theory. J Chem Phys 2008; 128:144502. [DOI: 10.1063/1.2883593] [Citation(s) in RCA: 86] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/23/2023] Open
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15
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Habershon S, Braams BJ, Manolopoulos DE. Quantum mechanical correlation functions, maximum entropy analytic continuation, and ring polymer molecular dynamics. J Chem Phys 2007; 127:174108. [DOI: 10.1063/1.2786451] [Citation(s) in RCA: 59] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
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16
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Ignaczak A, Schmickler W. Effects of friction and asymmetric inner sphere reorganization energy on the electron transfer reaction rate—Two-dimensional simulations. Electrochim Acta 2007. [DOI: 10.1016/j.electacta.2006.08.032] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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17
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Venkataraman C, Miller WH. Chemical reaction rates using the semiclassical Van Vleck initial value representation. J Chem Phys 2007; 126:094104. [PMID: 17362101 DOI: 10.1063/1.2567200] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A semiclassical initial value representation formulation using the Van Vleck [Proc. Natl. Acad. Sci. U.S.A. 14, 178 (1928)] propagator has been used to calculate the flux correlation function and thereby reaction rate constants. This Van Vleck formulation of the flux-flux correlation function is computationally as simple as the classical Wigner [Trans. Faraday Soc. 34, 29 (1938)] model. However, unlike the latter, it has the ability to capture quantum interference/coherence effects. Classical trajectories are evolved starting from the dividing surface that separates reactants and products, and are evolved negatively in time. This formulation has been tested on model problems ranging from the Eckart barrier, double well to the collinear H+H2.
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18
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Quantum–Classical Reaction Rate Theory. ACTA ACUST UNITED AC 2007. [DOI: 10.1007/978-3-540-34460-5_13] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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19
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Navrotskaya I, Geva E. Vibrational Energy Relaxation Rates of H2 and D2 in Liquid Argon via the Linearized Semiclassical Method. J Phys Chem A 2006; 111:460-7. [PMID: 17228894 DOI: 10.1021/jp066243g] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The vibrational energy relaxation (VER) rates for H2 and D2 in liquid argon (T=152 K, rho=1.45x1022 cm-3) are calculated using the linearized semiclassical (LSC) method (J. Phys. Chem. 2003, 107, 9059, 9070). The calculation is based on Fermi's golden rule. The VER rate constant is expressed in terms of the quantum-mechanical force-force correlation function, which is then estimated using the LSC method. A local harmonic approximation (LHA) is employed in order to compute the multidimensional Wigner integrals underlying the LSC approximation. The H2-Ar and D2-Ar interactions are described by the three-body potential of Bissonette et al. (J. Phys. Chem. A 1996, 105, 2639). The LHA-LSC-based VER rate constants for both D2 and H2 are found to be about 2-3 orders of magnitude slower than those obtained experimentally. However, their ratio agrees quantitatively with the corresponding experimental result. In contrast, the classical VER rate constants are found to be 8-9 orders of magnitude slower than those obtained experimentally, and their ratio is found to be qualitatively different from the corresponding experimental result.
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Affiliation(s)
- Irina Navrotskaya
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, USA
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20
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Shi Q, Geva E. A derivation of the mixed quantum-classical Liouville equation from the influence functional formalism. J Chem Phys 2006; 121:3393-404. [PMID: 15303902 DOI: 10.1063/1.1771641] [Citation(s) in RCA: 102] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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.
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Affiliation(s)
- Qiang Shi
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, USA
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21
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Braams BJ, Manolopoulos DE. On the short-time limit of ring polymer molecular dynamics. J Chem Phys 2006; 125:124105. [PMID: 17014164 DOI: 10.1063/1.2357599] [Citation(s) in RCA: 123] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We examine the short-time accuracy of a class of approximate quantum dynamical techniques that includes the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) methods. Both of these methods are based on the path integral molecular dynamics (PIMD) technique for calculating the exact static equilibrium properties of quantum mechanical systems. For Kubo-transformed real-time correlation functions involving operators that are linear functions of positions or momenta, the RPMD and (adiabatic) CMD approximations differ only in the choice of the artificial mass matrix of the system of ring polymer beads that is employed in PIMD. The obvious ansatz for a general method of this type is therefore to regard the elements of the PIMD (or Parrinello-Rahman) mass matrix as an adjustable set of parameters that can be chosen to improve the accuracy of the resulting approximation. We show here that this ansatz leads uniquely to the RPMD approximation when the criterion that is used to select the mass matrix is the short-time accuracy of the Kubo-transformed correlation function. In particular, we show that the leading error in the RPMD position autocorrelation function is O(t(8)) and the error in the velocity autocorrelation function is O(t(6)), for a general anharmonic potential. The corresponding errors in the CMD approximation are O(t(6)) and O(t(4)), respectively.
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Affiliation(s)
- Bastiaan J Braams
- Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30332, USA
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22
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Kim SY, Hammes-Schiffer S. Hybrid quantum/classical molecular dynamics for a proton transfer reaction coupled to a dissipative bath. J Chem Phys 2006; 124:244102. [PMID: 16821968 DOI: 10.1063/1.2206175] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A hybrid quantum/classical molecular dynamics approach is applied to a proton transfer reaction represented by a symmetric double well system coupled to a dissipative bath. In this approach, the proton is treated quantum mechanically and all bath modes are treated classically. The transition state theory rate constant is obtained from the potential of mean force, which is generated along a collective reaction coordinate with umbrella sampling techniques. The transmission coefficient, which accounts for dynamical recrossings of the dividing surface, is calculated with a reactive flux approach combined with the molecular dynamics with quantum transitions surface hopping method. The hybrid quantum/classical results agree well with numerically exact results in the spatial-diffusion-controlled regime, which is most relevant for proton transfer in proteins. This hybrid quantum/classical approach has already been shown to be computationally practical for studying proton transfer in large biological systems. These results have important implications for future applications to hydrogen transfer reactions in solution and proteins.
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Affiliation(s)
- Soo Young Kim
- Department of Chemistry, Pennsylvania State University, University Park, Pennsylvania 16802, USA
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23
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Zhao Y, Li X, Zheng Z, Liang W. Semiclassical calculation of nonadiabatic thermal rate constants: Application to condensed phase reactions. J Chem Phys 2006; 124:114508. [PMID: 16555902 DOI: 10.1063/1.2178323] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The nonadiabatic transition state theory proposed recently by Zhao et al. [J. Chem. Phys. 121, 8854 (2004)] is extended to calculate rate constants of complex systems by using the Monte Carlo and umbrella sampling methods. Surface hopping molecular dynamics technique is incorporated to take into account the dynamic recrossing effect. A nontrivial benchmark model of the nonadiabatic reaction in the condensed phase is used for the numerical test. It is found that our semiclassical results agree well with those produced by the rigorous quantum mechanical method. Comparing with available analytical approaches, we find that the simple statistical theory proposed by Straub and Berne [J. Chem. Phys. 87, 6111 (1987)] is applicable for a wide friction region although their formula is obtained using Landau-Zener [Phys. Z. Sowjetunion 2, 46 (1932); Proc. R. Soc. London, Ser. A 137, 696 (1932)] nonadiabatic transition probability along a one-dimensional diffusive coordinate. We also investigate how the nuclear tunneling events affect the dependence of the rate constant on the friction.
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Affiliation(s)
- Yi Zhao
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Chemical Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
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24
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The influence of nonbilinear system-bath coupling on quantum-mechanical activated rate processes. Chem Phys 2006. [DOI: 10.1016/j.chemphys.2005.05.044] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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25
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Ravitz O. On the computation of imaginary-time correlation functions beyond the ℏβ limit. Chem Phys 2006. [DOI: 10.1016/j.chemphys.2005.08.006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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26
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Kim H, Kapral R. Nonadiabatic quantum-classical reaction rates with quantum equilibrium structure. J Chem Phys 2005; 123:194108. [PMID: 16321077 DOI: 10.1063/1.2110140] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Time correlation function expressions for quantum reaction-rate coefficients are computed in a quantum-classical limit. This form for the correlation function retains the full quantum equilibrium structure of the system in the spectral density function but approximates the time evolution of the operator by quantum-classical Liouville dynamics. Approximate analytical expressions for the spectral density function, which incorporate quantum effects in the many-body environment and reaction coordinate, are derived. The results of numerical simulations of the reaction rate are presented for a reaction model in which a two-level system is coupled to a bistable oscillator which is, in turn, coupled to a bath of harmonic oscillators. The nonadiabatic quantum-classical dynamics is simulated in terms of an ensemble of surface-hopping trajectories and the effects of the quantum equilibrium structure on the reaction rate are discussed.
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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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27
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Rabani E, Krilov G, Reichman DR, Berne BJ. Transport properties of normal liquid helium: Comparison of various methodologies. J Chem Phys 2005; 123:184506. [PMID: 16292913 DOI: 10.1063/1.2109927] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We revisit the problem of self-diffusion in normal liquid helium above the lambda transition. Several different methods are applied to compute the velocity autocorrelation function. Since it is still impossible to determine the exact result for the velocity autocorrelation function from simulation, we appeal to the computation of short-time moments to determine the accuracy of the different approaches at short times. The main conclusion reached from our study is that both the quantum mode-coupling theory and the numerical analytic continuation approach must be regarded as a viable and competitive methods for the computation of dynamical properties of quantum systems.
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Affiliation(s)
- Eran Rabani
- School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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Predescu C. Moments of spectral functions: Monte Carlo evaluation and verification. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:056709. [PMID: 16383787 DOI: 10.1103/physreve.72.056709] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2005] [Indexed: 05/05/2023]
Abstract
The subject of the present study is the Monte Carlo path-integral evaluation of the moments of spectral functions. Such moments can be computed by formal differentiation of certain estimating functionals that are infinitely differentiable against time whenever the potential function is arbitrarily smooth. Here, I demonstrate that the numerical differentiation of the estimating functionals can be more successfully implemented by means of pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial interpolant), which utilize information from the entire interval . The algorithmic detail that leads to robust numerical approximations is the fact that the path-integral action and not the actual estimating functional are interpolated. Although the resulting approximation to the estimating functional is nonlinear, the derivatives can be computed from it in a fast and stable way by contour integration in the complex plane, with the help of the Cauchy integral formula (e.g., by Lyness' method). An interesting aspect of the present development is that Hamburger's conditions for a finite sequence of numbers to be a moment sequence provide the necessary and sufficient criteria for the computed data to be compatible with the existence of an inversion algorithm. Finally, the issue of appearance of the sign problem in the computation of moments, albeit in a milder form than for other quantities, is addressed.
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Affiliation(s)
- Cristian Predescu
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA.
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29
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Pollak E, Talkner P. Reaction rate theory: what it was, where is it today, and where is it going? CHAOS (WOODBURY, N.Y.) 2005; 15:26116. [PMID: 16035918 DOI: 10.1063/1.1858782] [Citation(s) in RCA: 157] [Impact Index Per Article: 8.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
A brief history is presented, outlining the development of rate theory during the past century. Starting from Arrhenius [Z. Phys. Chem. 4, 226 (1889)], we follow especially the formulation of transition state theory by Wigner [Z. Phys. Chem. Abt. B 19, 203 (1932)] and Eyring [J. Chem. Phys. 3, 107 (1935)]. Transition state theory (TST) made it possible to obtain quick estimates for reaction rates for a broad variety of processes even during the days when sophisticated computers were not available. Arrhenius' suggestion that a transition state exists which is intermediate between reactants and products was central to the development of rate theory. Although Wigner gave an abstract definition of the transition state as a surface of minimal unidirectional flux, it took almost half of a century until the transition state was precisely defined by Pechukas [Dynamics of Molecular Collisions B, edited by W. H. Miller (Plenum, New York, 1976)], but even this only in the realm of classical mechanics. Eyring, considered by many to be the father of TST, never resolved the question as to the definition of the activation energy for which Arrhenius became famous. In 1978, Chandler [J. Chem. Phys. 68, 2959 (1978)] finally showed that especially when considering condensed phases, the activation energy is a free energy, it is the barrier height in the potential of mean force felt by the reacting system. Parallel to the development of rate theory in the chemistry community, Kramers published in 1940 [Physica (Amsterdam) 7, 284 (1940)] a seminal paper on the relation between Einstein's theory of Brownian motion [Einstein, Ann. Phys. 17, 549 (1905)] and rate theory. Kramers' paper provided a solution for the effect of friction on reaction rates but left us also with some challenges. He could not derive a uniform expression for the rate, valid for all values of the friction coefficient, known as the Kramers turnover problem. He also did not establish the connection between his approach and the TST developed by the chemistry community. For many years, Kramers' theory was considered as providing a dynamic correction to the thermodynamic TST. Both of these questions were resolved in the 1980s when Pollak [J. Chem. Phys. 85, 865 (1986)] showed that Kramers' expression in the moderate to strong friction regime could be derived from TST, provided that the bath, which is the source of the friction, is handled at the same level as the system which is observed. This then led to the Mel'nikov-Pollak-Grabert-Hanggi [Mel'nikov and Meshkov, J. Chem. Phys. 85, 1018 (1986); Pollak, Grabert, and Hanggi, ibid. 91, 4073 (1989)] solution of the turnover problem posed by Kramers. Although classical rate theory reached a high level of maturity, its quantum analog leaves the theorist with serious challenges to this very day. As noted by Wigner [Trans. Faraday Soc. 34, 29 (1938)], TST is an inherently classical theory. A definite quantum TST has not been formulated to date although some very useful approximate quantum rate theories have been invented. The successes and challenges facing quantum rate theory are outlined. An open problem which is being investigated intensively is rate theory away from equilibrium. TST is no longer valid and cannot even serve as a conceptual guide for understanding the critical factors which determine rates away from equilibrium. The nonequilibrium quantum theory is even less well developed than the classical, and suffers from the fact that even today, we do not know how to solve the real time quantum dynamics for systems with "many" degrees of freedom.
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Affiliation(s)
- Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel
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30
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Rabani E, Reichman DR. QUANTUM MODE-COUPLING THEORY: Formulation and Applications to Normal and Supercooled Quantum Liquids. Annu Rev Phys Chem 2005; 56:157-85. [PMID: 15796699 DOI: 10.1146/annurev.physchem.56.092503.141138] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
▪ Abstract We review our recent efforts to formulate and study a mode-coupling approach to real-time dynamic fluctuations in quantum liquids. Comparison is made between the theory and recent neutron scattering experiments performed on liquid ortho-deuterium and para-hydrogen. We discuss extensions of the theory to supercooled and glassy states where quantum fluctuations compete with thermal fluctuations. Experimental scenarios for quantum glassy liquids are briefly discussed.
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Affiliation(s)
- Eran Rabani
- School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel.
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31
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Abstract
This Perspective presents a broad overview of the present status of theoretical capabilities for describing quantum dynamics in molecular systems with many degrees of freedom, e.g., chemical reactions in solution, clusters, solids, or biomolecular environments.
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Affiliation(s)
- William H Miller
- Department of Chemistry and K. S. Pitzer Center for Theoretical Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720-1460, USA.
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32
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Horikoshi A, Kinugawa K. Effective potential analytic continuation approach for real time quantum correlation functions involving nonlinear operators. J Chem Phys 2005; 122:174104. [PMID: 15910020 DOI: 10.1063/1.1888576] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023] Open
Abstract
We apply the effective potential analytic continuation (EPAC) method to the calculation of real time quantum correlation functions involving operators nonlinear in the position operator q. For a harmonic system the EPAC method provides the exact correlation function at all temperature ranges, while the other quantum dynamics methods, the centroid molecular dynamics and the ring polymer molecular dynamics, become worse at lower temperature. For an asymmetric anharmonic system, the EPAC correlation function is in very good agreement with the exact one at t = 0. When the time increases from zero, the EPAC method gives good coincidence with the exact result at lower temperature. Finally, we propose a simplified version of the EPAC method to reduce the computational cost required for the calculation of the standard effective potential.
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Affiliation(s)
- Atsushi Horikoshi
- Japan Science and Technology Agency and Department of Chemistry, Faculty of Science, Nara Women's University.
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33
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Abstract
We compute the quantum rate constant based on two extended stationary phase approximations to the imaginary-time formulation of the quantum rate theory. The optimized stationary phase approximation to the imaginary-time flux-flux correlation function employs the optimized quadratic reference system to overcome the inaccuracy of the quadratic expansion in the standard stationary phase approximation, and yields favorable agreements with instanton results for both adiabatic and nonadiabatic processes in dissipative and nondissipative systems. The integrated stationary phase approximation to the two-dimensional barrier free energy is particularly useful for adiabatic processes and demonstrates consistent results with the imaginary-time flux-flux correlation function approach. Our stationary phase methods do not require calculation of tunneling paths or stability matrices, and work equally well in the high-temperature and the low-temperature regimes. The numerical results suggest their general applicability for calibration of imaginary-time methods and for the calculation of quantum rate constants in systems with a large number of degrees of freedom.
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Affiliation(s)
- Shilong Yang
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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34
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Ceotto M, Yang S, Miller WH. Quantum reaction rate from higher derivatives of the thermal flux-flux autocorrelation function at time zero. J Chem Phys 2005; 122:44109. [PMID: 15740237 DOI: 10.1063/1.1839177] [Citation(s) in RCA: 30] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A quantum theory of thermal reaction rates is presented which may be viewed as an extension of the recently developed "quantum instanton" (QI) model [W. H. Miller, Y. Zhao, M. Ceotto, and S. Yang, J. Chem. Phys. 119, 1329 (2003)]. It is based on using higher derivatives of the flux-flux autocorrelation function C(t) (as given by Miller, Schwartz, and Tromp) at t=0 to construct a short time approximation for C(t). Tests of this theory on 1d and collinear reactions, both symmetric and asymmetric, show it to be more accurate than the original QI model, giving rate constants to approximately 5% for a wide range of temperature.
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Affiliation(s)
- Michele Ceotto
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California-Berkeley, Berkeley, CA 94720, USA
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35
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Yamamoto T, Miller WH. Path integral evaluation of the quantum instanton rate constant for proton transfer in a polar solvent. J Chem Phys 2005; 122:44106. [PMID: 15740234 DOI: 10.1063/1.1832598] [Citation(s) in RCA: 71] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/28/2023] Open
Abstract
The quantum instanton approximation for thermal rate constants, a type of quantum transition state theory (QTST), is applied to a model proton transfer reaction in liquid methyl chloride developed by Azzouz and Borgis. Monte Carlo path integral methods are used to carry out the calculations, and two other closely related QTST's, namely, the centroid-density and Hansen-Andersen QTST, are also evaluated for comparison using the present path integral approach. A technique is then introduced that calculates the kinetic isotope effect directly via thermodynamic integration of the rate with respect to hydrogen mass, which has the practical advantage of avoiding costly evaluation of the activation free energy. The present application to the Azzouz-Borgis problem shows that the above three types of QTST provide very similar results for the rate, within 30% of each other, which is nontrivial considering the totally different derivations of these QTSTs; the latter rates are also in reasonable agreement with some other previous results (e.g., obtained via molecular dynamics with quantum transitions), within a factor of approximately 2(7) for the H(D) transfer, thus significantly diminishing the possible range of the exact rates. In addition, it is revealed that a small but nonnegligible inconsistency exists in the parametrization of the Azzouz-Borgis model employed in previous studies, which resulted in the large apparent discrepancy in the calculated rates.
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Affiliation(s)
- Takeshi Yamamoto
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California-Berkeley, Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
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36
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Predescu C, Miller WH. Optimal Choice of Dividing Surface for the Computation of Quantum Reaction Rates. J Phys Chem B 2005; 109:6491-9. [PMID: 16851728 DOI: 10.1021/jp040593q] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
We consider the calculation of quantum mechanical rate constants for chemical reactions via algorithms that utilize short-time values of the symmetrized flux-flux correlation function. We argue that the dividing surface that makes optimal use of the short-time quantum information is the surface that minimizes the value at the origin of the symmetrized flux-flux correlation function. We also demonstrate that, in the classical limit, this quantum variational criterion produces the same dividing surface as Wigner's variational principle. Finally, we argue that the quantum variational criterion behaves in a nearly optimal fashion with respect to the minimization of the extent of re-crossing flux.
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Affiliation(s)
- Cristian Predescu
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA.
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37
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Rabani E, Miyazaki K, Reichman DR. Quantum mode-coupling theory for binary mixtures. J Chem Phys 2005; 122:34502. [PMID: 15740204 DOI: 10.1063/1.1832593] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We extend the quantum mode-coupling theory of neat liquids to the case of binary mixtures, in order to study supercooled liquids where quantum fluctuations may compete with thermal fluctuations. We apply the theory to a generic model of a binary mixture of Lennard-Jones particles. Our treatment may be used to study quantum aging and exotic glass melting scenarios in structural supercooled quantum liquids.
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Affiliation(s)
- Eran Rabani
- School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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38
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Predescu C. Reconstruction of thermally symmetrized quantum autocorrelation functions from imaginary-time data. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:066705. [PMID: 15697551 DOI: 10.1103/physreve.70.066705] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/29/2004] [Revised: 07/12/2004] [Indexed: 05/24/2023]
Abstract
In this paper, I propose a technique for recovering quantum dynamical information from imaginary-time data via the resolution of a one-dimensional Hamburger moment problem. It is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin. A general class of reconstruction algorithms is then identified, according to Theorem 3. The technique is advocated as especially effective for a certain class of quantum problems in continuum space, for which only a few moments are necessary. For such problems, it is argued that the derivatives at origin can be evaluated by Monte Carlo simulations via estimators of finite variances in the limit of an infinite number of path variables. Finally, a maximum entropy inversion algorithm for the Hamburger moment problem is utilized to compute the quantum rate of reaction for a one-dimensional symmetric Eckart barrier.
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Affiliation(s)
- Cristian Predescu
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California-Berkeley, Berkeley, CA 94720, USA.
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39
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Poulsen JA, Nyman G, Rossky PJ. Quantum Diffusion in Liquid Para-hydrogen: An Application of the Feynman−Kleinert Linearized Path Integral Approximation. J Phys Chem B 2004. [DOI: 10.1021/jp040425y] [Citation(s) in RCA: 67] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
| | - Gunnar Nyman
- Physical Chemistry, Gøteborg University, S-412-96, Göteborg, Sweden
| | - Peter J. Rossky
- Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry, University of Texas at Austin, Austin, Texas 78712
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40
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Hone TD, Voth GA. A centroid molecular dynamics study of liquidpara-hydrogen andortho-deuterium. J Chem Phys 2004; 121:6412-22. [PMID: 15446940 DOI: 10.1063/1.1780951] [Citation(s) in RCA: 89] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Centroid molecular dynamics (CMD) is applied to the study of collective and single-particle dynamics in liquid para-hydrogen at two state points and liquid ortho-deuterium at one state point. The CMD results are compared with the results of classical molecular dynamics, quantum mode coupling theory, a maximum entropy analytic continuation approach, pair-product forward- backward semiclassical dynamics, and available experimental results. The self-diffusion constants are in excellent agreement with the experimental measurements for all systems studied. Furthermore, it is shown that the method is able to adequately describe both the single-particle and collective dynamics of quantum liquids.
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Affiliation(s)
- Tyler D Hone
- Department of Chemistry and Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, USA
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41
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Shi Q, Geva E. Nonradiative Electronic Relaxation Rate Constants from Approximations Based on Linearizing the Path-Integral Forward−Backward Action. J Phys Chem A 2004. [DOI: 10.1021/jp049547g] [Citation(s) in RCA: 71] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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42
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Shi Q, Geva E. A semiclassical generalized quantum master equation for an arbitrary system-bath coupling. J Chem Phys 2004; 120:10647-58. [PMID: 15268091 DOI: 10.1063/1.1738109] [Citation(s) in RCA: 83] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
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.
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Affiliation(s)
- Qiang Shi
- Department of Chemistry and the FOCUS Center, University of Michigan, Ann Arbor, Michigan 48109-1055, USA
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43
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Yonetani Y, Kinugawa K. Centroid molecular dynamics approach to the transport properties of liquid para-hydrogen over the wide temperature range. J Chem Phys 2004; 120:10624-33. [PMID: 15268088 DOI: 10.1063/1.1735613] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Fundamental transport properties of liquid para-hydrogen (p-H(2)), i.e., diffusion coefficients, thermal conductivity, shear viscosity, and bulk viscosity, have been evaluated by means of the path integral centroid molecular dynamics (CMD) calculations. These transport properties have been obtained over the wide temperature range, 14-32 K. Calculated values of the diffusion coefficients and the shear viscosity are in good agreement with the experimental values at all the investigated temperatures. Although a relatively large deviation is found for the thermal conductivity, the calculated values are less than three times the amount of the experimental values at any temperature. On the other hand, the classical molecular dynamics has led all the transport properties to much larger deviation. For the bulk viscosity of liquid p-H(2), which was never known from experiments, the present CMD has given a clear temperature dependence. In addition, from the comparison based on the principle of corresponding states, it has been shown that the marked deviation of the transport properties of liquid p-H(2) from the feature which is expected from the molecular parameters is due to the quantum effect.
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Affiliation(s)
- Yoshiteru Yonetani
- Japan Science and Technology Corporation, Department of Chemistry, Faculty of Science, Nara Women's University, Nara 630-8506, Japan
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44
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Rabani E, Reichman DR. A fully self-consistent treatment of collective fluctuations in quantum liquids. J Chem Phys 2004; 120:1458-65. [PMID: 15268271 DOI: 10.1063/1.1631436] [Citation(s) in RCA: 47] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The problem of calculating collective density fluctuations in quantum liquids is revisited. A fully quantum mechanical self-consistent treatment based on a quantum mode-coupling theory is presented. The theory is compared with the maximum entropy analytic continuation approach and with available experimental results. The quantum mode-coupling theory provides semiquantitative results for both short and long time dynamics. The proper description of long time phenomena is important in future study of problems related to the physics of glassy quantum systems, and to the study of collective fluctuations in Bose fluids.
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Affiliation(s)
- Eran Rabani
- School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel.
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45
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Shi Q, Geva E. A new approach to calculating the memory kernel of the generalized quantum master equation for an arbitrary system–bath coupling. J Chem Phys 2003. [DOI: 10.1063/1.1624830] [Citation(s) in RCA: 91] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023] Open
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46
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Shi Q, Geva E. On the calculation of vibrational energy relaxation rate constants from centroid molecular dynamics simulations. J Chem Phys 2003. [DOI: 10.1063/1.1613636] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
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47
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Shi Q, Geva E. Semiclassical Theory of Vibrational Energy Relaxation in the Condensed Phase. J Phys Chem A 2003. [DOI: 10.1021/jp030497+] [Citation(s) in RCA: 143] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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48
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Cheney BG, Andersen HC. Dynamical corrections to quantum transition state theory. J Chem Phys 2003. [DOI: 10.1063/1.1570404] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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49
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Shi Q, Geva E. A relationship between semiclassical and centroid correlation functions. J Chem Phys 2003. [DOI: 10.1063/1.1564814] [Citation(s) in RCA: 122] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
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50
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Yamamoto T, Miller WH. Semiclassical calculation of thermal rate constants in full Cartesian space: The benchmark reaction D+H2→DH+H. J Chem Phys 2003. [DOI: 10.1063/1.1533081] [Citation(s) in RCA: 56] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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