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Lando GM, Giraud O, Ullmo D. Computing Quantum Mean Values in the Deep Chaotic Regime. PHYSICAL REVIEW LETTERS 2024; 132:260401. [PMID: 38996320 DOI: 10.1103/physrevlett.132.260401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/23/2023] [Revised: 12/29/2023] [Accepted: 05/17/2024] [Indexed: 07/14/2024]
Abstract
We study the time evolution of mean values of quantum operators in a regime plagued by two difficulties: the smallness of ℏ and the presence of strong and ubiquitous classical chaos. While numerics become too computationally expensive for purely quantum calculations as ℏ→0, methods that take advantage of the smallness of ℏ-that is, semiclassical methods-suffer from both conceptual and practical difficulties in the deep chaotic regime. We implement an approach which addresses these conceptual problems, leading to a deeper understanding of the origin of the interference contributions to the operator's mean value. We show that in the deep chaotic regime our approach is capable of unprecedented accuracy, while a standard semiclassical method (the Herman-Kluk propagator) produces only numerical noise. Our work paves the way to the development and employment of more efficient and accurate methods for quantum simulations of systems with strongly chaotic classical limits.
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Affiliation(s)
| | - Olivier Giraud
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
- MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Laboratory, Singapore, Singapore
- Centre for Quantum Technologies, National University of Singapore, 117543 Singapore, Singapore
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2
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Bonnet L. Semiclassical initial value representation: From Møller to Miller. II. J Chem Phys 2023; 158:114112. [PMID: 36948824 DOI: 10.1063/5.0137725] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/18/2023] Open
Abstract
As shown by W. H. Miller in a seminal article [J. Chem. Phys. 53, 3578 (1970)], the most convenient and accurate semiclassical (SC) theory of molecular scattering in action-angle coordinates is based on the initial value representation (IVR) and the use of shifted angles, which are different from the natural angles usually used in the quantum and classical treatments. Here, we show for an inelastic molecular collision that the initial and final shifted angles define three-segment classical paths that are exactly those involved in the classical-limit of Tannor-Weeks quantum scattering theory [J. Chem. Phys. 98, 3884 (1993)], provided that the translational wave packets |g+⟩ and |g-⟩ into play in this theory are both taken at |0⟩. Assuming this to be the case, using van Vleck propagators, and applying the stationary phase approximation, Miller's SCIVR expression of S-matrix elements is found, with an additional cut-off factor canceling the energetically forbidden transition probabilities. This factor, however, is close to unity in most practical cases. Furthermore, these developments show that the Møller operators underlie Miller's formulation, thus confirming, for molecular collisions, the results recently established in the simpler case of light-induced rotational transitions [L. Bonnet, J. Chem. Phys. 153, 174102 (2020)]. Last but not least, we show, based on the previous results, that for processes involving long-range anisotropic forces, implementing the Skinner-Miller method [Chem. Phys. Lett. 300, 20 (1999)] in shifted coordinates makes its predictions both easier and more accurate than in natural coordinates.
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Affiliation(s)
- Laurent Bonnet
- Univ. Bordeaux, CNRS, Bordeaux INP, ISM, UMR 5255, F-33400 Talence, France
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Malpathak S, Church MS, Ananth N. A Semiclassical Framework for Mixed Quantum Classical Dynamics. J Phys Chem A 2022; 126:6359-6375. [PMID: 36070472 DOI: 10.1021/acs.jpca.2c03467] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
Semiclassical (SC) approximations for quantum dynamic simulations in complex chemical systems range from rigorously accurate methods that are computationally expensive to methods that exhibit near-classical scaling with system size but are limited in their ability to describe quantum effects. In practical studies of high-dimensional reactions, neither extreme is the best choice: frequently a high-level quantum mechanical description is only required for a handful of modes, while the majority of environment modes that do not play a key role in the reactive event of interest are well served with a lower level of theory. In this feature, we introduce modified Filinov filtration as a powerful tool to construct mixed quantum-classical SC theories where different subsystems can be quantized to different extents without introducing ad hoc intersubsystem interaction terms. We demonstrate that these Filinov-based SC methods can systematically tune between quantum and classical limit SC behavior, offering a practical way forward to accurate and computationally efficient simulations of high-dimensional quantum processes.
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Affiliation(s)
- Shreyas Malpathak
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University Ithaca, New York 14853, United States
| | - Matthew S Church
- Department of Chemistry, Brown University, Providence, Rhode Island 02906, United States
| | - Nandini Ananth
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University Ithaca, New York 14853, United States
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Bonnet L. Semiclassical descriptions of rotational transitions in natural and shifted angles: Analysis of unexpected results. J Chem Phys 2021; 155:174103. [PMID: 34742210 DOI: 10.1063/5.0071227] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In the semiclassical theory of rotational transitions, S-matrix elements are expressed as integrals over initial and final angles of probability amplitudes calculated along the classical paths joining these angles, before final passage to an initial value representation [W. H. Miller, J. Phys. Chem. A 105, 2942 (2001)]. These angles can be either natural angles fixing the orientation of the rotor or angles shifted with respect to the previous ones so as to vary only within the interaction region causing the transitions. The two approaches, however, were recently shown to lead to different predictions. While the theory in natural angles lacks precision and exhibits unphysical behavior, the theory in shifted angles is much more accurate and physically well behaved [L. Bonnet, J. Chem. Phys. 153, 174102 (2020)]. The present work is devoted to the analysis of this unexpected finding.
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Affiliation(s)
- Laurent Bonnet
- Univ. Bordeaux, CNRS, Bordeaux INP, ISM, UMR 5255, F-33400 Talence, France
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Bonnet L. Erratum: "Semiclassical initial value representation: From Møller to Miller" [J. Chem. Phys. 153, 174102 (2020)]. J Chem Phys 2021; 155:059901. [PMID: 34364332 DOI: 10.1063/5.0063094] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- L Bonnet
- CNRS, ISM, UMR 5255, F-33400 Talence, France
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Rodríguez-Fernández A, Bonnet L, Crespos C, Larrégaray P, Díez Muiño R. When classical trajectories get to quantum accuracy: II. The scattering of rotationally excited H2 on Pd(111). Phys Chem Chem Phys 2020; 22:22805-22814. [DOI: 10.1039/d0cp02655g] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
The classical trajectory method in a quantum spirit assigns statistical weights to classical paths on the basis of two semiclassical corrections: Gaussian binning and the adiabaticity correction.
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Affiliation(s)
| | | | | | | | - Ricardo Díez Muiño
- Centro de Física de Materiales CFM/MPC (CSIC-UPV/EHU)
- 20018 Donostia-SanSebastián
- Spain
- Donostia International Physics Center (DIPC)
- 20018 Donostia-SanSebastián
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Bonnet L, Larrégaray P, Lara M, Launay JM. Theoretical Study of Barrierless Chemical Reactions Involving Nearly Elastic Rebound: The Case of S( 1D) + X 2, X = H, D. J Phys Chem A 2019; 123:6439-6454. [PMID: 31329443 DOI: 10.1021/acs.jpca.9b04938] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
For some values of the total angular momentum consistent with reaction, the title processes involve nonreactive trajectories proceeding through a single rebound mechanism during which the internal motion of the reagent diatom is nearly unperturbed. When such paths are in a significant amount, the classical reaction probability is found to be markedly lower than the quantum mechanical one. This finding was recently attributed to an unusual quantum effect called diffraction-mediated trapping, and a semiclassical correction was proposed in order to take into account this effect in the classical trajectory method. In the present work, we apply the resulting approach to the calculation of opacity functions as well as total and state-resolved integral cross sections (ICSs) and compare the values obtained with exact quantum ones, most of which are new. As the title reactions proceed through a deep insertion well, mean potential statistical calculations are also presented. Seven values of the collision energy, ranging from 30 to 1127 K, are considered. Two remarkable facts stand out: (i) The corrected classical treatment strongly improves the accuracy of the opacity function as compared to the usual classical treatment. When the entrance transition state is tight, however, those trajectories crossing it with a bending vibrational energy below the zero point energy must be discarded. (ii) The quantum opacity function, particularly its cutoff, is finely reproduced by the statistical approach. Consequently, the total ICS is also very well described by the two previous approximate methods. These, however, do not predict state-resolved ICSs with the same accuracy, proving thereby that (i) one or several genuine quantum effects involved in the dynamics are missed by the corrected classical treatment and (ii) the dynamics are not fully statistical.
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Affiliation(s)
- L Bonnet
- Université de Bordeaux, ISM , UMR 5255, F-33400 Talence , France.,CNRS , ISM , UMR 5255, F-33400 Talence , France
| | - P Larrégaray
- Université de Bordeaux, ISM , UMR 5255, F-33400 Talence , France.,CNRS , ISM , UMR 5255, F-33400 Talence , France
| | - M Lara
- Departamento de Química Física Aplicada, Facultad de Ciencias , Universidad Autónoma de Madrid , 28049 Madrid , Spain
| | - J-M Launay
- Institut de Physique de Rennes, UMR CNRS 6251 , Université de Rennes I , F-35042 Rennes , France
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Goel P, Stanton JF. Semiclassical transition state theory based on fourth order vibrational perturbation theory: Model system studies beyond symmetric Eckart barrier. J Chem Phys 2018; 149:134109. [DOI: 10.1063/1.5040978] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Prateek Goel
- Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA
| | - John F. Stanton
- Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA
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9
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Smith B, Akimov AV. Entangled trajectories Hamiltonian dynamics for treating quantum nuclear effects. J Chem Phys 2018; 148:144106. [DOI: 10.1063/1.5022573] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Brendan Smith
- Department of Chemistry, University at Buffalo, The State University of New York, Buffalo, New York 14260-3000, USA
| | - Alexey V. Akimov
- Department of Chemistry, University at Buffalo, The State University of New York, Buffalo, New York 14260-3000, USA
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Kreisbeck C, Kramer T, Molina RA. Time-dependent wave packet simulations of transport through Aharanov-Bohm rings with an embedded quantum dot. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2017; 29:155301. [PMID: 28195564 DOI: 10.1088/1361-648x/aa605d] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We have performed time-dependent wave packet simulations of realistic Aharonov-Bohm (AB) devices with a quantum dot embedded in one of the arms of the interferometer. The AB ring can function as a measurement device for the intrinsic transmission phase through the quantum dot, however, care has to be taken in analyzing the influence of scattering processes in the junctions of the interferometer arms. We consider a harmonic quantum dot and show how the Darwin-Fock spectrum emerges as a unique pattern in the interference fringes of the AB oscillations.
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Affiliation(s)
- C Kreisbeck
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, United States of America
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11
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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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12
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Akimov AV, Long R, Prezhdo OV. Coherence penalty functional: A simple method for adding decoherence in Ehrenfest dynamics. J Chem Phys 2014; 140:194107. [DOI: 10.1063/1.4875702] [Citation(s) in RCA: 70] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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13
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Park H, Shin C, Shin S. Wavepacket Correlation Function Approach for Nonadiabatic Reactions: Quasi-Jahn-Teller Model. B KOREAN CHEM SOC 2014. [DOI: 10.5012/bkcs.2014.35.4.1061] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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14
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Efficient importance sampling in semiclassical initial value representation calculations for time correlation functions. Theor Chem Acc 2014. [DOI: 10.1007/s00214-014-1448-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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15
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Conte R, Fu B, Kamarchik E, Bowman JM. A novel Gaussian Binning (1GB) analysis of vibrational state distributions in highly excited H2O from reactive quenching of OH* by H2. J Chem Phys 2013; 139:044104. [DOI: 10.1063/1.4816277] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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16
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Kryvohuz M. On the derivation of semiclassical expressions for quantum reaction rate constants in multidimensional systems. J Chem Phys 2013; 138:244114. [DOI: 10.1063/1.4811221] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Tao G, Miller WH. Time-dependent importance sampling in semi-classical initial value representation calculations for time correlation functions. III. A state-resolved implementation to electronically non-adiabatic dynamics. Mol Phys 2013. [DOI: 10.1080/00268976.2013.776712] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
Affiliation(s)
- Guohua Tao
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California , Berkeley, California, USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory , Berkeley, California, USA
| | - William H. Miller
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California , Berkeley, California, USA
- Chemical Sciences Division, Lawrence Berkeley National Laboratory , Berkeley, California, USA
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18
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Pan F, Tao G. Communication: importance sampling including path correlation in semiclassical initial value representation calculations for time correlation functions. J Chem Phys 2013; 138:091101. [PMID: 23485270 DOI: 10.1063/1.4794191] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Full semiclassical (SC) initial value representation (IVR) for time correlation functions involves a double phase space average over a set of two phase points, each of which evolves along a classical path. Conventionally, the two initial phase points are sampled independently for all degrees of freedom (DOF) in the Monte Carlo procedure. Here, we present an efficient importance sampling scheme by including the path correlation between the two initial phase points for the bath DOF, which greatly improves the performance of the SC-IVR calculations for large molecular systems. Satisfactory convergence in the study of quantum coherence in vibrational relaxation has been achieved for a benchmark system-bath model with up to 21 DOF.
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Affiliation(s)
- Feng Pan
- School of Advanced Materials, Peking University, Shenzhen Graduate School, Shenzhen 518055, China
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19
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Akimov AV, Prezhdo OV. Formulation of quantized Hamiltonian dynamics in terms of natural variables. J Chem Phys 2012; 137:224115. [DOI: 10.1063/1.4770224] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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20
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Tao G, Miller WH. Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. II. A simplified implementation. J Chem Phys 2012; 137:124105. [DOI: 10.1063/1.4752206] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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21
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Ivanov MV, Babikov D. Forward–backward propagation in the mixed quantum–classical theory for the collisional energy transfer. Chem Phys Lett 2012. [DOI: 10.1016/j.cplett.2012.03.042] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
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Ceotto M, Tantardini GF, Aspuru-Guzik A. Fighting the curse of dimensionality in first-principles semiclassical calculations: Non-local reference states for large number of dimensions. J Chem Phys 2011; 135:214108. [DOI: 10.1063/1.3664731] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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KONDORSKIY ALEXEY, NAKAMURA HIROKI. SEMICLASSICAL FROZEN GAUSSIAN PROPAGATION METHOD FOR ELECTRONICALLY NONADIABATIC CHEMICAL DYNAMICS: MØLLER OPERATOR FORMULATION AND INCORPORATION OF THE ZHU-NAKAMURA THEORY. JOURNAL OF THEORETICAL & COMPUTATIONAL CHEMISTRY 2011. [DOI: 10.1142/s0219633605001404] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
The title theory is developed by combining the Herman–Kluk semiclassical theory for adiabatic propagation on single potential energy surface and the Zhu–Nakamura theory for nonadiabatic transition. A fairly simple expression for the propagator based on classical trajectories is derived using the Møller operator formulation of scattering theory. The theory takes into account almost all quantum effects that occur during nonadiabatic transition, especially at low energies and is expected to be applicable to general chemical dynamics of high dimensions. Application to a two-dimensional model system shows that the theory works well for the propagation duration of several molecular vibrational periods and wide wave packet energy range.
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Affiliation(s)
- ALEXEY KONDORSKIY
- Department of Theoretical Studies, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan
- Lebedev Physical Institute, Russian Academy of Sciences, Leninsky pr., 53, Moscow, 119991, Russia
| | - HIROKI NAKAMURA
- Department of Theoretical Studies, Institute for Molecular Science, Myoaiji, Okazaki 444-8585, Japan
- Department of Functional Molecular Science, The Graduate University for Advanced Studies, Myodaiji, Okazaki 444-8585, Japan
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25
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Tao G, Miller WH. Time-dependent importance sampling in semiclassical initial value representation calculations for time correlation functions. J Chem Phys 2011; 135:024104. [DOI: 10.1063/1.3600656] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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26
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Segale D, Apkarian VA. Dissipative quantum coherent dynamics probed in phase-space: Electronically resonant 5-color 4-wave mixing on I2(B) in solid Kr. J Chem Phys 2011; 135:024203. [DOI: 10.1063/1.3598959] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- D. Segale
- Department of Chemistry, University of California, Irvine, California 92697-2025, USA
| | - V. A. Apkarian
- Department of Chemistry, University of California, Irvine, California 92697-2025, USA
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27
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Tatchen J, Pollak E, Tao G, Miller WH. Renormalization of the frozen Gaussian approximation to the quantum propagator. J Chem Phys 2011; 134:134104. [DOI: 10.1063/1.3573566] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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28
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Liu J, Miller WH. An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II. Thermal correlation functions. J Chem Phys 2011; 134:104102. [DOI: 10.1063/1.3555274] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Kaledin AL, McCurdy CW, Miller WH. A semiclassical correction for quantum mechanical energy levels. J Chem Phys 2010; 133:054101. [PMID: 20707520 DOI: 10.1063/1.3464318] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We propose a semiclassical method for correcting molecular energy levels obtained from a quantum mechanical variational calculation. A variational calculation gives the energy level (i.e., eigenvalue) as the expectation value of the molecular Hamiltonian <phi/H/phi>, where /phi> is the trial wave function. The true (i.e., exact) eigenvalue E can thus be expressed as this variational result plus a correction, i.e., E=<phi/H/phi>+DeltaE, the correction being due to the lack of exactness of the trial wave function. A formally exact expression for DeltaE is usually given (via Löwdin partitioning methodology) in terms of the Greens function of the Hamiltonian projected onto the orthogonal complement of /phi>. Formal treatment of this expression (using Brillouin-Wigner perturbation theory to infinite order) leads to an expression for DeltaE that involves matrix elements of the Greens function for the unprojected, i.e., full molecular Hamiltonian, which can then be approximated semiclassically. (Specifically, the Greens function is expressed as the Fourier transform of the quantum mechanical time evolution operator, e(-iHt/variant Planck's over 2pi), which in turn is approximated by using an initial value representation of semiclassical theory.) Calculations for several test problems (a one dimensional quartic potential, and vibrational energy levels of H(2)O and H(2)CO) clearly support our proposition that the error in the total eigenvalue E arises solely due to the semiclassical error in approximating DeltaE, which is usually a small fraction of the total energy E itself.
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Affiliation(s)
- Alexey L Kaledin
- Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, USA
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Tao G, Miller WH. Gaussian approximation for the structure function in semiclassical forward-backward initial value representations of time correlation functions. J Chem Phys 2009; 131:224107. [PMID: 20001024 DOI: 10.1063/1.3271241] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Initial value representations (IVRs) of semiclassical (SC) theory provide a general approach for adding quantum mechanical effects to classical molecular dynamics simulations of large molecular systems. Of the various versions of SC-IVR methodology for evaluating time correlation functions, the Fourier transform forward-backward (FB) approach is the simplest one that is able to describe true quantum coherence effects, so it is of considerable importance to find efficient and systematic ways for implementing it. It is shown in this paper that a Gaussian approximation for the "structure function"-the dependence of the correlation function on the (typically) momentum jump parameter-provides an efficient and accurate way for doing so. The approach is illustrated by an application to the time-dependent radial distribution function of I(2) (after photoexcitation) in a cluster of (up to 16) argon atoms.
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Affiliation(s)
- Guohua Tao
- Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720-1460, USA
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31
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Liu J, Miller WH, Paesani F, Zhang W, Case DA. Quantum dynamical effects in liquid water: A semiclassical study on the diffusion and the infrared absorption spectrum. J Chem Phys 2009; 131:164509. [PMID: 19894958 DOI: 10.1063/1.3254372] [Citation(s) in RCA: 71] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jian Liu
- Department of Chemistry and K. S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720-1460, USA
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32
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Ceotto M, Atahan S, Tantardini GF, Aspuru-Guzik A. Multiple coherent states for first-principles semiclassical initial value representation molecular dynamics. J Chem Phys 2009; 130:234113. [DOI: 10.1063/1.3155062] [Citation(s) in RCA: 73] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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33
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Jiang L, Babikov D. A reduced dimensionality model of ozone: Semiclassical treatment of van der Waals states. Chem Phys Lett 2009. [DOI: 10.1016/j.cplett.2009.04.082] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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34
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Miller WH. Electronically Nonadiabatic Dynamics via Semiclassical Initial Value Methods. J Phys Chem A 2009; 113:1405-15. [DOI: 10.1021/jp809907p] [Citation(s) in RCA: 130] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
Affiliation(s)
- William H. Miller
- Department of Chemistry and K. S. Pitzer Center for Theoretical Chemistry, University of California and Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720-1460
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35
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Tao G, Miller WH. Semiclassical description of vibrational quantum coherence in a three dimensional I[sub 2]Ar[sub n] (n≤6) cluster: A forward-backward initial value representation implementation. J Chem Phys 2009; 130:184108. [DOI: 10.1063/1.3132224] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/01/2023] Open
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36
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Liu J, Miller WH. Linearized semiclassical initial value time correlation functions with maximum entropy analytic continuation. J Chem Phys 2008; 129:124111. [PMID: 19045010 DOI: 10.1063/1.2981065] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Jian Liu
- Department of Chemistry and K. S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720-1460, USA
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37
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Gruenbaum SM, Loring RF. Semiclassical mean-trajectory approximation for nonlinear spectroscopic response functions. J Chem Phys 2008; 129:124510. [DOI: 10.1063/1.2978167] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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38
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Shalashilin DV, Burghardt I. Gaussian-based techniques for quantum propagation from the time-dependent variational principle: Formulation in terms of trajectories of coupled classical and quantum variables. J Chem Phys 2008; 129:084104. [DOI: 10.1063/1.2969101] [Citation(s) in RCA: 67] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/27/2023] Open
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39
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Liu J, Miller WH. Test of the consistency of various linearized semiclassical initial value time correlation functions in application to inelastic neutron scattering from liquid para-hydrogen. J Chem Phys 2008; 128:144511. [DOI: 10.1063/1.2889945] [Citation(s) in RCA: 52] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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40
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Gruenbaum SM, Loring RF. Interference and quantization in semiclassical response functions. J Chem Phys 2008; 128:124106. [DOI: 10.1063/1.2841943] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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41
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Pollak E. Continuum limit semiclassical initial value representation for dissipative systems. J Chem Phys 2007; 127:074505. [PMID: 17718618 DOI: 10.1063/1.2753151] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In this paper, we consider a dissipative system in which the system is coupled linearly to a harmonic bath. In the continuum limit, the bath is defined via a spectral density and the classical system dynamics is given in terms of a generalized Langevin equation. Using the path integral formulation and factorized initial conditions, it is well known that one can integrate out the harmonic bath, leaving only a path integral over the system degrees of freedom. However, the semiclassical initial value representation treatment of dissipative systems has usually been limited to a discretized treatment of the bath in terms of a finite number of bath oscillators. In this paper, the continuum limit of the semiclassical initial value representation is derived for dissipative systems. As in the path integral, the action is modified with an added nonlocal term, which expresses the influence of the bath on the dynamics. The first order correction term to the semiclassical initial value approximation is also derived in the continuum limit.
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Affiliation(s)
- Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel.
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42
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Hughes KH, Parry SM, Parlant G, Burghardt I. A Hybrid Hydrodynamic−Liouvillian Approach to Mixed Quantum−Classical Dynamics: Application to Tunneling in a Double Well. J Phys Chem A 2007; 111:10269-83. [PMID: 17685595 DOI: 10.1021/jp072642f] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
The hybrid quantum-classical approach of Burghardt and Parlant [Burghardt, I.; Parlant, G. J. Chem. Phys. 2004, 120, 3055], referred to here as the quantum-classical moment (QCM) approach, is demonstrated for the dynamics of a quantum double well coupled to a classical harmonic coordinate. The approach combines the quantum hydrodynamic and classical Liouvillian representations by the construction of a particular type of moments (that is, partial hydrodynamic moments) whose evolution is determined by a hierarchy of coupled equations. For pure states, which are at the center of the present study, this hierarchy terminates at the first order. In the Lagrangian picture, the deterministic trajectories result in dynamics which is Hamiltonian in the classical subspace, while the projection onto the quantum subspace evolves under a generalized hydrodynamic force. Importantly, this force also depends upon the classical (Q, P) variables. The present application demonstrates the tunneling dynamics in both the Eulerian and Lagrangian representations. The method is exact if the classical subspace is harmonic, as is the case for the systems studied here.
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Affiliation(s)
- Keith H Hughes
- School of Chemistry, University of Wales Bangor, Bangor, Gwynedd LL57 2UW, U.K.
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43
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Pollak E, Martin-Fierro E. New coherent state representation for the imaginary time propagator with applications to forward-backward semiclassical initial value representations of correlation functions. J Chem Phys 2007; 126:164107. [PMID: 17477589 DOI: 10.1063/1.2714518] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
There have been quite a few attempts in recent years to provide an initial value coherent state representation for the imaginary time propagator exp(-betaH). The most notable is the recent time evolving Gaussian approximation of Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] which may be considered as an expansion of the imaginary time propagator in terms of coherent states whose momentum is zero. In this paper, a similar but different expression is developed in which exp(-betaH) is represented in a series whose terms are weighted phase space averages of coherent states. Such a representation allows for the formulation of a new and simplified forward-backward semiclassical initial value representation expression for thermal correlation functions.
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Affiliation(s)
- Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
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44
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Martin-Fierro E, Pollak E. Semiclassical initial value series solution of the spin boson problem. J Chem Phys 2007; 126:164108. [PMID: 17477590 DOI: 10.1063/1.2714520] [Citation(s) in RCA: 31] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A numerical solution for the quantum dynamics of the spin boson problem is obtained using the semiclassical initial value series representation approach to the quantum dynamics. The zeroth order term of the series is computed using the new forward-backward representation for correlation functions presented in the preceding adjacent paper. This leads to a rapid convergence of the Monte Carlo sampling, as compared to previous attempts. The zeroth order results are already quite accurate. The first order term of the series is small, demonstrating the rapid convergence of the semiclassical initial value representation series. This is the first time that the first order term in the semiclassical initial value representation series has been converged for systems with the order of 50 degrees of freedom.
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Affiliation(s)
- Eva Martin-Fierro
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
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45
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Wang L, Pollak E. Frozen Gaussian Wavepacket Study of the Ground State of the He Atom. J Chem Theory Comput 2007; 3:344-9. [PMID: 26637022 DOI: 10.1021/ct600332v] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The Rayleigh-Ritz functional is used in conjunction with an approximate time evolution to improve ab initio estimates of ground-state energies. The improvement is due in part to the introduction of a novel variational "normalization function" for the approximate propagator. An additional variational parameter was introduced in the form of a constant shift energy of the Hamiltonian. The approximate propagator used was the frozen Gaussian propagator; however, the trajectories evolved on the coherent-state averaged Hamiltonian (Q representation). For Coulombic forces, this removes the singularity, easing the computation. An additional variational parameter was the width parameter used for the coherent states appearing in the frozen Gaussian propagator. Using an initial combination of nine Gaussian functions for He, with an initial energy of -2.5115 au, the variational method, with a very short time interval of integration, led to an improved energy of -2.81 ± 0.04 au.
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Affiliation(s)
- Ling Wang
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
| | - Eli Pollak
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
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46
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Kegerreis J, Makri N. Optimized Monte Carlo sampling in forward–backward semiclassical dynamics. J Comput Chem 2007; 28:818-24. [PMID: 17226835 DOI: 10.1002/jcc.20608] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Abstract
Forward-backward semiclassical dynamics (FBSD) provides a rigorous and powerful methodology for calculating time correlation functions in condensed phase systems characterized by substantial quantum mechanical effects associated with zero-point motion, quantum dispersion, or identical particle exchange symmetries. The efficiency of these simulations arises from the use of classical trajectories to capture all dynamical information. However, full quantization of the density operator makes these calculations rather expensive compared to fully classical molecular dynamics simulations. This article discusses the convergence properties of various correlation functions and introduces an optimal Monte Carlo sampling scheme that leads to a significant reduction of statistical error. A simple and efficient procedure for normalizing the FBSD results is also discussed. Illustrative examples on model systems are presented.
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Affiliation(s)
- Jeb Kegerreis
- Department of Chemistry, University of Illinois, 601 S. Goodwin Avenue, Urbana, Illinois 61801, USA
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48
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Zhang S, Pollak E. A prefactor free semiclassical initial value series representation of the propagator. J Chem Phys 2006; 121:3384-92. [PMID: 15303901 DOI: 10.1063/1.1772361] [Citation(s) in RCA: 57] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A new class of prefactor free semiclassical initial value representations (SCIVR) of the quantum propagator is presented. The derivation is based on the physically motivated demand, that on the average in phase space and in time, the propagator obey the exact quantum equation of motion. The resulting SCIVR series representation of the exact quantum propagator is also free of prefactors. When using a constant width parameter, the prefactor free SCIVR propagator is identical to the frozen Gaussian propagator of Heller [J. Chem. Phys. 75, 2923 (1981)]. A numerical study of the prefactor free SCIVR series is presented for scattering through a double slit potential, a system studied extensively previously by Gelabert et al. [J. Chem. Phys. 114, 2572 (2001)]. As a basis for comparison, the SCIVR series is also computed using the optimized Herman-Kluk SCIVR. We find that the sum of the zeroth order and the first order terms in the series suffice for an accurate determination of the diffraction pattern. The same exercise, but using the prefactor free propagator series needs also the second order term in the series, however the numerical effort is not greater than that needed for the Herman-Kluk propagator, since one does not need to compute the monodromy matrix elements at each point in time. The numerical advantage of the prefactor free propagator grows with increasing dimensionality of the problem.
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Affiliation(s)
- Shesheng Zhang
- Chemical Physics Department, Weizmann Institute of Science, 76100, Rehovot, Israel
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49
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Martin-Fierro E, Pollak E. Forward-backward semiclassical initial value series representation of quantum correlation functions. J Chem Phys 2006; 125:164104. [PMID: 17092060 DOI: 10.1063/1.2358985] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The forward-backward (FB) approximation as applied to semiclassical initial value representations (SCIVR's) has enabled the practical application of the SCIVR methodology to systems with many degrees of freedom. However, to date a systematic representation of the exact quantum dynamics in terms of the FB-SCIVR has proven elusive. In this paper, we provide a new derivation of a forward-backward phase space SCIVR expression (FBPS-SCIVR) derived previously by Thompson and Makri [Phys. Rev. E 59, R4729 (1999)]. This enables us to represent quantum correlation functions exactly in terms of a series whose leading order term is the FBPS-SCIVR expression. Numerical examples for systems with over 50 degrees of freedom are presented for the spin boson problem. Comparison of the FBPS-SCIVR with the numerically exact results of Wang [J. Chem. Phys. 113, 9948 (2000)] obtained using a multiconfigurational time dependent method shows that the leading order FBPS-SCIVR term already provides an excellent approximation.
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Affiliation(s)
- Eva Martin-Fierro
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovoth, Israel
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50
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Shao J, Pollak E. A new time evolving Gaussian series representation of the imaginary time propagator. J Chem Phys 2006; 125:133502. [PMID: 17029485 DOI: 10.1063/1.2207142] [Citation(s) in RCA: 36] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Frantsuzov and Mandelshtam [J. Chem. Phys. 121, 9247 (2004)] have recently demonstrated that a time evolving Gaussian approximation (TEGA) to the imaginary time propagator exp(-betaH) is useful for numerical computations of anharmonically coupled systems with many degrees of freedom. In this paper we derive a new exact series representation for the imaginary time propagator whose leading order term is the TEGA. One can thus use the TEGA not only as an approximation but also to obtain the exact imaginary time propagator. We also show how the TEGA may be generalized to provide a family of TEGA's. Finally, we find that the equations of motion governing the evolution of the center and width of the Gaussian may be thought of as introducing a quantum friction term to the classical evolution equations.
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Affiliation(s)
- Jiushu Shao
- Chemical Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel
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