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Nguyen H, Ng N, Lindoy LP, Park G, Millis AJ, Kin-Lic Chan G, Reichman DR. Correlation functions from tensor network influence functionals: The case of the spin-boson model. J Chem Phys 2024; 161:104111. [PMID: 39268824 DOI: 10.1063/5.0224880] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2024] [Accepted: 08/27/2024] [Indexed: 09/15/2024] Open
Abstract
We investigate the application of matrix product state (MPS) representations of the influence functionals (IFs) for the calculation of real-time equilibrium correlation functions in open quantum systems. Focusing specifically on the unbiased spin-boson model, we explore the use of IF-MPSs for complex time propagation, as well as IF-MPSs for constructing correlation functions in the steady state. We examine three different IF approaches: one based on the Kadanoff-Baym contour targeting correlation functions at all times, one based on a complex contour targeting the correlation function at a single time, and a steady state formulation, which avoids imaginary or complex times, while providing access to correlation functions at all times. We show that within the IF language, the steady state formulation provides a powerful approach to evaluate equilibrium correlation functions.
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Affiliation(s)
- Haimi Nguyen
- Department of Chemistry, Columbia University, New York, New York 10027, USA
| | - Nathan Ng
- Department of Chemistry, Columbia University, New York, New York 10027, USA
| | - Lachlan P Lindoy
- National Physical Laboratory, Teddington TW11 0LW, United Kingdom
| | - Gunhee Park
- Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125, USA
| | - Andrew J Millis
- Department of Physics, Columbia University, New York, New York 10027, USA
- Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
| | - Garnet Kin-Lic Chan
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA
| | - David R Reichman
- Department of Chemistry, Columbia University, New York, New York 10027, USA
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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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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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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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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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Aieta C, Ceotto M. A quantum method for thermal rate constant calculations from stationary phase approximation of the thermal flux-flux correlation function integral. J Chem Phys 2017; 146:214115. [DOI: 10.1063/1.4984099] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Chiara Aieta
- Dipartimento di Chimica, Università degli Studi di Milano, Via C. Golgi 19, 20133 Milano, Italy
| | - Michele Ceotto
- Dipartimento di Chimica, Università degli Studi di Milano, Via C. Golgi 19, 20133 Milano, Italy
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Rota R, Casulleras J, Mazzanti F, Boronat J. Quantum Monte Carlo estimation of complex-time correlations for the study of the ground-state dynamic structure function. J Chem Phys 2015; 142:114114. [DOI: 10.1063/1.4914995] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- R. Rota
- Dipartimento di Fisica and INO-CNR BEC Center, Università degli Studi di Trento, I-38123 Povo, Trento, Italy
| | - J. Casulleras
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
| | - F. Mazzanti
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
| | - J. Boronat
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
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Mandrà S, Schrier J, Ceotto M. Helium Isotope Enrichment by Resonant Tunneling through Nanoporous Graphene Bilayers. J Phys Chem A 2014; 118:6457-65. [PMID: 24854987 DOI: 10.1021/jp502548r] [Citation(s) in RCA: 24] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023]
Affiliation(s)
- Salvatore Mandrà
- Department of Physics, Università degli Studi di Milano , via Celoria 16, 20133 Milano, Italy
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Beutier J, Monteferrante M, Bonella S, Vuilleumier R, Ciccotti G. Gas phase infrared spectra via the phase integration quasi-classical method. MOLECULAR SIMULATION 2013. [DOI: 10.1080/08927022.2013.843776] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
Affiliation(s)
- J. Beutier
- Département de Chimie, UMR 8640 CNRS-ENS-UPMC, École Normale Supérieure, 24 rue Lhomond, 75005, Paris, France
| | - M. Monteferrante
- Dipartimento di Fisica and CNISM Unità 1, Università di Roma ‘La Sapienza’, Ple A. Moro 2, 00185, Rome, Italy
| | - S. Bonella
- Dipartimento di Fisica and CNISM Unità 1, Università di Roma ‘La Sapienza’, Ple A. Moro 2, 00185, Rome, Italy
| | - R. Vuilleumier
- Département de Chimie, UMR 8640 CNRS-ENS-UPMC, École Normale Supérieure, 24 rue Lhomond, 75005, Paris, France
| | - G. Ciccotti
- Dipartimento di Fisica and CNISM Unità 1, Università di Roma ‘La Sapienza’, Ple A. Moro 2, 00185, Rome, Italy
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Borgis D, Assaraf R, Rotenberg B, Vuilleumier R. Computation of pair distribution functions and three-dimensional densities with a reduced variance principle. Mol Phys 2013. [DOI: 10.1080/00268976.2013.838316] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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Hofer TS. From macromolecules to electrons-grand challenges in theoretical and computational chemistry. Front Chem 2013; 1:6. [PMID: 24790935 PMCID: PMC3982537 DOI: 10.3389/fchem.2013.00006] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/31/2012] [Accepted: 05/08/2013] [Indexed: 11/29/2022] Open
Affiliation(s)
- Thomas S. Hofer
- Theoretical Chemistry Division, Center for Chemistry and Biomedicine, Institute of General, Inorganic and Theoretical Chemistry, University of InnsbruckInnsbruck, Austria
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Basire M, Borgis D, Vuilleumier R. Computing Wigner distributions and time correlation functions using the quantum thermal bath method: application to proton transfer spectroscopy. Phys Chem Chem Phys 2013; 15:12591-601. [DOI: 10.1039/c3cp50493j] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Monteferrante M, Bonella S, Ciccotti G. Linearized symmetrized quantum time correlation functions calculation via phase pre-averaging. Mol Phys 2011. [DOI: 10.1080/00268976.2011.619506] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/17/2022]
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Georgescu I, Deckman J, Fredrickson LJ, Mandelshtam VA. Thermal Gaussian molecular dynamics for quantum dynamics simulations of many-body systems: Application to liquid para-hydrogen. J Chem Phys 2011; 134:174109. [PMID: 21548675 DOI: 10.1063/1.3585648] [Citation(s) in RCA: 24] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Ionut Georgescu
- Chemistry Department, University of California at Irvine, Irvine, California 92697, USA.
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