1
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Malpathak S, Ananth N. Semiclassical dynamics in Wigner phase space I: Adiabatic hybrid Wigner dynamics. J Chem Phys 2024; 161:094109. [PMID: 39234962 DOI: 10.1063/5.0223185] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2024] [Accepted: 08/12/2024] [Indexed: 09/06/2024] Open
Abstract
The Wigner phase space formulation of quantum mechanics is a complete framework for quantum dynamic calculations that elegantly highlights connections with classical dynamics. In this series of two articles, building upon previous efforts, we derive the full hierarchy of approximate semiclassical (SC) dynamic methods for adiabatic and non-adiabatic problems in Wigner phase space. In Paper I, focusing on adiabatic single surface processes, we derive the well-known double Herman-Kluk (DHK) approximation for real-time correlation functions in Wigner phase space and connect it to the linearized SC (LSC) approximation through a stationary phase approximation. We exploit this relationship to introduce a new hybrid SC method, termed Adiabatic Hybrid Wigner Dynamics (AHWD) that allows for a few important "system" degrees of freedom (dofs) to be treated at the DHK level, while treating the rest of the dofs (the "bath") at the LSC level. AHWD is shown to accurately capture quantum interference effects in models of coupled oscillators and the decoherence of vibrational probability density of a model I2 Morse oscillator coupled to an Ohmic thermal bath. We show that AHWD significantly mitigates the sign problem and employs reduced dimensional prefactors bringing calculations of complex system-bath problems within the reach of SC methods. Paper II focuses on extending this hybrid SC dynamics to nonadiabatic processes.
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Affiliation(s)
- Shreyas Malpathak
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
| | - Nandini Ananth
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
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2
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Bridge O, Lazzaroni P, Martinazzo R, Rossi M, Althorpe SC, Litman Y. Quantum rates in dissipative systems with spatially varying friction. J Chem Phys 2024; 161:024110. [PMID: 38984959 DOI: 10.1063/5.0216823] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2024] [Accepted: 06/20/2024] [Indexed: 07/11/2024] Open
Abstract
We investigate whether making the friction spatially dependent on the reaction coordinate introduces quantum effects into the thermal reaction rates for dissipative reactions. Quantum rates are calculated using the numerically exact multi-configuration time-dependent Hartree method, as well as the approximate ring-polymer molecular dynamics (RPMD), ring-polymer instanton methods, and classical molecular dynamics. By conducting simulations across a wide range of temperatures and friction strengths, we can identify the various regimes that govern the reactive dynamics. At high temperatures, in addition to the spatial-diffusion and energy-diffusion regimes predicted by Kramer's rate theory, a (coherent) tunneling-dominated regime is identified at low friction. At low temperatures, incoherent tunneling dominates most of Kramer's curve, except at very low friction, when coherent tunneling becomes dominant. Unlike in classical mechanics, the bath's influence changes the equilibrium time-independent properties of the system, leading to a complex interplay between spatially dependent friction and nuclear quantum effects even at high temperatures. More specifically, a realistic friction profile can lead to an increase (or decrease) of the quantum (classical) rates with friction within the spatial-diffusion regime, showing that classical and quantum rates display qualitatively different behaviors. Except at very low frictions, we find that RPMD captures most of the quantum effects in the thermal reaction rates.
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Affiliation(s)
- Oliver Bridge
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Paolo Lazzaroni
- MPI for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Rocco Martinazzo
- Department of Chemistry, Università degli Studi di Milano, Via Golgi 19, 20133 Milano, Italy
| | - Mariana Rossi
- MPI for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Stuart C Althorpe
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Yair Litman
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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3
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Upadhyayula S, Pollak E. ℏ2 Corrections to Semiclassical Transmission Coefficients. J Phys Chem A 2024; 128:3434-3448. [PMID: 38630022 PMCID: PMC11077496 DOI: 10.1021/acs.jpca.4c00452] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2024] [Revised: 03/21/2024] [Accepted: 03/21/2024] [Indexed: 05/03/2024]
Abstract
The uniform semiclassical expression for the energy-dependent transmission probability through a barrier has been a staple of reaction rate theory for almost 90 years. Yet, when using the classical Euclidean action, the transmission probability is identical to 1/2 when the energy equals the barrier height since the Euclidean action vanishes at this energy. This result is generally incorrect. It also leads to an inaccurate estimate of the leading order term in an ℏ2n expansion of the thermal transmission coefficient. The central result of this paper is that adding an ℏ2 dependent correction to the uniform semiclassical expression, whether as a constant action or as a shift in the energy scale, not only corrects this inaccuracy but also leads to a theory that is more accurate than the previous one for almost any energy. Shifting the energy scale is a generalization of the vibrational perturbation theory 2 (VPT2) and is much more accurate than the "standard" VPT2 theory, especially when the potential is asymmetric. Shifting the action by a constant is a generalization of a result obtained by Yasumori and Fueki (YF) only for the Eckart barrier. The resulting modified VPT2 and YF semiclassical theories are applied to the symmetric and asymmetric Eckart barrier, a Gaussian barrier, and a tanh barrier. The one-dimensional theories are also generalized to many-dimensional systems. Their effect on the thermal instanton theory is discussed.
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Affiliation(s)
- Sameernandan Upadhyayula
- Chemical and Biological Physics
Department, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Eli Pollak
- Chemical and Biological Physics
Department, Weizmann Institute of Science, Rehovot 76100, Israel
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4
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Pollak E. A personal perspective of the present status and future challenges facing thermal reaction rate theory. J Chem Phys 2024; 160:150902. [PMID: 38639316 DOI: 10.1063/5.0199557] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2024] [Accepted: 03/06/2024] [Indexed: 04/20/2024] Open
Abstract
Reaction rate theory has been at the center of physical chemistry for well over one hundred years. The evolution of the theory is not only of historical interest. Reliable and accurate computation of reaction rates remains a challenge to this very day, especially in view of the development of quantum chemistry methods, which predict the relevant force fields. It is still not possible to compute the numerically exact rate on the fly when the system has more than at most a few dozen anharmonic degrees of freedom, so one must consider various approximate methods, not only from the practical point of view of constructing numerical algorithms but also on conceptual and formal levels. In this Perspective, I present some of the recent analytical results concerning leading order terms in an ℏ2m series expansion of the exact rate and their implications on various approximate theories. A second aspect has to do with the crossover temperature between tunneling and thermal activation. Using a uniform semiclassical transmission probability rather than the "primitive" semiclassical theory leads to the conclusion that there is no divergence problem associated with a "crossover temperature." If one defines a semiclassical crossover temperature as the point at which the tunneling energy of the instanton equals the barrier height, then it is a factor of two higher than its previous estimate based on the "primitive" semiclassical approximation. In the low temperature tunneling regime, the uniform semiclassical theory as well as the "primitive" semiclassical theory were based on the classical Euclidean action of a periodic orbit on the inverted potential. The uniform semiclassical theory wrongly predicts that the "half-point," which is the energy at which the transmission probability equals 1/2, for any barrier potential, is always the barrier energy. We describe here how augmenting the Euclidean action with constant terms of order ℏ2 can significantly improve the accuracy of the semiclassical theory and correct this deficiency. This also leads to a deep connection with and improvement of vibrational perturbation theory. The uniform semiclassical theory also enables an extension of the quantum version of Kramers' turnover theory to temperatures below the "crossover temperature." The implications of these recent advances on various approximate methods used to date are discussed at length, leading to the conclusion that reaction rate theory will continue to challenge us both on conceptual and practical levels for years to come.
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Affiliation(s)
- Eli Pollak
- Chemical and Biological Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
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5
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Miyazaki K, Ananth N. Nonadiabatic simulations of photoisomerization and dissociation in ethylene using ab initio classical trajectories. J Chem Phys 2023; 159:124110. [PMID: 38127384 DOI: 10.1063/5.0163371] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2023] [Accepted: 09/05/2023] [Indexed: 12/23/2023] Open
Abstract
We simulate the nonadiabatic dynamics of photo-induced isomerization and dissociation in ethylene using ab initio classical trajectories in an extended phase space of nuclear and electronic variables. This is achieved by employing the linearized semiclassical initial value representation method for nonadiabatic dynamics, where discrete electronic states are mapped to continuous classical variables using either the Meyer-Miller-Stock-Thoss representation or a more recently introduced spin mapping approach. Trajectory initial conditions are sampled by constraining electronic state variables to a single initial excited state and by drawing nuclear phase space configurations from a Wigner distribution at a finite temperature. An ensemble of classical ab initio trajectories is then generated to compute thermal population correlation functions and analyze the mechanisms of isomerization and dissociation. Our results serve as a demonstration that this parameter-free semiclassical approach is computationally efficient and accurate, identifying mechanistic pathways in agreement with previous theoretical studies and also uncovering dissociation pathways observed experimentally.
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Affiliation(s)
- K Miyazaki
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - N Ananth
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
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6
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Gardner J, Corken D, Janke SM, Habershon S, Maurer RJ. Efficient implementation and performance analysis of the independent electron surface hopping method for dynamics at metal surfaces. J Chem Phys 2023; 158:064101. [PMID: 36792522 DOI: 10.1063/5.0137137] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Abstract
Independent electron surface hopping (IESH) is a computational algorithm for simulating the mixed quantum-classical molecular dynamics of adsorbate atoms and molecules interacting with metal surfaces. It is capable of modeling the nonadiabatic effects of electron-hole pair excitations on molecular dynamics. Here, we present a transparent, reliable, and efficient implementation of IESH, demonstrating its ability to predict scattering and desorption probabilities across a variety of systems, ranging from model Hamiltonians to full dimensional atomistic systems. We further show how the algorithm can be modified to account for the application of an external bias potential, comparing its accuracy to results obtained using the hierarchical quantum master equation. Our results show that IESH is a practical method for modeling coupled electron-nuclear dynamics at metal surfaces, especially for highly energetic scattering events.
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Affiliation(s)
- James Gardner
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Daniel Corken
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Svenja M Janke
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Scott Habershon
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Reinhard J Maurer
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
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7
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Poulsen JA, Nyman G. Computing reaction rates using a Wigner function based on a new effective frequency theory. Mol Phys 2023. [DOI: 10.1080/00268976.2023.2174361] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/09/2023]
Affiliation(s)
- Jens Aage Poulsen
- Department of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden
| | - Gunnar Nyman
- Department of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden
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8
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Chen Z, Yang Y. Incorporating Nuclear Quantum Effects in Molecular Dynamics with a Constrained Minimized Energy Surface. J Phys Chem Lett 2023; 14:279-286. [PMID: 36595586 DOI: 10.1021/acs.jpclett.2c02905] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
The accurate incorporation of nuclear quantum effects in large-scale molecular dynamics (MD) simulations remains a significant challenge. Recently, we combined constrained nuclear-electronic orbital (CNEO) theory with classical MD and obtained a new approach (CNEO-MD) that can accurately and efficiently incorporate nuclear quantum effects into classical simulations. In this Letter, we provide the theoretical foundation for CNEO-MD by developing an alternative formulation of the equations of motion for MD. In this new formulation, the expectation values of quantum nuclear positions evolve classically on an effective energy surface that is obtained from a constrained energy minimization procedure when solving for the quantum nuclear wave function, thus enabling the incorporation of nuclear quantum effects in classical MD simulations. For comparison with other existing approaches, we examined a series of model systems and found that this new MD approach is significantly more accurate than the conventional way of performing classical MD and generally outperforms centroid MD and ring-polymer MD in describing vibrations in these model systems.
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Affiliation(s)
- Zehua Chen
- Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, 1101 University Avenue, Madison, Wisconsin53706, United States
| | - Yang Yang
- Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin-Madison, 1101 University Avenue, Madison, Wisconsin53706, United States
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9
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Angiolari F, Huppert S, Spezia R. Quantum versus classical unimolecular fragmentation rate constants and activation energies at finite temperature from direct dynamics simulations. Phys Chem Chem Phys 2022; 24:29357-29370. [PMID: 36448557 DOI: 10.1039/d2cp03809a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Abstract
In the present work, we investigate how nuclear quantum effects modify the temperature dependent rate constants and, consequently, the activation energies in unimolecular reactions. In the reactions under study, nuclear quantum effects mainly stem from the presence of a large zero point energy. Thus, we investigate the behavior of methods compatible with direct dynamics simulations, the quantum thermal bath (QTB) and ring polymer molecular dynamics (RPMD). To this end, we first compare them with quantum reaction theory for a model Morse potential before extending this comparison to molecular models. Our results show that, in particular in the temperature range comparable with or lower than the zero point energy of the system, the RPMD method is able to correctly capture nuclear quantum effects on rate constants and activation energies. On the other hand, although the QTB provides a good description of equilibrium properties including zero-point energy effects, it largely overestimates the rate constants. The origin of the different behaviours is in the different distance distributions provided by the two methods and in particular how they differently describe the tails of such distributions. The comparison with transition state theory shows that RPMD can be used to study fragmentation of complex systems for which it may be difficult to determine the multiple reaction pathways and associated transition states.
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Affiliation(s)
- Federica Angiolari
- Sorbonne Université, Laboratoire de Chimie Théorique, UMR 7616 CNRS, 4 Place Jussieu, 75005 Paris, France.
| | - Simon Huppert
- Sorbonne Université, Institut de Nanosciences de Paris, UMR 7588 CNRS, 4 Place Jussieu, 75005 Paris, France
| | - Riccardo Spezia
- Sorbonne Université, Laboratoire de Chimie Théorique, UMR 7616 CNRS, 4 Place Jussieu, 75005 Paris, France.
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10
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Tan JA, Takahashi K. Vibrational spectrum of a 1D oscillator: The quantum, the Wigner, and the classical ways. J CHIN CHEM SOC-TAIP 2022. [DOI: 10.1002/jccs.202200398] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Jake A. Tan
- Institute of Atomic and Molecular Sciences Academia Sinica Taipei Taiwan
| | - Kaito Takahashi
- Institute of Atomic and Molecular Sciences Academia Sinica Taipei Taiwan
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11
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Liu Z, Xu W, Tuckerman ME, Sun X. Imaginary-Time Open-Chain Path-Integral Approach for Two-State Time Correlation Functions and Applications in Charge Transfer. J Chem Phys 2022; 157:114111. [DOI: 10.1063/5.0098162] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Quantum time correlation functions (TCFs) involving two states are important for describing nonadiabatic dynamical processes such as charge transfer. Based on a previous single-state method, we propose an imaginary-time open-chain path-integral (OCPI) approach for evaluating the two-state symmetrized TCFs. Expressing the forward and backward propagation on different electronic potential energy surfaces as a complex-time path integral, we then transform the path variables to average and difference variables such that the integration over the difference variables up to the second order can be performed analytically. The resulting expression for the symmetrized TCF is equivalent to sampling the open-chain configurations in an effective potential that corresponds to the average surface. Using importance sampling over the extended OCPI space via open path integral molecular dynamics, we tested the resulting path-integral approximation by calculating the Fermi's golden rule charge transfer rate constant within a widely-used spin-boson model. Comparing with the real-time linearized semiclassical method and analytical result, we show that the imaginary-time OCPI provides an accurate two-state symmetrized TCF and rate constant in the typical turnover region. It is shown that the first bead of the open chain corresponds to physical zero-time, and the endpoint bead corresponds to final time t; oscillations of the end-to-end distance perfectly match the nuclear mode frequency. The two-state OCPI scheme is seen to capture the tested model's electronic quantum coherence and nuclear quantum effects accurately.
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Affiliation(s)
- Zengkui Liu
- Division of Arts and Sciences, New York University Shanghai, China
| | - Wen Xu
- New York University Shanghai, China
| | - Mark E. Tuckerman
- Department of Chemistry and Courant Institute of Mathematical Sciences, New York University, United States of America
| | - Xiang Sun
- Division of Arts and Sciences, New York University Shanghai, China
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12
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Pollak E, Upadhyayula S, Liu J. Coherent state representation of thermal correlation functions with applications to rate theory. J Chem Phys 2022; 156:244101. [DOI: 10.1063/5.0088163] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022] Open
Abstract
A coherent state phase space representation of operators, based on the Husimi distribution, is used to derive an exact expression for the symmetrized version of thermal correlation functions. In addition to the time and temperature independent phase space representation of the two operators whose correlation function is of interest, the integrand includes a non-negative distribution function where only one imaginary time and one real time propagation are needed to compute it. The methodology is exemplified for the flux side correlation function used in rate theory. The coherent state representation necessitates the use of a smeared Gaussian flux operator whose coherent state phase space representation is identical to the classical flux expression. The resulting coherent state expression for the flux side correlation function has a number of advantages as compared to previous formulations. Since only one time propagation is needed, it is much easier to converge it with a semiclassical initial value representation. There is no need for forward–backward approximations, and in principle, the computation may be implemented on the fly. It also provides a route for analytic semiclassical approximations for the thermal rate, as exemplified by a computation of the transmission factor through symmetric and asymmetric Eckart barriers using a thawed Gaussian approximation for both imaginary and real time propagations. As a by-product, this example shows that one may obtain “good” tunneling rates using only above barrier classical trajectories even in the deep tunneling regime.
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Affiliation(s)
- Eli Pollak
- Chemical and Biological Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
| | - Sameernandan Upadhyayula
- Chemical and Biological Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
| | - 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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13
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He X, Wu B, Shang Y, Li B, Cheng X, Liu J. New phase space formulations and quantum dynamics approaches. WIRES COMPUTATIONAL MOLECULAR SCIENCE 2022. [DOI: 10.1002/wcms.1619] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
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 China
| | - Baihua Wu
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Youhao Shang
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Bingqi Li
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Xiangsong Cheng
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Jian Liu
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
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14
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Gardner J, Douglas-Gallardo OA, Stark WG, Westermayr J, Janke SM, Habershon S, Maurer RJ. NQCDynamics.jl: A Julia package for nonadiabatic quantum classical molecular dynamics in the condensed phase. J Chem Phys 2022; 156:174801. [PMID: 35525649 DOI: 10.1063/5.0089436] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
Accurate and efficient methods to simulate nonadiabatic and quantum nuclear effects in high-dimensional and dissipative systems are crucial for the prediction of chemical dynamics in the condensed phase. To facilitate effective development, code sharing, and uptake of newly developed dynamics methods, it is important that software implementations can be easily accessed and built upon. Using the Julia programming language, we have developed the NQCDynamics.jl package, which provides a framework for established and emerging methods for performing semiclassical and mixed quantum-classical dynamics in the condensed phase. The code provides several interfaces to existing atomistic simulation frameworks, electronic structure codes, and machine learning representations. In addition to the existing methods, the package provides infrastructure for developing and deploying new dynamics methods, which we hope will benefit reproducibility and code sharing in the field of condensed phase quantum dynamics. Herein, we present our code design choices and the specific Julia programming features from which they benefit. We further demonstrate the capabilities of the package on two examples of chemical dynamics in the condensed phase: the population dynamics of the spin-boson model as described by a wide variety of semiclassical and mixed quantum-classical nonadiabatic methods and the reactive scattering of H2 on Ag(111) using the molecular dynamics with electronic friction method. Together, they exemplify the broad scope of the package to study effective model Hamiltonians and realistic atomistic systems.
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Affiliation(s)
- James Gardner
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Oscar A Douglas-Gallardo
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Wojciech G Stark
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Julia Westermayr
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Svenja M Janke
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Scott Habershon
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
| | - Reinhard J Maurer
- Department of Chemistry, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
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15
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Poulsen JA, Nyman G. A Divergence-Free Wigner Transform of the Boltzmann Operator Based on an Effective Frequency Theory. J Phys Chem A 2021; 125:9209-9225. [PMID: 34636560 PMCID: PMC8543443 DOI: 10.1021/acs.jpca.1c05860] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022]
Abstract
![]()
The centroid effective
frequency representation of path integrals
as developed by Feynman and Kleinert was originally aimed at calculating
partition functions and related quantities in the canonical ensemble.
In its path integral formulation, only closed paths
were relevant. This formulation has been used by the present authors
in order to calculate the many-body Wigner function of the Boltzmann
operator, which includes also open paths. This usage of the theory
outside of the original intention can lead to mathematical divergence
issues for potentials with barriers, particularly at low temperature.
In the present paper, we modify the effective frequency theory of
Feynman and Kleinert by also including open paths in its variational
equations. In this way, a divergence-free approximation to the Boltzmann
operator matrix elements is derived. This generalized version of Feynman
and Kleinert’s formulation is thus more robust and can be applied
to all types of barriers at all temperatures. This new version is
used to calculate the Wigner functions of the Boltzmann operator for
a quartic oscillator and for a double well potential and both static
and dynamic properties are studied at several temperatures. The new
theory is found to be essentially as precise as the original one.
Its advantage is that it will always deliver a well-defined, even
if approximate, Wigner function, which can, for instance, be used
for sampling initial conditions for molecular dynamics simulations.
As will be discussed, the theory can be systematically improved by
including higher-order Fourier modes into the nonquadratic part of
the trial action.
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Affiliation(s)
- Jens Aage Poulsen
- Department of Chemistry and Molecular Biology, University of Gothenburg, SE 405 30 Gothenburg, Sweden
| | - Gunnar Nyman
- Department of Chemistry and Molecular Biology, University of Gothenburg, SE 405 30 Gothenburg, Sweden
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16
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Svensson SKM, Poulsen JA, Nyman G. Calculation of reaction rate constants from a classical Wigner model based on a Feynman path integral open polymer. Mol Phys 2021. [DOI: 10.1080/00268976.2021.1970264] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
| | - Jens Aage Poulsen
- Department of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden
| | - Gunnar Nyman
- Department of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden
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17
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Benson RL, Althorpe SC. On the "Matsubara heating" of overtone intensities and Fermi splittings. J Chem Phys 2021; 155:104107. [PMID: 34525826 DOI: 10.1063/5.0056829] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022] Open
Abstract
Classical molecular dynamics (MD) and imaginary-time path-integral dynamics methods underestimate the infrared absorption intensities of overtone and combination bands by typically an order of magnitude. Plé et al. [J. Chem. Phys. 155, 2863 (2021)] have shown that this is because such methods fail to describe the coupling of the centroid to the Matsubara dynamics of the fluctuation modes; classical first-order perturbation theory (PT) applied to the Matsubara dynamics is sufficient to recover most of the lost intensity in simple models and gives identical results to quantum (Rayleigh-Schrödinger) PT. Here, we show numerically that the results of this analysis can be used as post-processing correction factors, which can be applied to realistic (classical MD or path-integral dynamics) simulations of infrared spectra. We find that the correction factors recover most of the lost intensity in the overtone and combination bands of gas-phase water and ammonia and much of it for liquid water. We then re-derive and confirm the earlier PT analysis by applying canonical PT to Matsubara dynamics, which has the advantage of avoiding secular terms and gives a simple picture of the perturbed Matsubara dynamics in terms of action-angle variables. Collectively, these variables "Matsubara heat" the amplitudes of the overtone and combination vibrations of the centroid to what they would be in a classical system with the oscillators (of frequency Ωi) held at their quantum effective temperatures [of ℏΩi coth(βℏΩi/2)/2kB]. Numerical calculations show that a similar neglect of "Matsubara heating" causes path-integral methods to underestimate Fermi resonance splittings.
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Affiliation(s)
- Raz L Benson
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Stuart C Althorpe
- Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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18
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Plé T, Huppert S, Finocchi F, Depondt P, Bonella S. Anharmonic spectral features via trajectory-based quantum dynamics: A perturbative analysis of the interplay between dynamics and sampling. J Chem Phys 2021; 155:104108. [PMID: 34525824 DOI: 10.1063/5.0056824] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The performance of different approximate algorithms for computing anharmonic features in vibrational spectra is analyzed and compared on model and more realistic systems that present relevant nuclear quantum effects. The methods considered combine approximate sampling of the quantum thermal distribution with classical time propagation and include Matsubara dynamics, path integral dynamics approaches, linearized initial value representation, and the recently introduced adaptive quantum thermal bath. A perturbative analysis of these different methods enables us to account for the observed numerical performance on prototypes for overtones and combination bands and to draw qualitatively correct trends for the numerical results obtained for Fermi resonances. Our results prove that the unequal performances of these approaches often derive from the method employed to sample initial conditions and not, as usually assumed, from the lack of coherence in the time propagation. Furthermore, as confirmed by the analysis reported in Benson and Althorpe, J. Chem. Phys. 130, 194510 (2021), we demonstrate, both via the perturbative approach and numerically, that path integral dynamics methods fail to reproduce the intensities of these anharmonic features and follow purely classical trends with respect to their temperature behavior. Finally, the remarkably accurate performance of the adaptive quantum thermal bath approach is documented and motivated.
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Affiliation(s)
- Thomas Plé
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Simon Huppert
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Fabio Finocchi
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Philippe Depondt
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Sara Bonella
- CECAM Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne, Batochimie, Avenue Forel 2, 1015 Lausanne, Switzerland
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19
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Liu X, Zhang L, Liu J. Machine learning phase space quantum dynamics approaches. J Chem Phys 2021; 154:184104. [PMID: 34241027 DOI: 10.1063/5.0046689] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/06/2023] Open
Abstract
Derived from phase space expressions of the quantum Liouville theorem, equilibrium continuity dynamics is a category of trajectory-based phase space dynamics methods, which satisfies the two critical fundamental criteria: conservation of the quantum Boltzmann distribution for the thermal equilibrium system and being exact for any thermal correlation functions (even of nonlinear operators) in the classical and harmonic limits. The effective force and effective mass matrix are important elements in the equations of motion of equilibrium continuity dynamics, where only the zeroth term of an exact series expansion of the phase space propagator is involved. We introduce a machine learning approach for fitting these elements in quantum phase space, leading to a much more efficient integration of the equations of motion. Proof-of-concept applications to realistic molecules demonstrate that machine learning phase space dynamics approaches are possible as well as competent in producing reasonably accurate results with a modest computation effort.
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Affiliation(s)
- Xinzijian Liu
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Linfeng Zhang
- Beijing Institute of Big Data Research, 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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20
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Luo R, Yu K. Capturing the nuclear quantum effects in molecular dynamics for lattice thermal conductivity calculations: Using ice as example. J Chem Phys 2020; 153:194105. [DOI: 10.1063/5.0022013] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Ripeng Luo
- Tsinghua Shenzhen International Graduate School, Tsinghua-Berkeley Shenzhen Institute (TBSI), Tsinghua University, 1101 Xueyuan Road, Building C2, Shenzhen, Guangdong 518055, China
| | - Kuang Yu
- Tsinghua Shenzhen International Graduate School, Tsinghua-Berkeley Shenzhen Institute (TBSI), Tsinghua University, 1101 Xueyuan Road, Building C2, Shenzhen, Guangdong 518055, China
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21
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Zhang ZJ, Chen ZF, Liu J. Path integral Liouville dynamics simulations of vibrational spectra of formaldehyde and hydrogen peroxide. CHINESE J CHEM PHYS 2020. [DOI: 10.1063/1674-0068/cjcp2006099] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/29/2023]
Affiliation(s)
- Zhi-jun Zhang
- Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China
| | - Zi-fei Chen
- 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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22
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Svensson SKM, Poulsen JA, Nyman G. Classical Wigner model based on a Feynman path integral open polymer. J Chem Phys 2020; 152:094111. [DOI: 10.1063/1.5126183] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- S. Karl-Mikael Svensson
- Department of Chemistry and Molecular Biology, University of Gothenburg, SE 405 30 Gothenburg, Sweden
| | - Jens Aage Poulsen
- Department of Chemistry and Molecular Biology, University of Gothenburg, SE 405 30 Gothenburg, Sweden
| | - Gunnar Nyman
- Department of Chemistry and Molecular Biology, University of Gothenburg, SE 405 30 Gothenburg, Sweden
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23
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Benson RL, Trenins G, Althorpe SC. Which quantum statistics–classical dynamics method is best for water? Faraday Discuss 2020; 221:350-366. [DOI: 10.1039/c9fd00077a] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
We make a systematic comparison of TRPMD, CMD, QCMD and LSC-IVR by calculating the infrared spectrum of water in its gas, liquid and ice phases.
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Affiliation(s)
- Raz L. Benson
- Department of Chemistry
- University of Cambridge
- Cambridge
- UK
| | - George Trenins
- Department of Chemistry
- University of Cambridge
- Cambridge
- UK
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24
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Plé T, Huppert S, Finocchi F, Depondt P, Bonella S. Sampling the thermal Wigner density via a generalized Langevin dynamics. J Chem Phys 2019; 151:114114. [PMID: 31542021 DOI: 10.1063/1.5099246] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The Wigner thermal density is a function of considerable interest in the area of approximate (linearized or semiclassical) quantum dynamics where it is employed to generate initial conditions for the propagation of appropriate sets of classical trajectories. In this paper, we propose an original approach to compute the Wigner density based on a generalized Langevin equation. The stochastic dynamics is nontrivial in that it contains a coordinate-dependent friction coefficient and a generalized force that couples momenta and coordinates. These quantities are, in general, not known analytically and have to be estimated via auxiliary calculations. The performance of the new sampling scheme is tested on standard model systems with highly nonclassical features such as relevant zero point energy effects, correlation between momenta and coordinates, and negative parts of the Wigner density. In its current brute force implementation, the algorithm, whose convergence can be systematically checked, is accurate and has only limited overhead compared to schemes with similar characteristics. We briefly discuss potential ways to further improve its numerical efficiency.
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Affiliation(s)
- Thomas Plé
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Simon Huppert
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Fabio Finocchi
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Philippe Depondt
- Sorbonne Université, CNRS, Institut des NanoSciences de Paris, INSP, 4 Place Jussieu, F-75005 Paris, France
| | - Sara Bonella
- CECAM Centre Européen de Calcul Atomique et Moléculaire, École Polytechnique Fédérale de Lausanne, Batochimie, Avenue Forel 2, 1015 Lausanne, Switzerland
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25
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Trenins G, Willatt MJ, Althorpe SC. Path-integral dynamics of water using curvilinear centroids. J Chem Phys 2019. [DOI: 10.1063/1.5100587] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- George Trenins
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Michael J. Willatt
- Laboratory of Computational Science and Modeling, IMX, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Stuart C. Althorpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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26
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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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27
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Lu X, Wang X, Fu B, Zhang D. Theoretical Investigations of Rate Coefficients of H + H2O2 → OH + H2O on a Full-Dimensional Potential Energy Surface. J Phys Chem A 2019; 123:3969-3976. [DOI: 10.1021/acs.jpca.9b02526] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Xiaoxiao Lu
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, Dalian 116023, China
- Department of Chemical Physics, University of Science and Technology of China, Jinzhai Road 96, Hefei 230026, China
| | - Xingan Wang
- Department of Chemical Physics, University of Science and Technology of China, Jinzhai Road 96, Hefei 230026, China
| | - Bina Fu
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, Dalian 116023, China
| | - Donghui Zhang
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, Dalian 116023, China
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28
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Lu X, Meng Q, Wang X, Fu B, Zhang DH. Rate coefficients of the H + H2O2→ H2+ HO2reaction on an accurate fundamental invariant-neural network potential energy surface. J Chem Phys 2018; 149:174303. [PMID: 30409010 DOI: 10.1063/1.5063613] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Xiaoxiao Lu
- Department of Chemical Physics, University of Science and Technology of China, Jinzhai Road 96, Hefei 230026, China
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, Dalian 116023, China
| | - Qingyong Meng
- Department of Applied Chemistry, Northwestern Polytechnical University, Youyi West Road 127, Xi’an 710072, China
| | - Xingan Wang
- Department of Chemical Physics, University of Science and Technology of China, Jinzhai Road 96, Hefei 230026, China
| | - Bina Fu
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, Dalian 116023, China
| | - Dong H. Zhang
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, Dalian 116023, China
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29
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Trenins G, Althorpe SC. Mean-field Matsubara dynamics: Analysis of path-integral curvature effects in rovibrational spectra. J Chem Phys 2018; 149:014102. [DOI: 10.1063/1.5038616] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- George Trenins
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Stuart C. Althorpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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30
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Willatt MJ, Ceriotti M, Althorpe SC. Approximating Matsubara dynamics using the planetary model: Tests on liquid water and ice. J Chem Phys 2018; 148:102336. [PMID: 29544307 DOI: 10.1063/1.5004808] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Matsubara dynamics is the quantum-Boltzmann-conserving classical dynamics which remains when real-time coherences are taken out of the exact quantum Liouvillian [T. J. H. Hele et al., J. Chem. Phys. 142, 134103 (2015)]; because of a phase-term, it cannot be used as a practical method without further approximation. Recently, Smith et al. [J. Chem. Phys. 142, 244112 (2015)] developed a "planetary" model dynamics which conserves the Feynman-Kleinert (FK) approximation to the quantum-Boltzmann distribution. Here, we show that for moderately anharmonic potentials, the planetary dynamics gives a good approximation to Matsubara trajectories on the FK potential surface by decoupling the centroid trajectory from the locally harmonic Matsubara fluctuations, which reduce to a single phase-less fluctuation particle (the "planet"). We also show that the FK effective frequency can be approximated by a direct integral over these fluctuations, obviating the need to solve iterative equations. This modification, together with use of thermostatted ring-polymer molecular dynamics, allows us to test the planetary model on water (gas-phase, liquid, and ice) using the q-TIP4P/F potential surface. The "planetary" fluctuations give a poor approximation to the rotational/librational bands in the infrared spectrum, but a good approximation to the bend and stretch bands, where the fluctuation lineshape is found to be motionally narrowed by the vibrations of the centroid.
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Affiliation(s)
- Michael J Willatt
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Michele Ceriotti
- Laboratory of Computational Science and Modeling, Institute of Materials, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
| | - Stuart C Althorpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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31
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Cendagorta JR, Bačić Z, Tuckerman ME. An open-chain imaginary-time path-integral sampling approach to the calculation of approximate symmetrized quantum time correlation functions. J Chem Phys 2018; 148:102340. [PMID: 29544313 DOI: 10.1063/1.5005543] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
Abstract
We introduce a scheme for approximating quantum time correlation functions numerically within the Feynman path integral formulation. Starting with the symmetrized version of the correlation function expressed as a discretized path integral, we introduce a change of integration variables often used in the derivation of trajectory-based semiclassical methods. In particular, we transform to sum and difference variables between forward and backward complex-time propagation paths. Once the transformation is performed, the potential energy is expanded in powers of the difference variables, which allows us to perform the integrals over these variables analytically. The manner in which this procedure is carried out results in an open-chain path integral (in the remaining sum variables) with a modified potential that is evaluated using imaginary-time path-integral sampling rather than requiring the generation of a large ensemble of trajectories. Consequently, any number of path integral sampling schemes can be employed to compute the remaining path integral, including Monte Carlo, path-integral molecular dynamics, or enhanced path-integral molecular dynamics. We believe that this approach constitutes a different perspective in semiclassical-type approximations to quantum time correlation functions. Importantly, we argue that our approximation can be systematically improved within a cumulant expansion formalism. We test this approximation on a set of one-dimensional problems that are commonly used to benchmark approximate quantum dynamical schemes. We show that the method is at least as accurate as the popular ring-polymer molecular dynamics technique and linearized semiclassical initial value representation for correlation functions of linear operators in most of these examples and improves the accuracy of correlation functions of nonlinear operators.
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Affiliation(s)
| | - Zlatko Bačić
- Department of Chemistry, New York University, New York, New York 10003, USA
| | - Mark E Tuckerman
- Department of Chemistry, New York University, New York, New York 10003, USA
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32
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Robertson C, Habershon S. Harmonic-phase path-integral approximation of thermal quantum correlation functions. J Chem Phys 2018; 148:102316. [PMID: 29544325 DOI: 10.1063/1.5002189] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present an approximation to the thermal symmetric form of the quantum time-correlation function in the standard position path-integral representation. By transforming to a sum-and-difference position representation and then Taylor-expanding the potential energy surface of the system to second order, the resulting expression provides a harmonic weighting function that approximately recovers the contribution of the phase to the time-correlation function. This method is readily implemented in a Monte Carlo sampling scheme and provides exact results for harmonic potentials (for both linear and non-linear operators) and near-quantitative results for anharmonic systems for low temperatures and times that are likely to be relevant to condensed phase experiments. This article focuses on one-dimensional examples to provide insights into convergence and sampling properties, and we also discuss how this approximation method may be extended to many-dimensional systems.
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Affiliation(s)
- Christopher Robertson
- Department of Chemistry and Centre for Scientific Computing, University Of Warwick, Coventry CV4 7AL, United Kingdom
| | - Scott Habershon
- Department of Chemistry and Centre for Scientific Computing, University Of Warwick, Coventry CV4 7AL, United Kingdom
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33
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Affiliation(s)
- Xinzijian Liu
- Beijing National Laboratory For Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing, China
| | - Jian Liu
- Beijing National Laboratory For Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, Beijing, China
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34
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Shigeta Y, Harada R, Sato R, Kitoh-Nishioka H, Bui TKM, Sato A, Tokiwa T, Kyan A, Ishii Y, Kimatsuka M, Yamasaki S, Kayanuma M, Shoji M. Classical cumulant dynamics for statistical chemical physics. MOLECULAR SIMULATION 2017. [DOI: 10.1080/08927022.2017.1315770] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
Affiliation(s)
- Yasuteru Shigeta
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Ryuhei Harada
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Ryuma Sato
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | | | - Thi Kieu My Bui
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Akimasa Sato
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Takaki Tokiwa
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Akane Kyan
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Yuki Ishii
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Masato Kimatsuka
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Sotaro Yamasaki
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Megumi Kayanuma
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
| | - Mitsuo Shoji
- Center for Computational Sciences, University of Tsukuba, Tsukuba, Japan
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35
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Nagashima H, Tsuda S, Tsuboi N, Hayashi AK, Tokumasu T. A molecular dynamics study of nuclear quantum effect on diffusivity of hydrogen molecule. J Chem Phys 2017; 147:024501. [PMID: 28711051 DOI: 10.1063/1.4991732] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this paper, the nuclear quantum effect of the hydrogen molecule on its diffusivity was analyzed using the molecular dynamics (MD) method. The centroid MD (CMD) method was applied to reproduce the time evolution of the molecules. The diffusion coefficient of hydrogen was calculated using the Green-Kubo method over a wide temperature region, and the temperature dependence of the quantum effect of the hydrogen molecule on its diffusivity was addressed. The calculated results were compared with classical MD results based on the principle of corresponding state (PCS). It was confirmed that the difference in the diffusion coefficient calculated in the CMD and classical MD methods was small, and the PCS appears to be satisfied on the temperature dependence of the diffusion coefficient, even though the quantum effect of the hydrogen molecules was taken into account. It was clarified that this result did not suggest that the quantum effect on the diffusivity of the hydrogen molecule was small but that the two changes in the intermolecular interaction of hydrogen due to the quantum effect offset each other. Moreover, it was found that this tendency was related to the temperature dependence of the ratio of the kinetic energy of the quantum fluctuational motion to the classical kinetic energy.
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Affiliation(s)
- H Nagashima
- Department of Mechanical Systems Engineering, University of the Ryukuys, Okinawa, Senbaru, Nishihara-cho, Okinawa 903-0213, Japan
| | - S Tsuda
- Department of Mechanical Engineering, Kyushu University, Fukuoka 819-0395, Japan
| | - N Tsuboi
- Department of Mechanical and Control Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan
| | - A K Hayashi
- Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara 229-8558, Japan
| | - T Tokumasu
- Institute of Fluid Science, Tohoku University, Sendai 980-8577, Japan
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36
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Affiliation(s)
- Timothy J. H. Hele
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY, USA
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37
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Montoya-Castillo A, Reichman DR. Approximate but accurate quantum dynamics from the Mori formalism. II. Equilibrium time correlation functions. J Chem Phys 2017; 146:084110. [DOI: 10.1063/1.4975388] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - David R. Reichman
- Department of Chemistry, Columbia University, New York, New York 10027, USA
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38
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Montoya-Castillo A, Reichman DR. Path integral approach to the Wigner representation of canonical density operators for discrete systems coupled to harmonic baths. J Chem Phys 2017; 146:024107. [DOI: 10.1063/1.4973646] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - David R. Reichman
- Department of Chemistry, Columbia University, New York, New York 10027, USA
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39
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Welsch R, Song K, Shi Q, Althorpe SC, Miller TF. Non-equilibrium dynamics from RPMD and CMD. J Chem Phys 2016; 145:204118. [DOI: 10.1063/1.4967958] [Citation(s) in RCA: 42] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Affiliation(s)
- Ralph Welsch
- Division of Chemistry and Chemical Engineering, California Institute of Technology,1200 E. California Blvd., Pasadena, California 91125, USA
| | - Kai 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
| | - Stuart C. Althorpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Thomas F. Miller
- Division of Chemistry and Chemical Engineering, California Institute of Technology,1200 E. California Blvd., Pasadena, California 91125, USA
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40
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Hasegawa T. Communication: Constant uncertainty molecular dynamics: A simple and efficient algorithm to incorporate quantum nature into a real-time molecular dynamics simulation. J Chem Phys 2016; 145:171101. [DOI: 10.1063/1.4966917] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Taisuke Hasegawa
- Department of Chemistry, Graduate School of Science, Kyoto University, Sakyoku, Kyoto 606-8502, Japan
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41
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Mielke SL, Truhlar DG. A whole-path importance-sampling scheme for Feynman path integral calculations of absolute partition functions and free energies. J Chem Phys 2016; 144:034110. [PMID: 26801023 DOI: 10.1063/1.4939869] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022] Open
Abstract
Using Feynman path integrals, a molecular partition function can be written as a double integral with the inner integral involving all closed paths centered at a given molecular configuration, and the outer integral involving all possible molecular configurations. In previous work employing Monte Carlo methods to evaluate such partition functions, we presented schemes for importance sampling and stratification in the molecular configurations that constitute the path centroids, but we relied on free-particle paths for sampling the path integrals. At low temperatures, the path sampling is expensive because the paths can travel far from the centroid configuration. We now present a scheme for importance sampling of whole Feynman paths based on harmonic information from an instantaneous normal mode calculation at the centroid configuration, which we refer to as harmonically guided whole-path importance sampling (WPIS). We obtain paths conforming to our chosen importance function by rejection sampling from a distribution of free-particle paths. Sample calculations on CH4 demonstrate that at a temperature of 200 K, about 99.9% of the free-particle paths can be rejected without integration, and at 300 K, about 98% can be rejected. We also show that it is typically possible to reduce the overhead associated with the WPIS scheme by sampling the paths using a significantly lower-order path discretization than that which is needed to converge the partition function.
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Affiliation(s)
- Steven L Mielke
- Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant St. S.E., Minneapolis, Minnesota 55455-0431, USA
| | - Donald G Truhlar
- Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant St. S.E., Minneapolis, Minnesota 55455-0431, USA
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Liu J, Zhang Z. Path integral Liouville dynamics: Applications to infrared spectra of OH, water, ammonia, and methane. J Chem Phys 2016; 144:034307. [DOI: 10.1063/1.4939953] [Citation(s) in RCA: 27] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- 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
- State Key Joint Laboratory of Environmental Simulation and Pollution Control, College of Environmental Sciences and Engineering, Peking University, Beijing 100871, China
| | - Zhijun Zhang
- 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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Meng Q, Chen J, Zhang DH. Communication: Rate coefficients of the H + CH4 → H2 + CH3 reaction from ring polymer molecular dynamics on a highly accurate potential energy surface. J Chem Phys 2015; 143:101102. [DOI: 10.1063/1.4930860] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Qingyong Meng
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, 116023 Dalian, China
| | - Jun Chen
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, 116023 Dalian, China
| | - Dong H. Zhang
- State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Zhongshan Road 457, 116023 Dalian, China
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44
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Hele TJH, Suleimanov YV. Should thermostatted ring polymer molecular dynamics be used to calculate thermal reaction rates? J Chem Phys 2015; 143:074107. [DOI: 10.1063/1.4928599] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Timothy J. H. Hele
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Yury V. Suleimanov
- Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi St., Nicosia 2121, Cyprus
- Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139, USA
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45
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Petersen J, Kay KG. Wave packet propagation across barriers by semiclassical initial value methods. J Chem Phys 2015; 143:014107. [PMID: 26156465 DOI: 10.1063/1.4923221] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.
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Affiliation(s)
- Jakob Petersen
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
| | - Kenneth G Kay
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
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Abstract
We show a new imaginary time path integral based method--path integral Liouville dynamics (PILD), which can be derived from the equilibrium Liouville dynamics [J. Liu and W. H. Miller, J. Chem. Phys. 134, 104101 (2011)] in the Wigner phase space. Numerical tests of PILD with the simple (white noise) Langevin thermostat have been made for two strongly anharmonic model problems. Since implementation of PILD does not request any specific form of the potential energy surface, the results suggest that PILD offers a potentially useful approach for general condensed phase molecular systems to have the two important properties: conserves the quantum canonical distribution and recovers exact thermal correlation functions (of even nonlinear operators, i.e., nonlinear functions of position or momentum operators) in the classical, high temperature, and harmonic limits.
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Affiliation(s)
- 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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Hele TJH, Willatt MJ, Muolo A, Althorpe SC. Boltzmann-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynamics”. J Chem Phys 2015; 142:134103. [DOI: 10.1063/1.4916311] [Citation(s) in RCA: 71] [Impact Index Per Article: 7.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Timothy J. H. Hele
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Michael J. Willatt
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Andrea Muolo
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
| | - Stuart C. Althorpe
- Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
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Beutier J, Borgis D, Vuilleumier R, Bonella S. Computing thermal Wigner densities with the phase integration method. J Chem Phys 2014; 141:084102. [DOI: 10.1063/1.4892597] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
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49
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Jang S, Sinitskiy AV, Voth GA. Can the ring polymer molecular dynamics method be interpreted as real time quantum dynamics? J Chem Phys 2014. [DOI: 10.1063/1.4870717] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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50
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Bai S, Xie W, Zhu L, Shi Q. Calculation of absorption spectra involving multiple excited states: Approximate methods based on the mixed quantum classical Liouville equation. J Chem Phys 2014; 140:084105. [DOI: 10.1063/1.4866367] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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