1
|
Lu CY, Lee TY, Chou CC. Moving boundary truncated grid method for electronic nonadiabatic dynamics. J Chem Phys 2022; 156:044107. [DOI: 10.1063/5.0078909] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Chun-Yaung Lu
- Texas Advanced Computing Center, The University of Texas at Austin, Austin, Texas 78758, USA
| | - Tsung-Yen Lee
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Chia-Chun Chou
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
| |
Collapse
|
2
|
Lee TY, Lu CY, Chou CC. Moving Boundary Truncated Grid Method: Application to the Time Evolution of Distribution Functions in Phase Space. J Phys Chem A 2021; 125:476-491. [PMID: 33372780 DOI: 10.1021/acs.jpca.0c09525] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The moving boundary truncated grid (TG) method, previously developed to integrate the time-dependent Schrödinger equation and the imaginary time Schrödinger equation, is extended to the time evolution of distribution functions in phase space. A variable number of phase space grid points in the Eulerian representation are used to integrate the equation of motion for the distribution function, and the boundaries of the TG are adaptively determined as the distribution function evolves in time. Appropriate grid points are activated and deactivated for propagation of the distribution function, and no advance information concerning the dynamics in phase space is required. The TG method is used to integrate the equations of motion for phase space distribution functions, including the Klein-Kramers, Wigner-Moyal, and modified Caldeira-Leggett equations. Even though the initial distribution function is nonnegative, the solutions to the Wigner-Moyal and modified Caldeira-Leggett equations may develop negative basins in phase space originating from interference effects. Trajectory-based methods for propagation of the distribution function do not permit the formation of negative regions. However, the TG method can correctly capture the negative basins. Comparisons between the computational results obtained from the full grid and TG calculations demonstrate that the TG method not only significantly reduces the computational effort but also permits accurate propagation of various distribution functions in phase space.
Collapse
Affiliation(s)
- Tsung-Yen Lee
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
| | - Chun-Yaung Lu
- Texas Advanced Computing Center, The University of Texas at Austin, Austin, Texas 78758, United States
| | - Chia-Chun Chou
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
| |
Collapse
|
3
|
Nelson TR, White AJ, Bjorgaard JA, Sifain AE, Zhang Y, Nebgen B, Fernandez-Alberti S, Mozyrsky D, Roitberg AE, Tretiak S. Non-adiabatic Excited-State Molecular Dynamics: Theory and Applications for Modeling Photophysics in Extended Molecular Materials. Chem Rev 2020; 120:2215-2287. [PMID: 32040312 DOI: 10.1021/acs.chemrev.9b00447] [Citation(s) in RCA: 220] [Impact Index Per Article: 55.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/22/2022]
Abstract
Optically active molecular materials, such as organic conjugated polymers and biological systems, are characterized by strong coupling between electronic and vibrational degrees of freedom. Typically, simulations must go beyond the Born-Oppenheimer approximation to account for non-adiabatic coupling between excited states. Indeed, non-adiabatic dynamics is commonly associated with exciton dynamics and photophysics involving charge and energy transfer, as well as exciton dissociation and charge recombination. Understanding the photoinduced dynamics in such materials is vital to providing an accurate description of exciton formation, evolution, and decay. This interdisciplinary field has matured significantly over the past decades. Formulation of new theoretical frameworks, development of more efficient and accurate computational algorithms, and evolution of high-performance computer hardware has extended these simulations to very large molecular systems with hundreds of atoms, including numerous studies of organic semiconductors and biomolecules. In this Review, we will describe recent theoretical advances including treatment of electronic decoherence in surface-hopping methods, the role of solvent effects, trivial unavoided crossings, analysis of data based on transition densities, and efficient computational implementations of these numerical methods. We also emphasize newly developed semiclassical approaches, based on the Gaussian approximation, which retain phase and width information to account for significant decoherence and interference effects while maintaining the high efficiency of surface-hopping approaches. The above developments have been employed to successfully describe photophysics in a variety of molecular materials.
Collapse
Affiliation(s)
- Tammie R Nelson
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| | - Alexander J White
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| | - Josiah A Bjorgaard
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| | - Andrew E Sifain
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States.,U.S. Army Research Laboratory , Aberdeen Proving Ground , Maryland 21005 , United States
| | - Yu Zhang
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| | - Benjamin Nebgen
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| | | | - Dmitry Mozyrsky
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| | - Adrian E Roitberg
- Department of Chemistry , University of Florida , Gainesville , Florida 32611 , United States
| | - Sergei Tretiak
- Theoretical Division , Los Alamos National Laboratory , Los Alamos , New Mexico 87545 , United States
| |
Collapse
|
4
|
Chang BY, Shin S, Malinovsky VS, Sola IR. Grid-Based Ehrenfest Model To Study Electron-Nuclear Processes. J Phys Chem A 2019; 123:7171-7176. [PMID: 31314529 DOI: 10.1021/acs.jpca.9b05214] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The two-dimensional electron-nuclear Schrödinger equation using soft-core Coulomb potentials has been a cornerstone for modeling and predicting the behavior of one-active-electron diatomic molecules, particularly for processes where both bound and continuum states are important. The model, however, is computationally expensive to extend to more electron or nuclear coordinates. Here we propose use of the Ehrenfest approach to treat the nuclear motion, while the electronic motion is still solved by quantum propagation on a grid. In this work, we present results for a one-dimensional treatment of H2+, where the quantum and semiclassical dynamics can be directly compared, showing remarkably good agreement for a variety of situations. The advantage of the Ehrenfest approach is that it can be easily extended to treat as many nuclear degrees of freedom as needed.
Collapse
Affiliation(s)
- Bo Y Chang
- School of Chemistry , Seoul National University , Seoul 08826 , Republic of Korea
| | - Seokmin Shin
- School of Chemistry , Seoul National University , Seoul 08826 , Republic of Korea
| | | | - Ignacio R Sola
- Departamento de Química Física , Universidad Complutense , 28040 Madrid , Spain
| |
Collapse
|
5
|
Complex-valued derivative propagation method with approximate Bohmian trajectories: Application to electronic nonadiabatic dynamics. Chem Phys Lett 2018. [DOI: 10.1016/j.cplett.2018.03.051] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
|
6
|
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
| |
Collapse
|
7
|
Antipov SV, Bhattacharyya S, El Hage K, Xu ZH, Meuwly M, Rothlisberger U, Vaníček J. Ultrafast dynamics induced by the interaction of molecules with electromagnetic fields: Several quantum, semiclassical, and classical approaches. STRUCTURAL DYNAMICS (MELVILLE, N.Y.) 2017; 4:061509. [PMID: 29376107 PMCID: PMC5758379 DOI: 10.1063/1.4996559] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 07/17/2017] [Accepted: 10/11/2017] [Indexed: 06/07/2023]
Abstract
Several strategies for simulating the ultrafast dynamics of molecules induced by interactions with electromagnetic fields are presented. After a brief overview of the theory of molecule-field interaction, we present several representative examples of quantum, semiclassical, and classical approaches to describe the ultrafast molecular dynamics, including the multiconfiguration time-dependent Hartree method, Bohmian dynamics, local control theory, semiclassical thawed Gaussian approximation, phase averaging, dephasing representation, molecular mechanics with proton transfer, and multipolar force fields. In addition to the general overview, some focus is given to the description of nuclear quantum effects and to the direct dynamics, in which the ab initio energies and forces acting on the nuclei are evaluated on the fly. Several practical applications, performed within the framework of the Swiss National Center of Competence in Research "Molecular Ultrafast Science and Technology," are presented: These include Bohmian dynamics description of the collision of H with H2, local control theory applied to the photoinduced ultrafast intramolecular proton transfer, semiclassical evaluation of vibrationally resolved electronic absorption, emission, photoelectron, and time-resolved stimulated emission spectra, infrared spectroscopy of H-bonding systems, and multipolar force fields applications in the condensed phase.
Collapse
Affiliation(s)
- Sergey V Antipov
- Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Swarnendu Bhattacharyya
- Laboratory of Computational Chemistry and Biochemistry, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Krystel El Hage
- Department of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland
| | - Zhen-Hao Xu
- Department of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland
| | - Markus Meuwly
- Department of Chemistry, University of Basel, Klingelbergstrasse 80, CH-4056 Basel, Switzerland
| | - Ursula Rothlisberger
- Laboratory of Computational Chemistry and Biochemistry, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| | - Jiří Vaníček
- Laboratory of Theoretical Physical Chemistry, Institut des Sciences et Ingénierie Chimiques, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
| |
Collapse
|
8
|
|
9
|
High JS, Rego LGC, Jakubikova E. Quantum Dynamics Simulations of Excited State Energy Transfer in a Zinc–Free-Base Porphyrin Dyad. J Phys Chem A 2016; 120:8075-8084. [DOI: 10.1021/acs.jpca.6b05739] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Judah S. High
- Department
of Chemistry, North Carolina State University, Raleigh, North Carolina 27695, United States
| | - Luis G. C. Rego
- Department
of Physics, Federal University of Santa Catarina, Florianopolis 88040-900, Brazil
| | - Elena Jakubikova
- Department
of Chemistry, North Carolina State University, Raleigh, North Carolina 27695, United States
| |
Collapse
|
10
|
Heaps CW, Mazziotti DA. Accurate non-adiabatic quantum dynamics from pseudospectral sampling of time-dependent Gaussian basis sets. J Chem Phys 2016. [DOI: 10.1063/1.4959872] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Charles W. Heaps
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| | - David A. Mazziotti
- Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
| |
Collapse
|
11
|
Curchod BFE, Agostini F, Gross EKU. An exact factorization perspective on quantum interferences in nonadiabatic dynamics. J Chem Phys 2016; 145:034103. [DOI: 10.1063/1.4958637] [Citation(s) in RCA: 37] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
12
|
Chou CC. Complex-valued derivative propagation method with approximate Bohmian trajectories for quantum barrier scattering. Chem Phys 2015. [DOI: 10.1016/j.chemphys.2015.06.008] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
|
13
|
Chou CC. Ground state energy from the single trajectory propagation of the Schrödinger–Langevin equation. Chem Phys Lett 2015. [DOI: 10.1016/j.cplett.2015.05.043] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
|
14
|
Albareda G, Bofill JM, Tavernelli I, Huarte-Larrañaga F, Illas F, Rubio A. Conditional Born-Oppenheimer Dynamics: Quantum Dynamics Simulations for the Model Porphine. J Phys Chem Lett 2015; 6:1529-1535. [PMID: 26263307 DOI: 10.1021/acs.jpclett.5b00422] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We report a new theoretical approach to solve adiabatic quantum molecular dynamics halfway between wave function and trajectory-based methods. The evolution of a N-body nuclear wave function moving on a 3N-dimensional Born-Oppenheimer potential-energy hyper-surface is rewritten in terms of single-nuclei wave functions evolving nonunitarily on a 3-dimensional potential-energy surface that depends parametrically on the configuration of an ensemble of generally defined trajectories. The scheme is exact and, together with the use of trajectory-based statistical techniques, can be exploited to circumvent the calculation and storage of many-body quantities (e.g., wave function and potential-energy surface) whose size scales exponentially with the number of nuclear degrees of freedom. As a proof of concept, we present numerical simulations of a 2-dimensional model porphine where switching from concerted to sequential double proton transfer (and back) is induced quantum mechanically.
Collapse
Affiliation(s)
- Guillermo Albareda
- †Institut de Química Teòrica i Computacional, Universitat de Barcelona, Barcelona 08028, Spain
- ‡Departament de Química Física, Universitat de Barcelona, Barcelona 08028, Spain
| | - Josep Maria Bofill
- †Institut de Química Teòrica i Computacional, Universitat de Barcelona, Barcelona 08028, Spain
- §Departament de Química Orgànica, Universitat de Barcelona, Barcelona 08028, Spain
| | - Ivano Tavernelli
- ∥IBM Research Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
| | - Fermin Huarte-Larrañaga
- †Institut de Química Teòrica i Computacional, Universitat de Barcelona, Barcelona 08028, Spain
- ‡Departament de Química Física, Universitat de Barcelona, Barcelona 08028, Spain
| | - Francesc Illas
- †Institut de Química Teòrica i Computacional, Universitat de Barcelona, Barcelona 08028, Spain
- ‡Departament de Química Física, Universitat de Barcelona, Barcelona 08028, Spain
| | - Angel Rubio
- ⊥Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
| |
Collapse
|
15
|
Reimers JR, McKemmish LK, McKenzie RH, Hush NS. Non-adiabatic effects in thermochemistry, spectroscopy and kinetics: the general importance of all three Born–Oppenheimer breakdown corrections. Phys Chem Chem Phys 2015. [DOI: 10.1039/c5cp02238j] [Citation(s) in RCA: 36] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
Analytical and numerical solutions describing Born–Oppenheimer breakdown in a simple, widely applicable, model depict shortcomings in modern computational methods.
Collapse
Affiliation(s)
- Jeffrey R. Reimers
- International Centre for Quantum and Molecular Structure
- College of Sciences, Shanghai University
- Shanghai 200444
- China
- School of Mathematical and Physical Sciences
| | - Laura K. McKemmish
- Department of Physics and Astronomy
- University College London
- London
- UK
- School of Chemistry
| | - Ross H. McKenzie
- School of Mathematics and Physics
- The University of Queensland
- Australia
| | - Noel S. Hush
- School of Chemistry
- The University of Sydney
- Sydney
- Australia
- School of Molecular Biosciences
| |
Collapse
|
16
|
Chou CC. Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: application to the photodissociation dynamics of NOCl. J Chem Phys 2014; 140:104307. [PMID: 24628169 DOI: 10.1063/1.4867636] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneously integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.
Collapse
Affiliation(s)
- Chia-Chun Chou
- Department of Chemistry, National Tsing Hua University, Hsinchu 30013, Taiwan
| |
Collapse
|
17
|
Petersen J, Kay KG. Complex time paths for semiclassical wave packet propagation with complex trajectories. J Chem Phys 2014; 141:054114. [PMID: 25106577 DOI: 10.1063/1.4891918] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The use of complex-valued trajectories in semiclassical wave packet methods can lead to problems that prevent calculation of the wave function in certain regions of the configuration space. We investigate this so-called bald spot problem in the context of generalized Gaussian wave packet dynamics. The analysis shows that the bald spot phenomenon is essentially due to the complex nature of the initial conditions for the trajectories. It is, therefore, expected to be a general feature of several semiclassical methods that rely on trajectories with such initial conditions. A bald region is created when a trajectory, needed to calculate the wave function at a given time, reaches a singularity of the potential energy function in the complex plane at an earlier, real time. This corresponds to passage of a branch point singularity across the real axis of the complex time plane. The missing portions of the wave function can be obtained by deforming the time path for the integration of the equations of motion into the complex plane so that the singularity is circumvented. We present examples of bald spots, singularity times, and suitable complex time paths for one-dimensional barrier transmission in the Eckart and Gaussian systems. Although the bald regions for the Eckart system are often localized, they are found to be semi-infinite for the Gaussian system. For the case of deep tunneling, the bald regions for both systems may encompass the entire portion of space occupied by the transmitted wave packet. Thus, the use of complex time paths becomes essential for a treatment of barrier tunneling.
Collapse
Affiliation(s)
- Jakob Petersen
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Kenneth G Kay
- Department of Chemistry, Bar-Ilan University, Ramat-Gan 52900, Israel
| |
Collapse
|
18
|
Wang L, Trivedi D, Prezhdo OV. Global Flux Surface Hopping Approach for Mixed Quantum-Classical Dynamics. J Chem Theory Comput 2014; 10:3598-605. [DOI: 10.1021/ct5003835] [Citation(s) in RCA: 103] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/20/2023]
Affiliation(s)
- Linjun Wang
- Department
of Chemistry, University of Rochester, Rochester, New York 14627, United States
| | - Dhara Trivedi
- Department of Physics & Astronomy, University of Rochester, Rochester, New York 14627, United States
| | - Oleg V. Prezhdo
- Department
of Chemistry, University of Rochester, Rochester, New York 14627, United States
- Department of Physics & Astronomy, University of Rochester, Rochester, New York 14627, United States
| |
Collapse
|
19
|
Zamstein N, Tannor DJ. Communication: Overcoming the root search problem in complex quantum trajectory calculations. J Chem Phys 2014; 140:041105. [DOI: 10.1063/1.4862898] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
20
|
|
21
|
Zamstein N, Tannor DJ. Non-adiabatic molecular dynamics with complex quantum trajectories. II. The adiabatic representation. J Chem Phys 2013; 137:22A518. [PMID: 23249055 DOI: 10.1063/1.4739846] [Citation(s) in RCA: 40] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present a complex quantum trajectory method for treating non-adiabatic dynamics. Each trajectory evolves classically on a single electronic surface but with complex position and momentum. The equations of motion are derived directly from the time-dependent Schrödinger equation, and the population exchange arises naturally from amplitude-transfer terms. In this paper the equations of motion are derived in the adiabatic representation to complement our work in the diabatic representation [N. Zamstein and D. J. Tannor, J. Chem. Phys. 137, 22A517 (2012)]. We apply our method to two benchmark models introduced by John Tully [J. Chem. Phys. 93, 1061 (1990)], and get very good agreement with converged quantum-mechanical calculations. Specifically, we show that decoherence (spatial separation of wavepackets on different surfaces) is already contained in the equations of motion and does not require ad hoc augmentation.
Collapse
Affiliation(s)
- Noa Zamstein
- Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
| | | |
Collapse
|
22
|
|