1
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Liu Z, Lyu N, Hu Z, Zeng H, Batista VS, Sun X. Benchmarking various nonadiabatic semiclassical mapping dynamics methods with tensor-train thermo-field dynamics. J Chem Phys 2024; 161:024102. [PMID: 38980091 DOI: 10.1063/5.0208708] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/16/2024] [Accepted: 06/20/2024] [Indexed: 07/10/2024] Open
Abstract
Accurate quantum dynamics simulations of nonadiabatic processes are important for studies of electron transfer, energy transfer, and photochemical reactions in complex systems. In this comparative study, we benchmark various approximate nonadiabatic dynamics methods with mapping variables against numerically exact calculations based on the tensor-train (TT) representation of high-dimensional arrays, including TT-KSL for zero-temperature dynamics and TT-thermofield dynamics for finite-temperature dynamics. The approximate nonadiabatic dynamics methods investigated include mixed quantum-classical Ehrenfest mean-field and fewest-switches surface hopping, linearized semiclassical mapping dynamics, symmetrized quasiclassical dynamics, the spin-mapping method, and extended classical mapping models. Different model systems were evaluated, including the spin-boson model for nonadiabatic dynamics in the condensed phase, the linear vibronic coupling model for electronic transition through conical intersections, the photoisomerization model of retinal, and Tully's one-dimensional scattering models. Our calculations show that the optimal choice of approximate dynamical method is system-specific, and the accuracy is sensitively dependent on the zero-point-energy parameter and the initial sampling strategy for the mapping variables.
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Affiliation(s)
- Zengkui Liu
- Division of Arts and Sciences, NYU Shanghai, 567 West Yangsi Road, Shanghai 200124, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- Department of Chemistry, New York University, New York, New York 10003, USA
| | - Ningyi Lyu
- Division of Arts and Sciences, NYU Shanghai, 567 West Yangsi Road, Shanghai 200124, China
- Department of Chemistry, Yale University, P.O. Box 208107, New Haven, Connecticut 06520-8107, USA
| | - Zhubin Hu
- Division of Arts and Sciences, NYU Shanghai, 567 West Yangsi Road, Shanghai 200124, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China
| | - Hao Zeng
- Division of Arts and Sciences, NYU Shanghai, 567 West Yangsi Road, Shanghai 200124, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China
| | - Victor S Batista
- Department of Chemistry, Yale University, P.O. Box 208107, New Haven, Connecticut 06520-8107, USA
| | - Xiang Sun
- Division of Arts and Sciences, NYU Shanghai, 567 West Yangsi Road, Shanghai 200124, China
- NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
- Department of Chemistry, New York University, New York, New York 10003, USA
- State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China
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2
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Gandolfi M, Ceotto M. Molecular Dynamics of Artificially Pair-Decoupled Systems: An Accurate Tool for Investigating the Importance of Intramolecular Couplings. J Chem Theory Comput 2023; 19:6093-6108. [PMID: 37698951 PMCID: PMC10536992 DOI: 10.1021/acs.jctc.3c00553] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2023] [Indexed: 09/14/2023]
Abstract
We propose a numerical technique to accurately simulate the vibrations of organic molecules in the gas phase, when pairs of atoms (or, in general, groups of degrees of freedom) are artificially decoupled, so that their motion is instantaneously decorrelated. The numerical technique we have developed is a symplectic integration algorithm that never requires computation of the force but requires estimates of the Hessian matrix. The theory we present to support our technique postulates a pair-decoupling Hamiltonian function, which parametrically depends on a decoupling coefficient α ∈ [0, 1]. The closer α is to 0, the more decoupled the selected atoms. We test the correctness of our numerical method on small molecular systems, and we apply it to study the vibrational spectroscopic features of salicylic acid at the Density Functional Theory ab initio level on a fitted potential. Our pair-decoupled simulations of salicylic acid show that decoupling hydrogen-bonded atoms do not significantly influence the frequencies of stretching modes, but enhance enormously the out-of-plane wagging and twisting motions of the hydroxyl and carboxyl groups to the point that the carboxyl and hydroxyl groups may overcome high potential energy barriers and change the salicylic acid conformation after a short simulation time. In addition, we found that the acidity of salicylic acid is more influenced by the dynamical couplings of the proton of the carboxylic group with the carbon ring than with the hydroxyl group.
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Affiliation(s)
- Michele Gandolfi
- Dipartimento di Chimica, Università degli Studi di Milano, via Golgi 19, 20133 Milano, Italy
| | - Michele Ceotto
- Dipartimento di Chimica, Università degli Studi di Milano, via Golgi 19, 20133 Milano, Italy
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3
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Cook L, Runeson JE, Richardson JO, Hele TJH. Which Algorithm Best Propagates the Meyer-Miller-Stock-Thoss Mapping Hamiltonian for Non-Adiabatic Dynamics? J Chem Theory Comput 2023; 19:6109-6125. [PMID: 37704193 PMCID: PMC10536990 DOI: 10.1021/acs.jctc.3c00709] [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: 06/28/2023] [Indexed: 09/15/2023]
Abstract
A common strategy to simulate mixed quantum-classical dynamics is by propagating classical trajectories with mapping variables, often using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian or the related spin-mapping approach. When mapping the quantum subsystem, the coupled dynamics reduce to a set of equations of motion to integrate. Several numerical algorithms have been proposed, but a thorough performance comparison appears to be lacking. Here, we compare three time-propagation algorithms for the MMST Hamiltonian: the Momentum Integral (MInt) (J. Chem. Phys., 2018, 148, 102326), the Split-Liouvillian (SL) (Chem. Phys., 2017, 482, 124-134), and the algorithm in J. Chem. Phys., 2012, 136, 084101 that we refer to as the Degenerate Eigenvalue (DE) algorithm due to the approximation required during derivation. We analyze the accuracy of individual trajectories, correlation functions, energy conservation, symplecticity, Liouville's theorem, and the computational cost. We find that the MInt algorithm is the only rigorously symplectic algorithm. However, comparable accuracy at a lower computational cost can be obtained with the SL algorithm. The approximation implicitly made within the DE algorithm conserves energy poorly, even for small timesteps, and thus leads to slightly different results. These results should guide future mapping-variable simulations.
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Affiliation(s)
- Lauren
E. Cook
- Department
of Chemistry, University College London, Christopher Ingold Building, London WC1H 0AJ, U.K.
| | - Johan E. Runeson
- Department
of Chemistry and Applied Biosciences, ETH
Zürich, Zürich 8093, Switzerland
| | - Jeremy O. Richardson
- Department
of Chemistry and Applied Biosciences, ETH
Zürich, Zürich 8093, Switzerland
| | - Timothy J. H. Hele
- Department
of Chemistry, University College London, Christopher Ingold Building, London WC1H 0AJ, U.K.
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4
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Amati G, Runeson JE, Richardson JO. On detailed balance in nonadiabatic dynamics: From spin spheres to equilibrium ellipsoids. J Chem Phys 2023; 158:064113. [PMID: 36792511 DOI: 10.1063/5.0137828] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
Trajectory-based methods that propagate classical nuclei on multiple quantum electronic states are often used to simulate nonadiabatic processes in the condensed phase. A long-standing problem of these methods is their lack of detailed balance, meaning that they do not conserve the equilibrium distribution. In this article, we investigate ideas for restoring detailed balance in mixed quantum-classical systems by tailoring the previously proposed spin-mapping approach to thermal equilibrium. We find that adapting the spin magnitude can recover the correct long-time populations but is insufficient to conserve the full equilibrium distribution. The latter can however be achieved by a more flexible mapping of the spin onto an ellipsoid, which is constructed to fulfill detailed balance for arbitrary potentials. This ellipsoid approach solves the problem of negative populations that has plagued previous mapping approaches and can therefore be applied also to strongly asymmetric and anharmonic systems. Because it conserves the thermal distribution, the method can also exploit efficient sampling schemes used in standard molecular dynamics, which drastically reduces the number of trajectories needed for convergence. The dynamics does however still have mean-field character, as is observed most clearly by evaluating reaction rates in the golden-rule limit. This implies that although the ellipsoid mapping provides a rigorous framework, further work is required to find an accurate classical-trajectory approximation that captures more properties of the true quantum dynamics.
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Affiliation(s)
- Graziano Amati
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
| | - Johan E Runeson
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
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5
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Bossion D, Chowdhury SN, Huo P. Non-adiabatic ring polymer molecular dynamics in the phase space of the SU(N) Lie group. J Chem Phys 2023; 158:044123. [PMID: 36725494 DOI: 10.1063/5.0133970] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/12/2023] Open
Abstract
We derive the non-adiabatic ring polymer molecular dynamics (RPMD) approach in the phase space of the SU(N) Lie Group. This method, which we refer to as the spin mapping non-adiabatic RPMD (SM-NRPMD), is based on the spin-mapping formalism for the electronic degrees of freedom (DOFs) and ring polymer path-integral description for the nuclear DOFs. Using the Stratonovich-Weyl transform for the electronic DOFs and the Wigner transform for the nuclear DOFs, we derived an exact expression of the Kubo-transformed time-correlation function (TCF). We further derive the spin mapping non-adiabatic Matsubara dynamics using the Matsubara approximation that removes the high frequency nuclear normal modes in the TCF and derive the SM-NRPMD approach from the non-adiabatic Matsubara dynamics by discarding the imaginary part of the Liouvillian. The SM-NRPMD method has numerical advantages compared to the original NRPMD method based on the Meyer-Miller-Stock-Thoss (MMST) mapping formalism due to a more natural mapping using the SU(N) Lie Group that preserves the symmetry of the original system. We numerically compute the Kubo-transformed position auto-correlation function and electronic population correlation function for three-state model systems. The numerical results demonstrate the accuracy of the SM-NRPMD method, which outperforms the original MMST-based NRPMD. We envision that the SM-NRPMD method will be a powerful approach to simulate electronic non-adiabatic dynamics and nuclear quantum effects accurately.
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Affiliation(s)
- Duncan Bossion
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Sutirtha N Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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6
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Lin K, Peng J, Xu C, Gu FL, Lan Z. Trajectory Propagation of Symmetrical Quasi-classical Dynamics with Meyer-Miller Mapping Hamiltonian Using Machine Learning. J Phys Chem Lett 2022; 13:11678-11688. [PMID: 36511563 DOI: 10.1021/acs.jpclett.2c02159] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
The long short-term memory recurrent neural network (LSTM-RNN) approach is applied to realize the trajectory-based nonadiabatic dynamics within the framework of the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian (MM-SQC). After construction, the LSTM-RNN model allows us to propagate the entire trajectory evolutions of all involved degrees of freedoms (DOFs) from initial conditions. The proposed idea is proven to be reliable and accurate in the simulations of the dynamics of several site-exciton electron-phonon coupling models and three Tully's scattering models. It indicates that the LSTM-RNN model perfectly captures the dynamical information on the trajectory evolution in the MM-SQC dynamics. Our work proposes a novel machine learning approach in the simulation of trajectory-based nonadiabatic dynamic of complex systems with a large number of DOFs.
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Affiliation(s)
- Kunni Lin
- School of Chemistry, South China Normal University, Guangzhou 510006, P. R. China
- MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, P. R. China
| | - Jiawei Peng
- School of Chemistry, South China Normal University, Guangzhou 510006, P. R. China
- MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, P. R. China
| | - Chao Xu
- MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, P. R. China
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety, School of Environment, South China Normal University, Guangzhou 510006, P. R. China
| | - Feng Long Gu
- MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, P. R. China
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety, School of Environment, South China Normal University, Guangzhou 510006, P. R. China
| | - Zhenggang Lan
- MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, P. R. China
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety, School of Environment, South China Normal University, Guangzhou 510006, P. R. China
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7
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Bossion DL, Ying W, Chowdhury S, Huo P. Non-adiabatic Mapping Dynamics in the Phase Space of the ${SU}(N)$ Lie Group. J Chem Phys 2022; 157:084105. [DOI: 10.1063/5.0094893] [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
We present the rigorous theoretical framework of the generalized spin mapping representation for non-adiabatic dynamics. Our work is based up a new mapping formalism recently introduced by Runeson and Richardson in [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich-Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom, and derived an exact expression of the time-correlation function as well as the exact quantum Liouvillian for the non-adiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this Linearized method is tested using non-adiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables and connect the previous methods in a clear and concise manner.
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Affiliation(s)
| | | | - Sutirtha Chowdhury
- Department of Chemistry, University of Rochester, United States of America
| | - Pengfei Huo
- Department of Chemsitry, University of Rochester Department of Chemistry, United States of America
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8
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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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9
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Saller MAC, Lai Y, Geva E. An Accurate Linearized Semiclassical Approach for Calculating Cavity-Modified Charge Transfer Rate Constants. J Phys Chem Lett 2022; 13:2330-2337. [PMID: 35245071 DOI: 10.1021/acs.jpclett.2c00122] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
We show that combining the linearized semiclasscial approximation with Fermi's golden rule (FGR) rate theory gives rise to a general-purpose cost-effective and scalable computational framework that can accurately capture the cavity-induced rate enhancement of charge transfer reactions that occurs when the molecular system is placed inside a microcavity. Both partial linearization with respect to the nuclear and photonic degrees of freedom and full linerization with respect to nuclear, photonic, and electronic degrees of freedom (the latter within the mapping Hamiltonian approach) are shown to be highly accurate, provided that the Wigner transforms of the product (WoP) of operators at the initial time is not replaced by the product of their Wigner transforms. We also show that the partial linearization method yields the quantum-mechanically exact cavity-modified FGR rate constant for a model system in which the donor and acceptor potential energy surfaces are harmonic and identical except for a shift in the equilibrium energy and geometry, if WoP is applied.
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Affiliation(s)
- Maximilian A C Saller
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Yifan Lai
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Eitan Geva
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
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10
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Ananth N. Path Integrals for Nonadiabatic Dynamics: Multistate Ring Polymer Molecular Dynamics. Annu Rev Phys Chem 2022; 73:299-322. [PMID: 35081325 DOI: 10.1146/annurev-physchem-082620-021809] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
This review focuses on a recent class of path-integral-based methods that simulate nonadiabatic dynamics in the condensed phase using only classical molecular dynamics trajectories in an extended phase space. Specifically, a semiclassical mapping protocol is used to derive an exact, continuous, Cartesian variable path-integral representation for the canonical partition function of a system in which multiple electronic states are coupled to nuclear degrees of freedom. Building on this exact statistical foundation, multistate ring polymer molecular dynamics methods are developed for the approximate calculation of real-time thermal correlation functions. The remarkable promise of these multistate ring polymer methods, their successful applications, and their limitations are discussed in detail.Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
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Affiliation(s)
- Nandini Ananth
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York, USA;
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11
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Mannouch JR, Richardson JO. A partially linearized spin-mapping approach for simulating nonlinear optical spectra. J Chem Phys 2022; 156:024108. [PMID: 35032975 DOI: 10.1063/5.0077744] [Citation(s) in RCA: 15] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present a partially linearized method based on spin-mapping for computing both linear and nonlinear optical spectra. As observables are obtained from ensembles of classical trajectories, the approach can be applied to the large condensed-phase systems that undergo photosynthetic light-harvesting processes. In particular, the recently derived spin partially linearized density matrix method has been shown to exhibit superior accuracy in computing population dynamics compared to other related classical-trajectory methods. Such a method should also be ideally suited to describing the quantum coherences generated by interaction with light. We demonstrate that this is, indeed, the case by calculating the nonlinear optical response functions relevant for the pump-probe and 2D photon-echo spectra for a Frenkel biexciton model and the Fenna-Matthews-Olsen light-harvesting complex. One especially desirable feature of our approach is that the full spectrum can be decomposed into its constituent components associated with the various Liouville-space pathways, offering a greater insight beyond what can be directly obtained from experiments.
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12
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Choi S, Vaníček J. High-order geometric integrators for representation-free Ehrenfest dynamics. J Chem Phys 2021; 155:124104. [PMID: 34598577 DOI: 10.1063/5.0061878] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
Ehrenfest dynamics is a useful approximation for ab initio mixed quantum-classical molecular dynamics that can treat electronically nonadiabatic effects. Although a severe approximation to the exact solution of the molecular time-dependent Schrödinger equation, Ehrenfest dynamics is symplectic, is time-reversible, and conserves exactly the total molecular energy as well as the norm of the electronic wavefunction. Here, we surpass apparent complications due to the coupling of classical nuclear and quantum electronic motions and present efficient geometric integrators for "representation-free" Ehrenfest dynamics, which do not rely on a diabatic or adiabatic representation of electronic states and are of arbitrary even orders of accuracy in the time step. These numerical integrators, obtained by symmetrically composing the second-order splitting method and exactly solving the kinetic and potential propagation steps, are norm-conserving, symplectic, and time-reversible regardless of the time step used. Using a nonadiabatic simulation in the region of a conical intersection as an example, we demonstrate that these integrators preserve the geometric properties exactly and, if highly accurate solutions are desired, can be even more efficient than the most popular non-geometric integrators.
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Affiliation(s)
- Seonghoon Choi
- Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Jiří Vaníček
- Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
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13
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Hu D, Xie Y, Peng J, Lan Z. On-the-Fly Symmetrical Quasi-Classical Dynamics with Meyer-Miller Mapping Hamiltonian for the Treatment of Nonadiabatic Dynamics at Conical Intersections. J Chem Theory Comput 2021; 17:3267-3279. [PMID: 34028268 DOI: 10.1021/acs.jctc.0c01249] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The on-the-fly version of the symmetrical quasi-classical dynamics method based on the Meyer-Miller mapping Hamiltonian (SQC/MM) is implemented to study the nonadiabatic dynamics at conical intersections of polyatomic systems. The current on-the-fly implementation of the SQC/MM method is based on the adiabatic representation and the dressed momentum. To include the zero-point energy (ZPE) correction of the electronic mapping variables, we employ both the γ-adjusted and γ-fixed approaches. Nonadiabatic dynamics of the methaniminium cation (CH2NH2+) and azomethane are simulated using the on-the-fly SQC/MM method. For CH2NH2+, both ZPE correction approaches give reasonable and consistent results. However, for azomethane, the γ-adjusted version of the SQC/MM dynamics behaves much better than the γ-fixed version. Further analysis indicates that it is always recommended to use the γ-adjusted SQC/MM dynamics in the on-the-fly simulation of photoinduced dynamics of polyatomic systems, particularly when the excited state is well separated from the ground state in the Franck-Condon region. This work indicates that the on-the-fly SQC/MM method is a powerful simulation protocol to deal with the nonadiabatic dynamics of realistic polyatomic systems.
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Affiliation(s)
- Deping Hu
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety & MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, China.,School of Environment, South China Normal University, Guangzhou 510006, China
| | - Yu Xie
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety & MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, China.,School of Environment, South China Normal University, Guangzhou 510006, China
| | - Jiawei Peng
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety & MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, China.,School of Environment, South China Normal University, Guangzhou 510006, China
| | - Zhenggang Lan
- SCNU Environmental Research Institute, Guangdong Provincial Key Laboratory of Chemical Pollution and Environmental Safety & MOE Key Laboratory of Environmental Theoretical Chemistry, South China Normal University, Guangzhou 510006, China.,School of Environment, South China Normal University, Guangzhou 510006, China
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14
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Chowdhury SN, Huo P. Non-adiabatic Matsubara dynamics and non-adiabatic ring-polymer molecular dynamics. J Chem Phys 2021; 154:124124. [PMID: 33810665 DOI: 10.1063/5.0042136] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023] Open
Abstract
We present the non-adiabatic Matsubara dynamics, a general framework for computing the time-correlation function (TCF) of electronically non-adiabatic systems. This new formalism is derived based on the generalized Kubo-transformed TCF using the Wigner representation for both the nuclear degrees of freedom and the electronic mapping variables. By dropping the non-Matsubara nuclear normal modes in the quantum Liouvillian and explicitly integrating these modes out from the expression of the TCF, we derived the non-adiabatic Matsubara dynamics approach. Further making the approximation to drop the imaginary part of the Matsubara Liouvillian and enforce the nuclear momentum integral to be real, we arrived at the non-adiabatic ring-polymer molecular dynamics (NRPMD) approach. We have further justified the capability of NRPMD for simulating the non-equilibrium TCF. This work provides the rigorous theoretical foundation for several recently proposed state-dependent RPMD approaches and offers a general framework for developing new non-adiabatic quantum dynamics methods in the future.
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Affiliation(s)
- Sutirtha N Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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15
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Chowdhury SN, Mandal A, Huo P. Ring polymer quantization of the photon field in polariton chemistry. J Chem Phys 2021; 154:044109. [PMID: 33514102 DOI: 10.1063/5.0038330] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023] Open
Abstract
We use the ring polymer (RP) representation to quantize the radiation field inside an optical cavity to investigate polariton quantum dynamics. Using a charge transfer model coupled to an optical cavity, we demonstrate that the RP quantization of the photon field provides accurate rate constants of the polariton mediated electron transfer reaction compared to Fermi's golden rule. Because RP quantization uses extended phase space to describe the photon field, it significantly reduces the computational costs compared to the commonly used Fock state description of the radiation field. Compared to the other quasi-classical descriptions of the photon field, such as the classical Wigner based mean-field Ehrenfest model, the RP representation provides a much more accurate description of the polaritonic quantum dynamics because it alleviates the potential quantum distribution leakage problem associated with the photonic degrees of freedom (DOF). This work demonstrates the possibility of using the ring polymer description to treat the quantized radiation field in polariton chemistry, offering an accurate and efficient approach for future investigations in cavity quantum electrodynamics.
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Affiliation(s)
- Sutirtha N Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Arkajit Mandal
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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16
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Gao X, Geva E. Improving the Accuracy of Quasiclassical Mapping Hamiltonian Methods by Treating the Window Function Width as an Adjustable Parameter. J Phys Chem A 2020; 124:11006-11016. [DOI: 10.1021/acs.jpca.0c09750] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Xing Gao
- School of Materials, Sun Yat-sen University, Shenzhen, Guangdong 518100, China
| | - Eitan Geva
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
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17
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Gao X, Saller MAC, Liu Y, Kelly A, Richardson JO, Geva E. Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Electronically Nonadiabatic Molecular Dynamics. J Chem Theory Comput 2020; 16:2883-2895. [DOI: 10.1021/acs.jctc.9b01267] [Citation(s) in RCA: 37] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Xing Gao
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | | | - Yudan Liu
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
| | - Aaron Kelly
- Department of Chemistry, Dalhousie University, 15000 Halifax, Nova Scotia, Canada
| | | | - Eitan Geva
- Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, United States
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18
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Runeson JE, Richardson JO. Generalized spin mapping for quantum-classical dynamics. J Chem Phys 2020; 152:084110. [DOI: 10.1063/1.5143412] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/24/2023] Open
Affiliation(s)
- Johan E. Runeson
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
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19
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Saller MAC, Kelly A, Richardson JO. Improved population operators for multi-state nonadiabatic dynamics with the mixed quantum-classical mapping approach. Faraday Discuss 2020; 221:150-167. [DOI: 10.1039/c9fd00050j] [Citation(s) in RCA: 29] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Application to the 7-state Frenkel-exciton Hamiltonian for the Fenna–Matthews–Olson complex shows that using a different representation of the electronic population operators can drastically improve the accuracy of the quasiclassical mapping approach without increasing the computational effort.
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Affiliation(s)
| | - Aaron Kelly
- Department of Chemistry
- Dalhousie University
- Halifax
- Canada
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20
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Chowdhury SN, Huo P. State dependent ring polymer molecular dynamics for investigating excited nonadiabatic dynamics. J Chem Phys 2019; 150:244102. [DOI: 10.1063/1.5096276] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sutirtha N. Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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21
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Affiliation(s)
- Wei Fang
- School of Physics and Collaborative Innovation Centre of Quantum Matter, Peking University, Beijing, People's Republic of China
- Thomas Young Centre, London Centre for Nanotechnology, and Department of Physics and Astronomy, University College London, London, UK
- Laboratory of Physical Chemistry, ETH Zurich, Zurich, Switzerland
| | - Ji Chen
- Department of Electronic Structure Theory, Max Plank Institute for Solid State Research, Stuttgart, Germany
| | - Yexin Feng
- School of Physics and Electronics, Hunan University, Changsha, People's Republic of China
| | - Xin-Zheng Li
- School of Physics and Collaborative Innovation Centre of Quantum Matter, Peking University, Beijing, People's Republic of China
- State Key Laboratory for Artificial Microstructure and Mesoscopic Physics, Peking University, Beijing, People's Republic of China
| | - Angelos Michaelides
- Thomas Young Centre, London Centre for Nanotechnology, and Department of Physics and Astronomy, University College London, London, UK
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22
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Saller MAC, Kelly A, Richardson JO. On the identity of the identity operator in nonadiabatic linearized semiclassical dynamics. J Chem Phys 2019; 150:071101. [DOI: 10.1063/1.5082596] [Citation(s) in RCA: 37] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - Aaron Kelly
- Department of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada
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23
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Karsten S, Ivanov SD, Bokarev SI, Kühn O. Simulating vibronic spectra via Matsubara-like dynamics: Coping with the sign problem. J Chem Phys 2018; 149:194103. [DOI: 10.1063/1.5046874] [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)
- Sven Karsten
- Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
| | - Sergei D. Ivanov
- Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
| | - Sergey I. Bokarev
- Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
| | - Oliver Kühn
- Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
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24
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Xie Y, Zheng J, Lan Z. Performance evaluation of the symmetrical quasi-classical dynamics method based on Meyer-Miller mapping Hamiltonian in the treatment of site-exciton models. J Chem Phys 2018; 149:174105. [DOI: 10.1063/1.5047002] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yu Xie
- Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences, Qingdao 266101, Shandong, China
- The Environmental Research Institute; MOE Key Laboratory of Theoretical Chemistry of Environment, South China Normal University, Guangzhou 510006, China
| | - Jie Zheng
- Industrial Research Institute of Nonwovens and Technical Textiles, College of Textiles and Clothing, Qingdao University, Qingdao 266071, China
| | - Zhenggang Lan
- Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences, Qingdao 266101, Shandong, China
- The Environmental Research Institute; MOE Key Laboratory of Theoretical Chemistry of Environment, South China Normal University, Guangzhou 510006, China
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25
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Collet E, Cammarata M. Disentangling Ultrafast Electronic and Structural Dynamics with X-Ray Lasers. Chemistry 2018; 24:15696-15705. [DOI: 10.1002/chem.201802105] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2018] [Indexed: 12/18/2022]
Affiliation(s)
- Eric Collet
- Univ Rennes; CNRS, IPR (Institut de Physique de Rennes)-UMR 6251; 35000 Rennes France
| | - Marco Cammarata
- Univ Rennes; CNRS, IPR (Institut de Physique de Rennes)-UMR 6251; 35000 Rennes France
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26
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Wang W, Dong W, Jiang D, Ning Z. Water molecules migration at oil–paper interface under the coupling fields of electric and temperature: a molecular dynamics study. Mol Phys 2018. [DOI: 10.1080/00268976.2018.1488007] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
Affiliation(s)
- Wei Wang
- Beijing Key Laboratory of High Voltage and Electromagnetic Compatibility, North China Electric Power University, Beijing, People’s Republic of China
| | - Wenyan Dong
- Beijing Key Laboratory of High Voltage and Electromagnetic Compatibility, North China Electric Power University, Beijing, People’s Republic of China
| | - Da Jiang
- Beijing Key Laboratory of High Voltage and Electromagnetic Compatibility, North China Electric Power University, Beijing, People’s Republic of China
| | - Zhongzheng Ning
- Beijing Key Laboratory of High Voltage and Electromagnetic Compatibility, North China Electric Power University, Beijing, People’s Republic of China
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27
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28
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Mattiat J, Richardson JO. Effects of tunnelling and asymmetry for system-bath models of electron transfer. J Chem Phys 2018; 148:102311. [PMID: 29544261 DOI: 10.1063/1.5001116] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We apply the newly derived nonadiabatic golden-rule instanton theory to asymmetric models describing electron-transfer in solution. The models go beyond the usual spin-boson description and have anharmonic free-energy surfaces with different values for the reactant and product reorganization energies. The instanton method gives an excellent description of the behaviour of the rate constant with respect to asymmetry for the whole range studied. We derive a general formula for an asymmetric version of the Marcus theory based on the classical limit of the instanton and find that this gives significant corrections to the standard Marcus theory. A scheme is given to compute this rate based only on equilibrium simulations. We also compare the rate constants obtained by the instanton method with its classical limit to study the effect of tunnelling and other quantum nuclear effects. These quantum effects can increase the rate constant by orders of magnitude.
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Affiliation(s)
- Johann Mattiat
- Laboratory of Physical Chemistry, ETH Zurich, 8093 Zurich, Switzerland
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29
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Church MS, Hele TJH, Ezra GS, Ananth N. Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation. J Chem Phys 2018; 148:102326. [DOI: 10.1063/1.5005557] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Matthew S. Church
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Timothy J. H. Hele
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Gregory S. Ezra
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Nandini Ananth
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
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30
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Karsten S, Ivanov SD, Bokarev SI, Kühn O. Quasi-classical approaches to vibronic spectra revisited. J Chem Phys 2018; 148:102337. [DOI: 10.1063/1.5011764] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Sven Karsten
- Institute of Physics, Rostock University, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
| | - Sergei D. Ivanov
- Institute of Physics, Rostock University, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
| | - Sergey I. Bokarev
- Institute of Physics, Rostock University, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
| | - Oliver Kühn
- Institute of Physics, Rostock University, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany
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31
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Pierre S, Duke JR, Hele TJH, Ananth N. A mapping variable ring polymer molecular dynamics study of condensed phase proton-coupled electron transfer. J Chem Phys 2017; 147:234103. [DOI: 10.1063/1.4986517] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Affiliation(s)
- Sadrach Pierre
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Jessica R. Duke
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Timothy J. H. Hele
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
| | - Nandini Ananth
- Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853, USA
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32
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Chowdhury SN, Huo P. Coherent state mapping ring polymer molecular dynamics for non-adiabatic quantum propagations. J Chem Phys 2017; 147:214109. [DOI: 10.1063/1.4995616] [Citation(s) in RCA: 37] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sutirtha N. Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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33
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Karsten S, Bokarev SI, Aziz SG, Ivanov SD, Kühn O. A time-correlation function approach to nuclear dynamical effects in X-ray spectroscopy. J Chem Phys 2017; 146:224203. [DOI: 10.1063/1.4984930] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sven Karsten
- Institute of Physics, Rostock University, Universitätsplatz 3, 18055 Rostock, Germany
| | - Sergey I. Bokarev
- Institute of Physics, Rostock University, Universitätsplatz 3, 18055 Rostock, Germany
| | - Saadullah G. Aziz
- Chemistry Department, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia
| | - Sergei D. Ivanov
- Institute of Physics, Rostock University, Universitätsplatz 3, 18055 Rostock, Germany
| | - Oliver Kühn
- Institute of Physics, Rostock University, Universitätsplatz 3, 18055 Rostock, Germany
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34
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Penfold TJ. Accelerating direct quantum dynamics using graphical processing units. Phys Chem Chem Phys 2017; 19:19601-19608. [DOI: 10.1039/c7cp01473b] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023]
Abstract
The direct dynamics variational multi-configurational Gaussian (DD-vMCG) method is combined with electronic structure calculations accelerated by Graphical Processing Units (GPUs). This is used to identify GPU acceleration will have a significant effect for both ground and excited state simulations.
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Affiliation(s)
- T. J. Penfold
- School of Chemistry
- Newcastle University
- Newcastle upon Tyne
- UK
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35
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Hickson KM, Suleimanov YV. An experimental and theoretical investigation of the C(1D) + D2 reaction. Phys Chem Chem Phys 2017; 19:480-486. [DOI: 10.1039/c6cp07381f] [Citation(s) in RCA: 37] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Rate constants derived from ring polymer molecular dynamics calculations confirm the validity of this method for studying low-temperature complex-forming reactions
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Affiliation(s)
- Kevin M. Hickson
- Université de Bordeaux
- Institut des Sciences Moléculaires
- F-33400 Talence
- France
- CNRS
| | - Yury V. Suleimanov
- Computation-based Science and Technology Research Center
- Cyprus Institute
- Nicosia 2121
- Cyprus
- Department of Chemical Engineering
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