1
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Malpathak S, Ananth N. Semiclassical dynamics in Wigner phase space II: Nonadiabatic hybrid Wigner dynamics. J Chem Phys 2024; 161:094110. [PMID: 39234964 DOI: 10.1063/5.0223187] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2024] [Accepted: 08/12/2024] [Indexed: 09/06/2024] Open
Abstract
We present an approximate semiclassical (SC) framework for mixed quantized dynamics in Wigner phase space in a two-part series. In the first article, we introduced the Adiabatic Hybrid Wigner Dynamics (AHWD) method that allows for a few important "system" degrees of freedom to be quantized using high-level double Herman-Kluk SC theory while describing the rest (the "bath") using classical-limit linearized SC theory. In this second article, we extend our hybrid Wigner dynamics to nonadiabatic processes. The resulting Nonadiabatic Hybrid Wigner Dynamics (NHWD) has two variants that differ in the choice of degrees of freedom to be quantized. Specifically, we introduce NHWD(E) where only the electronic state variables are quantized and the NHWD(V) where both electronic state variables and a handful of strongly coupled nuclear modes are quantized. We show that while NHWD(E) proves accurate for a wide range of scattering models and spin-boson models, systems where a few nuclear modes are strongly coupled to electronic states require NHWD(V) to accurately capture the long-time dynamics. Taken together, we show that AHWD and NHWD represent a new framework for SC simulations of high-dimensional systems with significant quantum effects.
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Affiliation(s)
- Shreyas Malpathak
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
| | - Nandini Ananth
- Department of Chemistry and Chemical Biology, Baker Laboratory, Cornell University, Ithaca, New York 14853, USA
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2
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Sayer T, Montoya-Castillo A. Generalized quantum master equations can improve the accuracy of semiclassical predictions of multitime correlation functions. J Chem Phys 2024; 161:011101. [PMID: 38949578 DOI: 10.1063/5.0219205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2024] [Accepted: 06/12/2024] [Indexed: 07/02/2024] Open
Abstract
Multitime quantum correlation functions are central objects in physical science, offering a direct link between the experimental observables and the dynamics of an underlying model. While experiments such as 2D spectroscopy and quantum control can now measure such quantities, the accurate simulation of such responses remains computationally expensive and sometimes impossible, depending on the system's complexity. A natural tool to employ is the generalized quantum master equation (GQME), which can offer computational savings by extending reference dynamics at a comparatively trivial cost. However, dynamical methods that can tackle chemical systems with atomistic resolution, such as those in the semiclassical hierarchy, often suffer from poor accuracy, limiting the credence one might lend to their results. By combining work on the accuracy-boosting formulation of semiclassical memory kernels with recent work on the multitime GQME, here we show for the first time that one can exploit a multitime semiclassical GQME to dramatically improve both the accuracy of coarse mean-field Ehrenfest dynamics and obtain orders of magnitude efficiency gains.
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Affiliation(s)
- Thomas Sayer
- Department of Chemistry, University of Colorado Boulder, Boulder, Colorado 80309, USA
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3
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Huang L, Shi Z, Wang L. Detailed Complementary Consistency: Wave Function Tells Particle How to Hop, Particle Tells Wave Function How to Collapse. J Phys Chem Lett 2024; 15:6771-6781. [PMID: 38912973 DOI: 10.1021/acs.jpclett.4c01313] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/25/2024]
Abstract
In mixed quantum-classical dynamics, the quantum subsystem can have both wave function and particle-like descriptions. However, they may yield inconsistent results for the expectation value of the same physical quantity. We here propose a novel detailed complementary consistency (DCC) method based on the principle of detailed internal consistency. Namely, the wave function along each trajectory tells the particle how to hop, while the particle tells the wave function how to collapse based on active states in the trajectory ensemble. As benchmarked in a diverse array of representative models with localized nonadiabatic couplings, DCC not only achieves fully consistent results (i.e., identical populations calculated based on wave functions and active states) but also closely reproduces the exact quantum results. Due to the high performance, our new DCC method has great potential to give a consistent and accurate mixed quantum-classical description of general nonadiabatic dynamics after further development.
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Affiliation(s)
- Lei Huang
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
| | - Zhecun Shi
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
| | - Linjun Wang
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
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4
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Shu Y, Truhlar DG. Generalized Semiclassical Ehrenfest Method: A Route to Wave Function-Free Photochemistry and Nonadiabatic Dynamics with Only Potential Energies and Gradients. J Chem Theory Comput 2024; 20:4396-4426. [PMID: 38819014 DOI: 10.1021/acs.jctc.4c00424] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/01/2024]
Abstract
We reconsider recent methods by which direct dynamics calculations of electronically nonadiabatic processes can be carried out while requiring only adiabatic potential energies and their gradients. We show that these methods can be understood in terms of a new generalization of the well-known semiclassical Ehrenfest method. This is convenient because it eliminates the need to evaluate electronic wave functions and their matrix elements along the mixed quantum-classical trajectories. The new approximations and procedures enabling this advance are the curvature-driven approximation to the time-derivative coupling, the generalized semiclassical Ehrenfest method, and a new gradient correction scheme called the time-derivative matrix (TDM) scheme. When spin-orbit coupling is present, one can carry out dynamics calculations in the fully adiabatic basis using potential energies and gradients calculated without spin-orbit coupling plus the spin-orbit coupling matrix elements. Even when spin-orbit coupling is neglected, the method is useful because it allows calculations by electronic structure methods for which nonadiabatic coupling vectors are unavailable. In order to place the new considerations in context, the article starts out with a review of background material on trajectory surface hopping, the semiclassical Ehrenfest scheme, and methods for incorporating decoherence. We consider both internal conversion and intersystem crossing. We also review several examples from our group of successful applications of the curvature-driven approximation.
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Affiliation(s)
- Yinan Shu
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
| | - Donald G Truhlar
- Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431, United States
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5
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Qiu T, Bian X, Tao Z, Subotnik JE. A fast and smooth one-electron approach for investigating charge transfer states and D1-D0 crossings for systems with odd numbers of electrons. J Chem Phys 2024; 160:214115. [PMID: 38832731 DOI: 10.1063/5.0209036] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2024] [Accepted: 05/13/2024] [Indexed: 06/05/2024] Open
Abstract
We propose an efficient version of ensemble Hartree-Fock/density functional theory to calculate a set of two charge-transfer states for systems with odd-numbers of electrons. The approach does require definitions of donor/acceptor fragments; however, the approach is not very sensitive to such definitions-even in the limit of very strong electronic coupling. The key ansatz is that, by mandating that the vector space spanned by the active orbitals projects equally onto the donor and acceptor fragments, such a constraint eliminates all intra-molecular local excitations and makes it far easier to generate potential energy surfaces that are smooth over a wide region of configuration space. The method is fast, working with only two electron configurations, and should be useful for ab initio non-adiabatic dynamics in the near future.
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Affiliation(s)
- Tian Qiu
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Xuezhi Bian
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Zhen Tao
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Joseph E Subotnik
- Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
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6
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Kang M, Nuomin H, Chowdhury SN, Yuly JL, Sun K, Whitlow J, Valdiviezo J, Zhang Z, Zhang P, Beratan DN, Brown KR. Seeking a quantum advantage with trapped-ion quantum simulations of condensed-phase chemical dynamics. Nat Rev Chem 2024; 8:340-358. [PMID: 38641733 DOI: 10.1038/s41570-024-00595-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 03/07/2024] [Indexed: 04/21/2024]
Abstract
Simulating the quantum dynamics of molecules in the condensed phase represents a longstanding challenge in chemistry. Trapped-ion quantum systems may serve as a platform for the analog-quantum simulation of chemical dynamics that is beyond the reach of current classical-digital simulation. To identify a 'quantum advantage' for these simulations, performance analysis of both analog-quantum simulation on noisy hardware and classical-digital algorithms is needed. In this Review, we make a comparison between a noisy analog trapped-ion simulator and a few choice classical-digital methods on simulating the dynamics of a model molecular Hamiltonian with linear vibronic coupling. We describe several simple Hamiltonians that are commonly used to model molecular systems, which can be simulated with existing or emerging trapped-ion hardware. These Hamiltonians may serve as stepping stones towards the use of trapped-ion simulators for systems beyond the reach of classical-digital methods. Finally, we identify dynamical regimes in which classical-digital simulations seem to have the weakest performance with respect to analog-quantum simulations. These regimes may provide the lowest hanging fruit to make the most of potential quantum advantages.
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Affiliation(s)
- Mingyu Kang
- Duke Quantum Center, Duke University, Durham, NC, USA.
- Department of Physics, Duke University, Durham, NC, USA.
| | - Hanggai Nuomin
- Department of Chemistry, Duke University, Durham, NC, USA
| | | | - Jonathon L Yuly
- Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA
| | - Ke Sun
- Duke Quantum Center, Duke University, Durham, NC, USA
- Department of Physics, Duke University, Durham, NC, USA
| | - Jacob Whitlow
- Duke Quantum Center, Duke University, Durham, NC, USA
- Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA
| | - Jesús Valdiviezo
- Kenneth S. Pitzer Theory Center, University of California, Berkeley, CA, USA
- Department of Chemistry, University of California, Berkeley, CA, USA
- Departamento de Ciencias, Sección Química, Pontificia Universidad Católica del Perú, Lima, Peru
| | - Zhendian Zhang
- Department of Chemistry, Duke University, Durham, NC, USA
| | - Peng Zhang
- Department of Chemistry, Duke University, Durham, NC, USA
| | - David N Beratan
- Department of Physics, Duke University, Durham, NC, USA.
- Department of Chemistry, Duke University, Durham, NC, USA.
- Department of Biochemistry, Duke University, Durham, NC, USA.
| | - Kenneth R Brown
- Duke Quantum Center, Duke University, Durham, NC, USA.
- Department of Physics, Duke University, Durham, NC, USA.
- Department of Chemistry, Duke University, Durham, NC, USA.
- Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA.
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7
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Guo X, Li G, Shi Z, Wang L. Surface Hopping with Reliable Wave Function by Introducing Auxiliary Wave Packets to Trajectory Branching. J Phys Chem Lett 2024:3345-3353. [PMID: 38498301 DOI: 10.1021/acs.jpclett.4c00437] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/20/2024]
Abstract
It is well-known that the widely utilized fewest switches surface hopping method suffers from the severe overcoherence problem, and thus adiabatic populations calculated by wave functions are generally inferior to those based on active states. More importantly, to achieve a complete description of nonadiabatic dynamics, the density matrix is essential. In this paper, we present an auxiliary branching corrected surface hopping (A-BCSH) method that introduces auxiliary wave packets (WPs) on the adiabatic potential energy surfaces for trajectory branching. Both rapid and gradual separation of WP components on different surfaces are characterized, and thus the correct decoherence time along each trajectory is captured. As demonstrated in the three standard Tully models, A-BCSH exhibits excellent internal consistency. Namely, close adiabatic populations are obtained based on both wave functions and active states. In particular, A-BCSH successfully obtains a reliable time-dependent spatial distribution of the density matrix, which relies only on electronic wave functions. Due to its high performance, our A-BCSH method provides a new and highly promising perspective on further development of more consistent surface hopping with reliable wave function.
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Affiliation(s)
- Xin Guo
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
| | - Guijie Li
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
| | - Zhecun Shi
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
| | - Linjun Wang
- Key Laboratory of Excited-State Materials of Zhejiang Province, Department of Chemistry, Zhejiang University, Hangzhou 310058, China
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8
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Makri N. Kink Sum for Long-Memory Small Matrix Path Integral Dynamics. J Phys Chem B 2024. [PMID: 38437738 DOI: 10.1021/acs.jpcb.3c08282] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/06/2024]
Abstract
The small matrix decomposition of the real-time path integral (SMatPI) allows for numerically exact and efficient propagation of the reduced density matrix (RDM) for system-bath Hamiltonians. Its high efficiency lies in the small size of the SMatPI matrices employed in the iterative algorithm, whose size is equal to that of the full RDM. By avoiding the storage and multiplication of large tensors, the SMatPI algorithm is applicable in multistate systems under a variety of conditions. The main computational effort is the evaluation of path sums within the entangled memory length to construct the SMatPI matrices. A number of methods are available for this task, each with its own favorable parameter regime, but calculations with strong system-bath coupling and long memory at low temperatures remain out of reach. The present paper evaluates the path sums by binning the paths (in forward time only) based on their amplitudes, which depend on the number and type of kinks they contain. The algorithm is very efficient, leading to a dramatic acceleration of path sums and significantly extending the accessible memory length in the most challenging regimes.
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Affiliation(s)
- Nancy Makri
- Departments of Chemistry and Physics, University of Illinois, 505 S. Mathews Avenue, Urbana, Illinois 61801, United States
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9
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Calderón LF, Triviño H, Pachón LA. Quantum to Classical Cavity Chemistry Electrodynamics. J Phys Chem Lett 2023; 14:11725-11734. [PMID: 38112558 DOI: 10.1021/acs.jpclett.3c02870] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/21/2023]
Abstract
Polaritonic chemistry has ushered in new avenues for controlling molecular dynamics. However, two key questions remain: (i) Can classical light sources elicit the same effects as certain quantum light sources on molecular systems? (ii) Can semiclassical treatments of light-matter interactions capture nontrivial quantum effects observed in molecular dynamics? This work presents a quantum-classical approach addressing issues of realizing cavity chemistry effects without actual cavities. It also highlights the limitations of the standard semiclassical light-matter interaction. It is demonstrated that classical light sources can mimic quantum effects up to the second order of light-matter interaction provided that the mean-field contribution, the symmetrized two-time correlation function, and the linear response function are the same in both situations. Numerical simulations show that the quantum-classical method aligns more closely with exact quantum molecular-only dynamics for quantum light states such as Fock states, superpositions of Fock states, and vacuum squeezed states than does the conventional semiclassical approach.
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Affiliation(s)
- Leonardo F Calderón
- Grupo de Física Teórica y Matemática Aplicada, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia; Calle 70 No. 52-21, 500001 Medellín, Colombia
- Grupo de Física Computacional en Materia Condensada, Escuela de Física, Facultad de Ciencias, Universidad Industrial de Santander UIS; Cra 27 Calle 9 Ciudad Universitaria, 680002 Bucaramanga, Colombia
| | - Humberto Triviño
- Grupo de Física Teórica y Matemática Aplicada, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia; Calle 70 No. 52-21, 500001 Medellín, Colombia
| | - Leonardo A Pachón
- Grupo de Física Teórica y Matemática Aplicada, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia; Calle 70 No. 52-21, 500001 Medellín, Colombia
- Grupo de Física Atómica y Molecular, Instituto de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Antioquia; Calle 70 No. 52-21, 500001 Medellín, Colombia
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10
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Amati G, Mannouch JR, Richardson JO. Detailed balance in mixed quantum-classical mapping approaches. J Chem Phys 2023; 159:214114. [PMID: 38054513 DOI: 10.1063/5.0176291] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2023] [Accepted: 11/07/2023] [Indexed: 12/07/2023] Open
Abstract
The violation of detailed balance poses a serious problem for the majority of current quasiclassical methods for simulating nonadiabatic dynamics. In order to analyze the severity of the problem, we predict the long-time limits of the electronic populations according to various quasiclassical mapping approaches by applying arguments from classical ergodic theory. Our analysis confirms that regions of the mapping space that correspond to negative populations, which most mapping approaches introduce in order to go beyond the Ehrenfest approximation, pose the most serious issue for reproducing the correct thermalization behavior. This is because inverted potentials, which arise from negative electronic populations entering the nuclear force, can result in trajectories unphysically accelerating off to infinity. The recently developed mapping approach to surface hopping (MASH) provides a simple way of avoiding inverted potentials while retaining an accurate description of the dynamics. We prove that MASH, unlike any other quasiclassical approach, is guaranteed to describe the exact thermalization behavior of all quantum-classical systems, confirming it as one of the most promising methods for simulating nonadiabatic dynamics in real condensed-phase systems.
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Affiliation(s)
- Graziano Amati
- Department of Chemistry and Applied Biosciences, ETH Zürich, 8093 Zürich, Switzerland
| | - Jonathan R Mannouch
- Department of Chemistry and Applied Biosciences, ETH Zürich, 8093 Zürich, Switzerland
| | - Jeremy O Richardson
- Department of Chemistry and Applied Biosciences, ETH Zürich, 8093 Zürich, Switzerland
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11
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Tully JC. Ehrenfest dynamics with quantum mechanical nuclei. Chem Phys Lett 2023. [DOI: 10.1016/j.cplett.2023.140396] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 03/06/2023]
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12
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Sergi A, Lamberto D, Migliore A, Messina A. Quantum-Classical Hybrid Systems and Ehrenfest's Theorem. ENTROPY (BASEL, SWITZERLAND) 2023; 25:e25040602. [PMID: 37190388 PMCID: PMC10138123 DOI: 10.3390/e25040602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2023] [Revised: 03/17/2023] [Accepted: 03/30/2023] [Indexed: 05/17/2023]
Abstract
The conceptual analysis of quantum mechanics brings to light that a theory inherently consistent with observations should be able to describe both quantum and classical systems, i.e., quantum-classical hybrids. For example, the orthodox interpretation of measurements requires the transient creation of quantum-classical hybrids. Despite its limitations in defining the classical limit, Ehrenfest's theorem makes the simplest contact between quantum and classical mechanics. Here, we generalized the Ehrenfest theorem to bipartite quantum systems. To study quantum-classical hybrids, we employed a formalism based on operator-valued Wigner functions and quantum-classical brackets. We used this approach to derive the form of the Ehrenfest theorem for quantum-classical hybrids. We found that the time variation of the average energy of each component of the bipartite system is equal to the average of the symmetrized quantum dissipated power in both the quantum and the quantum-classical case. We expect that these theoretical results will be useful both to analyze quantum-classical hybrids and to develop self-consistent numerical algorithms for Ehrenfest-type simulations.
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Affiliation(s)
- Alessandro Sergi
- Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università degli Studi di Messina, viale F. Stagno d'Alcontres 31, 98166 Messina, Italy
- Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa
| | - Daniele Lamberto
- Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università degli Studi di Messina, viale F. Stagno d'Alcontres 31, 98166 Messina, Italy
| | - Agostino Migliore
- Department of Chemical Sciences, University of Padova, Via Marzolo 1, 35131 Padova, Italy
| | - Antonino Messina
- Dipartimento di Matematica ed Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
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13
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Montoya-Castillo A, Markland TE. A derivation of the conditions under which bosonic operators exactly capture fermionic structure and dynamics. J Chem Phys 2023; 158:094112. [PMID: 36889969 DOI: 10.1063/5.0138664] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/15/2023] Open
Abstract
The dynamics of many-body fermionic systems are important in problems ranging from catalytic reactions at electrochemical surfaces to transport through nanojunctions and offer a prime target for quantum computing applications. Here, we derive the set of conditions under which fermionic operators can be exactly replaced by bosonic operators that render the problem amenable to a large toolbox of dynamical methods while still capturing the correct dynamics of n-body operators. Importantly, our analysis offers a simple guide on how one can exploit these simple maps to calculate nonequilibrium and equilibrium single- and multi-time correlation functions essential in describing transport and spectroscopy. We use this to rigorously analyze and delineate the applicability of simple yet effective Cartesian maps that have been shown to correctly capture the correct fermionic dynamics in select models of nanoscopic transport. We illustrate our analytical results with exact simulations of the resonant level model. Our work provides new insights as to when one can leverage the simplicity of bosonic maps to simulate the dynamics of many-electron systems, especially those where an atomistic representation of nuclear interactions becomes essential.
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Affiliation(s)
| | - Thomas E Markland
- Department of Chemistry, Stanford University, Stanford, California 94305, USA
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14
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Shu Y, Truhlar DG. Decoherence and Its Role in Electronically Nonadiabatic Dynamics. J Chem Theory Comput 2023; 19:380-395. [PMID: 36622843 PMCID: PMC9878713 DOI: 10.1021/acs.jctc.2c00988] [Citation(s) in RCA: 14] [Impact Index Per Article: 14.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Indexed: 01/10/2023]
Abstract
Decoherence is the tendency of a time-evolved reduced density matrix for a subsystem to assume a form corresponding to a statistical ensemble of states rather than a coherent combination of pure-state wave functions. When a molecular process involves changes in the electronic state and the coordinates of the nuclei, as in ultraviolet or visible light photochemistry or electronically inelastic collisions, the reduced density matrix of the electronic subsystem suffers decoherence, due to its interaction with the nuclear subsystem. We present the background necessary to conceptualize this decoherence; in particular, we discuss the density matrix description of pure states and mixed states, and we discuss pointer states and decoherence time. We then discuss how decoherence is treated in the coherent switching with decay of mixing algorithm and the trajectory surface hopping method for semiclassical calculations of electronically nonadiabatic processes.
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Affiliation(s)
- Yinan Shu
- Department of Chemistry,
Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota55455-0431, United States
| | - Donald G. Truhlar
- Department of Chemistry,
Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota55455-0431, United States
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15
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Gelin MF, Chen L, Domcke W. Equation-of-Motion Methods for the Calculation of Femtosecond Time-Resolved 4-Wave-Mixing and N-Wave-Mixing Signals. Chem Rev 2022; 122:17339-17396. [PMID: 36278801 DOI: 10.1021/acs.chemrev.2c00329] [Citation(s) in RCA: 20] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/25/2023]
Abstract
Femtosecond nonlinear spectroscopy is the main tool for the time-resolved detection of photophysical and photochemical processes. Since most systems of chemical interest are rather complex, theoretical support is indispensable for the extraction of the intrinsic system dynamics from the detected spectroscopic responses. There exist two alternative theoretical formalisms for the calculation of spectroscopic signals, the nonlinear response-function (NRF) approach and the spectroscopic equation-of-motion (EOM) approach. In the NRF formalism, the system-field interaction is assumed to be sufficiently weak and is treated in lowest-order perturbation theory for each laser pulse interacting with the sample. The conceptual alternative to the NRF method is the extraction of the spectroscopic signals from the solutions of quantum mechanical, semiclassical, or quasiclassical EOMs which govern the time evolution of the material system interacting with the radiation field of the laser pulses. The NRF formalism and its applications to a broad range of material systems and spectroscopic signals have been comprehensively reviewed in the literature. This article provides a detailed review of the suite of EOM methods, including applications to 4-wave-mixing and N-wave-mixing signals detected with weak or strong fields. Under certain circumstances, the spectroscopic EOM methods may be more efficient than the NRF method for the computation of various nonlinear spectroscopic signals.
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Affiliation(s)
- Maxim F Gelin
- School of Science, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Lipeng Chen
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany
| | - Wolfgang Domcke
- Department of Chemistry, Technical University of Munich, D-85747 Garching,Germany
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16
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Amati G, Saller MAC, Kelly A, Richardson JO. Quasiclassical approaches to the generalized quantum master equation. J Chem Phys 2022; 157:234103. [PMID: 36550031 DOI: 10.1063/5.0124028] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022] Open
Abstract
The formalism of the generalized quantum master equation (GQME) is an effective tool to simultaneously increase the accuracy and the efficiency of quasiclassical trajectory methods in the simulation of nonadiabatic quantum dynamics. The GQME expresses correlation functions in terms of a non-Markovian equation of motion, involving memory kernels that are typically fast-decaying and can therefore be computed by short-time quasiclassical trajectories. In this paper, we study the approximate solution of the GQME, obtained by calculating the kernels with two methods: Ehrenfest mean-field theory and spin-mapping. We test the approaches on a range of spin-boson models with increasing energy bias between the two electronic levels and place a particular focus on the long-time limits of the populations. We find that the accuracy of the predictions of the GQME depends strongly on the specific technique used to calculate the kernels. In particular, spin-mapping outperforms Ehrenfest for all the systems studied. The problem of unphysical negative electronic populations affecting spin-mapping is resolved by coupling the method with the master equation. Conversely, Ehrenfest in conjunction with the GQME can predict negative populations, despite the fact that the populations calculated from direct dynamics are positive definite.
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Affiliation(s)
- Graziano Amati
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
| | | | - Aaron Kelly
- The Hamburg Center for Ultrafast Imaging, Universität Hamburg, Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, 22761 Hamburg, Germany
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17
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DECIDE: A Deterministic Mixed Quantum-Classical Dynamics Approach. APPLIED SCIENCES-BASEL 2022. [DOI: 10.3390/app12147022] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Mixed quantum-classical dynamics provides an efficient way of simulating the dynamics of quantum subsystems coupled to many-body environments. Many processes, including proton-transfer reactions, electron-transfer reactions, and vibrational energy transport, for example, take place in such open systems. The most accurate algorithms for performing mixed quantum-classical simulations require very large ensembles of trajectories to obtain converged expectation values, which is computationally prohibitive for quantum subsystems containing even a few degrees of freedom. The recently developed “Deterministic evolution of coordinates with initial decoupled equations” (DECIDE) method has demonstrated high accuracy and low computational cost for a host of model systems; however, these applications relied on representing the equations of motion in subsystem and adiabatic energy bases. While these representations are convenient for certain systems, the position representation is convenient for many other systems, including systems undergoing proton- and electron-transfer reactions. Thus, in this review, we provide a step-by-step derivation of the DECIDE approach and demonstrate how to cast the DECIDE equations in a quantum harmonic oscillator position basis for a simple one-dimensional (1D) hydrogen bond model. After integrating the DECIDE equations of motion on this basis, we show that the total energy of the system is conserved for this model and calculate various quantities of interest. Limitations of casting the equations in an incomplete basis are also discussed.
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18
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He X, Wu B, Shang Y, Li B, Cheng X, Liu J. New phase space formulations and quantum dynamics approaches. WIRES COMPUTATIONAL MOLECULAR SCIENCE 2022. [DOI: 10.1002/wcms.1619] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/05/2022]
Affiliation(s)
- Xin He
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Baihua Wu
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Youhao Shang
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Bingqi Li
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Xiangsong Cheng
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
| | - Jian Liu
- Beijing National Laboratory for Molecular Sciences Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University Beijing China
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19
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Runeson JE, Lawrence JE, Mannouch JR, Richardson JO. Explaining the Efficiency of Photosynthesis: Quantum Uncertainty or Classical Vibrations? J Phys Chem Lett 2022; 13:3392-3399. [PMID: 35404611 PMCID: PMC9036581 DOI: 10.1021/acs.jpclett.2c00538] [Citation(s) in RCA: 11] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2022] [Accepted: 04/01/2022] [Indexed: 06/14/2023]
Abstract
Photosynthetic organisms are known to use a mechanism of vibrationally assisted exciton energy transfer to efficiently harvest energy from light. The importance of quantum effects in this mechanism is a long-standing topic of debate, which has traditionally focused on the role of excitonic coherences. Here, we address another recent claim: that the efficient energy transfer in the Fenna-Matthews-Olson complex relies on nuclear quantum uncertainty and would not function if the vibrations were classical. We present a counter-example to this claim, showing by trajectory-based simulations that a description in terms of quantum electrons and classical nuclei is indeed sufficient to describe the funneling of energy to the reaction center. We analyze and compare these findings to previous classical-nuclear approximations that predicted the absence of an energy funnel and conclude that the key difference and the reason for the discrepancy is the ability of the trajectories to properly account for Newton's third law.
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Affiliation(s)
- Johan E. Runeson
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
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20
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Braver Y, Valkunas L, Gelzinis A. Stark absorption and Stark fluorescence spectroscopies: Theory and simulations. J Chem Phys 2021; 155:244101. [PMID: 34972359 DOI: 10.1063/5.0073962] [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
Stark spectroscopy experiments are widely used to study the properties of molecular systems, particularly those containing charge-transfer (CT) states. However, due to the small transition dipole moments and large static dipole moments of the CT states, the standard interpretation of the Stark absorption and Stark fluorescence spectra in terms of the Liptay model may be inadequate. In this work, we provide a theoretical framework for calculations of Stark absorption and Stark fluorescence spectra and propose new methods of simulations that are based on the quantum-classical theory. In particular, we use the forward-backward trajectory solution and a variant of the Poisson bracket mapping equation, which have been recently adapted for the calculation of conventional (field-free) absorption and fluorescence spectra. For comparison, we also apply the recently proposed complex time-dependent Redfield theory, while exact results are obtained using the hierarchical equations of motion approach. We show that the quantum-classical methods produce accurate results for a wide range of systems, including those containing CT states. The CT states contribute significantly to the Stark spectra, and the standard Liptay formalism is shown to be inapplicable for the analysis of spectroscopic data in those cases. We demonstrate that states with large static dipole moments may cause a pronounced change in the total fluorescence yield of the system in the presence of an external electric field. This effect is correctly captured by the quantum-classical methods, which should therefore prove useful for further studies of Stark spectra of real molecular systems. As an example, we calculate the Stark spectra for the Fenna-Matthews-Olson complex of green sulfur bacteria.
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Affiliation(s)
- Yakov Braver
- Faculty of Physics, Institute of Chemical Physics, Vilnius University, Saulėtekio Ave. 9-III, LT-10222 Vilnius, Lithuania and Department of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio Ave. 3, LT-10257 Vilnius, Lithuania
| | - Leonas Valkunas
- Faculty of Physics, Institute of Chemical Physics, Vilnius University, Saulėtekio Ave. 9-III, LT-10222 Vilnius, Lithuania and Department of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio Ave. 3, LT-10257 Vilnius, Lithuania
| | - Andrius Gelzinis
- Faculty of Physics, Institute of Chemical Physics, Vilnius University, Saulėtekio Ave. 9-III, LT-10222 Vilnius, Lithuania and Department of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio Ave. 3, LT-10257 Vilnius, Lithuania
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21
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Albareda G, Lively K, Sato SA, Kelly A, Rubio A. Conditional Wave Function Theory: A Unified Treatment of Molecular Structure and Nonadiabatic Dynamics. J Chem Theory Comput 2021; 17:7321-7340. [PMID: 34752108 PMCID: PMC8675140 DOI: 10.1021/acs.jctc.1c00772] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2021] [Indexed: 11/28/2022]
Abstract
We demonstrate that a conditional wave function theory enables a unified and efficient treatment of the equilibrium structure and nonadiabatic dynamics of correlated electron-ion systems. The conditional decomposition of the many-body wave function formally recasts the full interacting wave function of a closed system as a set of lower-dimensional (conditional) coupled "slices". We formulate a variational wave function ansatz based on a set of conditional wave function slices and demonstrate its accuracy by determining the structural and time-dependent response properties of the hydrogen molecule. We then extend this approach to include time-dependent conditional wave functions and address paradigmatic nonequilibrium processes including strong-field molecular ionization, laser-driven proton transfer, and nuclear quantum effects induced by a conical intersection. This work paves the road for the application of conditional wave function theory in equilibrium and out-of-equilibrium ab initio molecular simulations of finite and extended systems.
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Affiliation(s)
- Guillermo Albareda
- Nano-Bio
Spectroscopy Group and European Theoretical Spectroscopy Facility
(ETSF), Universidad del País Vasco
(UPV/EHU), Av. Tolosa
72, 20018 San Sebastian, Spain
- Institute
of Theoretical and Computational Chemistry, University of Barcelona, Martí i Franquès 1-11, 08028 Barcelona, Spain
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Kevin Lively
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Centre for Ultrafast Imaging, University
of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - Shunsuke A. Sato
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Sciences, University of
Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
| | - Aaron Kelly
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Centre for Ultrafast Imaging, University
of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
- Department
of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada
| | - Angel Rubio
- Nano-Bio
Spectroscopy Group and European Theoretical Spectroscopy Facility
(ETSF), Universidad del País Vasco
(UPV/EHU), Av. Tolosa
72, 20018 San Sebastian, Spain
- Max
Planck Institute for the Structure and Dynamics of Matter and Center
for Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany
- The
Hamburg Centre for Ultrafast Imaging, University
of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
- Center
for Computational Quantum Physics (CCQ), Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, United
States
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22
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Braver Y, Valkunas L, Gelzinis A. Quantum-Classical Approach for Calculations of Absorption and Fluorescence: Principles and Applications. J Chem Theory Comput 2021; 17:7157-7168. [PMID: 34618457 PMCID: PMC8719324 DOI: 10.1021/acs.jctc.1c00777] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2021] [Indexed: 01/20/2023]
Abstract
Absorption and fluorescence spectroscopy techniques provide a wealth of information on molecular systems. The simulations of such experiments remain challenging, however, despite the efforts put into developing the underlying theory. An attractive method of simulating the behavior of molecular systems is provided by the quantum-classical theory─it enables one to keep track of the state of the bath explicitly, which is needed for accurate calculations of fluorescence spectra. Unfortunately, until now there have been relatively few works that apply quantum-classical methods for modeling spectroscopic data. In this work, we seek to provide a framework for the calculations of absorption and fluorescence lineshapes of molecular systems using the methods based on the quantum-classical Liouville equation, namely, the forward-backward trajectory solution (FBTS) and the non-Hamiltonian variant of the Poisson bracket mapping equation (PBME-nH). We perform calculations on a molecular dimer and the photosynthetic Fenna-Matthews-Olson complex. We find that in the case of absorption, the FBTS outperforms PBME-nH, consistently yielding highly accurate results. We next demonstrate that for fluorescence calculations, the method of choice is a hybrid approach, which we call PBME-nH-Jeff, that utilizes the effective coupling theory [Gelzinis, A.; J. Chem. Phys. 2020, 152, 051103] to estimate the excited state equilibrium density operator. Thus, we find that FBTS and PBME-nH-Jeff are excellent candidates for simulating, respectively, absorption and fluorescence spectra of real molecular systems.
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Affiliation(s)
- Yakov Braver
- Institute
of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio Avenue 9-III, LT-10222 Vilnius, Lithuania
- Department
of Molecular Compound Physics, Center for
Physical Sciences and Technology, Saulėtekio Avenue 3, LT-10257 Vilnius, Lithuania
| | - Leonas Valkunas
- Institute
of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio Avenue 9-III, LT-10222 Vilnius, Lithuania
- Department
of Molecular Compound Physics, Center for
Physical Sciences and Technology, Saulėtekio Avenue 3, LT-10257 Vilnius, Lithuania
| | - Andrius Gelzinis
- Institute
of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio Avenue 9-III, LT-10222 Vilnius, Lithuania
- Department
of Molecular Compound Physics, Center for
Physical Sciences and Technology, Saulėtekio Avenue 3, LT-10257 Vilnius, Lithuania
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23
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Chowdhury SN, Huo P. Non-adiabatic Matsubara dynamics and non-adiabatic ring-polymer molecular dynamics. J Chem Phys 2021; 154:124124. [PMID: 33810665 DOI: 10.1063/5.0042136] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/06/2023] Open
Abstract
We present the non-adiabatic Matsubara dynamics, a general framework for computing the time-correlation function (TCF) of electronically non-adiabatic systems. This new formalism is derived based on the generalized Kubo-transformed TCF using the Wigner representation for both the nuclear degrees of freedom and the electronic mapping variables. By dropping the non-Matsubara nuclear normal modes in the quantum Liouvillian and explicitly integrating these modes out from the expression of the TCF, we derived the non-adiabatic Matsubara dynamics approach. Further making the approximation to drop the imaginary part of the Matsubara Liouvillian and enforce the nuclear momentum integral to be real, we arrived at the non-adiabatic ring-polymer molecular dynamics (NRPMD) approach. We have further justified the capability of NRPMD for simulating the non-equilibrium TCF. This work provides the rigorous theoretical foundation for several recently proposed state-dependent RPMD approaches and offers a general framework for developing new non-adiabatic quantum dynamics methods in the future.
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Affiliation(s)
- Sutirtha N Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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24
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Kim HW, Rhee YM. Two-oscillator mapping modification of the Poisson bracket mapping equation formulation of the quantum-classical Liouville equation. J Chem Phys 2020; 153:214103. [PMID: 33291890 DOI: 10.1063/5.0027799] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022] Open
Abstract
Mapping basis solutions provide efficient ways for simulating mixed quantum-classical (MQC) dynamics in complex systems by matching multiple quantum states of interest to some fictitious physical states. Recently, various MQC methods were devised such that two harmonic oscillators are employed to represent each electronic state, showing improvements over one-oscillator-based methods. Here, we introduce and analyze newly modified mapping approximations of the quantum-classical Liouville equation (QCLE) using two oscillators for each electronic state. We design two separate mapping relations that we can adopt toward simulating dynamics and computing expectation values. Through the process, two MQC methods can be constructed, one of which actually reproduces the population dynamics of the forward and backward trajectory solution of QCLE. By applying the methods to spin-boson systems with a range of parameters, we find out that the choice of mapping relations greatly affects the simulation results. We also show that further improvement is possible through using modified identity operator formulations. Our findings may be helpful in constructing improved MQC methods in the future.
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Affiliation(s)
- Hyun Woo Kim
- Center for Molecular Modeling and Simulation, Korea Research Institute of Chemical Technology (KRICT), Daejeon 34114, South Korea
| | - Young Min Rhee
- Department of Chemistry, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, South Korea
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25
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Mannouch JR, Richardson JO. A partially linearized spin-mapping approach for nonadiabatic dynamics. I. Derivation of the theory. J Chem Phys 2020; 153:194109. [DOI: 10.1063/5.0031168] [Citation(s) in RCA: 13] [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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26
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Mannouch JR, Richardson JO. A partially linearized spin-mapping approach for nonadiabatic dynamics. II. Analysis and comparison with related approaches. J Chem Phys 2020; 153:194110. [DOI: 10.1063/5.0031173] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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27
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Braver Y, Valkunas L, Gelzinis A. Benchmarking the forward–backward trajectory solution of the quantum-classical Liouville equation. J Chem Phys 2020; 152:214116. [DOI: 10.1063/5.0006538] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Yakov Braver
- Institute of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio Ave. 9-III, LT-10222 Vilnius, Lithuania
- Department of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio Ave. 3, LT-10257 Vilnius, Lithuania
| | - Leonas Valkunas
- Institute of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio Ave. 9-III, LT-10222 Vilnius, Lithuania
- Department of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio Ave. 3, LT-10257 Vilnius, Lithuania
| | - Andrius Gelzinis
- Institute of Chemical Physics, Faculty of Physics, Vilnius University, Saulėtekio Ave. 9-III, LT-10222 Vilnius, Lithuania
- Department of Molecular Compound Physics, Center for Physical Sciences and Technology, Saulėtekio Ave. 3, LT-10257 Vilnius, Lithuania
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28
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Runeson JE, Richardson JO. Generalized spin mapping for quantum-classical dynamics. J Chem Phys 2020; 152:084110. [DOI: 10.1063/1.5143412] [Citation(s) in RCA: 27] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/24/2023] Open
Affiliation(s)
- Johan E. Runeson
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
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29
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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: 213] [Impact Index Per Article: 53.3] [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.
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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
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30
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Runeson JE, Richardson JO. Spin-mapping approach for nonadiabatic molecular dynamics. J Chem Phys 2019; 151:044119. [DOI: 10.1063/1.5100506] [Citation(s) in RCA: 33] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Johan E. Runeson
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
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31
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Chowdhury SN, Huo P. State dependent ring polymer molecular dynamics for investigating excited nonadiabatic dynamics. J Chem Phys 2019; 150:244102. [DOI: 10.1063/1.5096276] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sutirtha N. Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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32
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Bose A, Makri N. Coherent State-Based Path Integral Methodology for Computing the Wigner Phase Space Distribution. J Phys Chem A 2019; 123:4284-4294. [DOI: 10.1021/acs.jpca.9b00758] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Amartya Bose
- Department of Chemistry, University of Illinois, 505 S. Mathews Avenue, Urbana, Illinois 61801, United States
| | - Nancy Makri
- Department of Chemistry, University of Illinois, 505 S. Mathews Avenue, Urbana, Illinois 61801, United States
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33
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Carli M, Turelli M, Faccioli P. Microscopic calculation of absorption spectra of macromolecules: An analytic approach. J Chem Phys 2019; 150:144103. [PMID: 30981270 DOI: 10.1063/1.5084120] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We develop a cross-disciplinary approach to analytically compute optical response functions of open macromolecular systems by exploiting the mathematical formalism of quantum field theory (QFT). Indeed, the entries of the density matrix for the electronic excitations interacting with their open dissipative environment are mapped into vacuum-to-vacuum Green's functions in a fictitious relativistic closed quantum system. We show that by re-summing appropriate self-energy diagrams in this dual QFT, it is possible to obtain analytic expressions for the response functions in Mukamel's theory. This yields physical insight into the structure and dynamics of vibronic resonances, since their frequency and width is related to fundamental physical constants and microscopic model parameters. For illustration, we apply this scheme to compute the linear absorption spectrum of the Fenna-Matthews-Olson light harvesting complex, comparing analytic calculations, numerical Monte Carlo simulations, and experimental data.
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Affiliation(s)
- Matteo Carli
- Physics Department of Trento University, Via Sommarive 14, Povo, Trento 38123, Italy
| | - Michele Turelli
- Physics Department of Trento University, Via Sommarive 14, Povo, Trento 38123, Italy
| | - Pietro Faccioli
- Physics Department of Trento University, Via Sommarive 14, Povo, Trento 38123, Italy
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34
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Thapa MJ, Fang W, Richardson JO. Nonadiabatic quantum transition-state theory in the golden-rule limit. I. Theory and application to model systems. J Chem Phys 2019; 150:104107. [DOI: 10.1063/1.5081108] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Manish J. Thapa
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
| | - Wei Fang
- Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
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35
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Saller MAC, Kelly A, Richardson JO. On the identity of the identity operator in nonadiabatic linearized semiclassical dynamics. J Chem Phys 2019; 150:071101. [DOI: 10.1063/1.5082596] [Citation(s) in RCA: 37] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
| | - Aaron Kelly
- Department of Chemistry, Dalhousie University, Halifax, Nova Scotia B3H 4R2, Canada
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36
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Liu J, Hsieh CY, Segal D, Hanna G. Heat transfer statistics in mixed quantum-classical systems. J Chem Phys 2018; 149:224104. [PMID: 30553258 DOI: 10.1063/1.5066025] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023] Open
Abstract
The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronic devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions related to the bath time scale and subsystem-bath coupling strength. In particular, we adopt the full counting statistics approach to derive a general expression for the moment generating function of heat in systems whose dynamics are described by the quantum-classical Liouville equation (QCLE). From this expression, one can deduce expressions for the dynamics of the average heat and heat current, which may be evaluated using numerical simulations. Due to the approximate nature of the QCLE, we also find that the steady state fluctuation symmetry holds up to order ℏ for systems whose subsystem-bath couplings and baths go beyond bilinear and harmonic, respectively. To demonstrate the approach, we consider the nonequilibrium spin boson model and simulate its time-dependent average heat and heat current under various conditions.
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Affiliation(s)
- Junjie Liu
- Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada
| | - Chang-Yu Hsieh
- Singapore-MIT Alliance for Research and Technology (SMART) Center, 1 Create Way, Singapore 138602, Singapore
| | - Dvira Segal
- Department of Chemistry and Centre for Quantum Information and Quantum Control, University of Toronto, 80 Saint George St., Toronto, Ontario M5S 3H6, Canada
| | - Gabriel Hanna
- Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada
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37
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Schile AJ, Limmer DT. Studying rare nonadiabatic dynamics with transition path sampling quantum jump trajectories. J Chem Phys 2018; 149:214109. [DOI: 10.1063/1.5058281] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Addison J. Schile
- Department of Chemistry, University of California, Berkeley, California 94618, USA
- Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94618, USA
| | - David T. Limmer
- Department of Chemistry, University of California, Berkeley, California 94618, USA
- Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94618, USA
- Kavli Energy NanoSciences Institute, University of California, Berkeley, California 94618, USA
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38
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Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths. Symmetry (Basel) 2018. [DOI: 10.3390/sym10100518] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein.
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39
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Bose A, Makri N. Wigner Distribution by Adiabatic Switching in Normal Mode or Cartesian Coordinates and Molecular Applications. J Chem Theory Comput 2018; 14:5446-5458. [DOI: 10.1021/acs.jctc.8b00179] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Amartya Bose
- Department of Chemistry, University of Illinois, Urbana, Illinois 61801, United States
| | - Nancy Makri
- Department of Chemistry, University of Illinois, Urbana, Illinois 61801, United States
- Department of Physics, University of Illinois, Urbana, Illinois 61801, United States
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40
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Liu J, Hanna G. Efficient and Deterministic Propagation of Mixed Quantum-Classical Liouville Dynamics. J Phys Chem Lett 2018; 9:3928-3933. [PMID: 29947224 DOI: 10.1021/acs.jpclett.8b01619] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We propose a highly efficient mixed quantum-classical molecular dynamics scheme based on a solution of the quantum-classical Liouville equation (QCLE). By casting the equations of motion for the quantum subsystem and classical bath degrees of freedom onto an approximate set of coupled first-order differential equations for c-numbers, this scheme propagates the composite system in time deterministically in terms of independent classical-like trajectories. To demonstrate its performance, we apply the method to the spin-boson model, a photoinduced electron transfer model, and a Fenna-Matthews-Olsen complex model, and find excellent agreement out to long times with the numerically exact results, using several orders of magnitude fewer trajectories than surface-hopping solutions of the QCLE. Owing to its accuracy and efficiency, this method promises to be very useful for studying the dynamics of mixed quantum-classical systems.
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Affiliation(s)
- Junjie Liu
- Department of Chemistry , University of Alberta , Edmonton , Alberta T6G 2G2 , Canada
| | - Gabriel Hanna
- Department of Chemistry , University of Alberta , Edmonton , Alberta T6G 2G2 , Canada
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41
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Jain A, Subotnik JE. Vibrational Energy Relaxation: A Benchmark for Mixed Quantum–Classical Methods. J Phys Chem A 2017; 122:16-27. [DOI: 10.1021/acs.jpca.7b09018] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Amber Jain
- Department of Chemistry, University of Pennsylvania, 231 South 34th Street, Philadelphia, Pennsylvania 19104, United States
| | - Joseph E. Subotnik
- Department of Chemistry, University of Pennsylvania, 231 South 34th Street, Philadelphia, Pennsylvania 19104, United States
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42
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Chowdhury SN, Huo P. Coherent state mapping ring polymer molecular dynamics for non-adiabatic quantum propagations. J Chem Phys 2017; 147:214109. [DOI: 10.1063/1.4995616] [Citation(s) in RCA: 37] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Sutirtha N. Chowdhury
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, 120 Trustee Road, Rochester, New York 14627, USA
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43
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Ikeda T, Tanimura Y. Probing photoisomerization processes by means of multi-dimensional electronic spectroscopy: The multi-state quantum hierarchical Fokker-Planck equation approach. J Chem Phys 2017; 147:014102. [DOI: 10.1063/1.4989537] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Affiliation(s)
- Tatsushi Ikeda
- Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
| | - Yoshitaka Tanimura
- Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
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44
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Kar A, Franco I. Quantifying fermionic decoherence in many-body systems. J Chem Phys 2017; 146:214107. [PMID: 28595395 PMCID: PMC5648582 DOI: 10.1063/1.4984128] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2017] [Accepted: 05/13/2017] [Indexed: 01/01/2023] Open
Abstract
Practical measures of electronic decoherence, called distilled purities, that are applicable to many-body systems are introduced. While usual measures of electronic decoherence such as the purity employ the full N-particle density matrix which is generally unavailable, the distilled purities are based on the r-body reduced density matrices (r-RDMs) which are more accessible quantities. The r-body distilled purities are derivative quantities of the previously introduced r-body reduced purities [I. Franco and H. Appel, J. Chem. Phys. 139, 094109 (2013)] that measure the non-idempotency of the r-RDMs. Specifically, the distilled purities exploit the structure of the reduced purities to extract coherences between Slater determinants with integer occupations defined by a given single-particle basis that compose an electronic state. In this way, the distilled purities offer a practical platform to quantify coherences in a given basis that can be used to analyze the quantum dynamics of many-electron systems. Exact expressions for the one-body and two-body distilled purities are presented and the utility of the approach is exemplified via an analysis of the dynamics of oligo-acetylene as described by the Su-Schrieffer-Heeger Hamiltonian. Last, the advantages and limitations of the purity, reduced purity, and distilled purity as measures of electronic coherence are discussed.
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Affiliation(s)
- Arnab Kar
- Department of Chemistry, University of Rochester, Rochester, New York 14627, USA
| | - Ignacio Franco
- Department of Chemistry, University of Rochester, Rochester, New York 14627, USA
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45
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Ryabinkin IG, Izmaylov AF. Mixed Quantum-Classical Dynamics Using Collective Electronic Variables: A Better Alternative to Electronic Friction Theories. J Phys Chem Lett 2017; 8:440-444. [PMID: 28036176 DOI: 10.1021/acs.jpclett.6b02712] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
An accurate description of nonadiabatic dynamics of molecular species on metallic surfaces poses a serious computational challenge associated with a multitude of closely spaced electronic states. We propose a mixed quantum-classical scheme that addresses this challenge by introducing collective electronic variables. These variables are defined through analytic block-diagonalization applied to the time-dependent Hamiltonian matrix governing the electronic dynamics. We compare our scheme with a simplified Ehrenfest approach and with a full-memory electronic friction model on a 1D "adatom + atomic chain" model. Our simulations demonstrate that collective-mode dynamics with only a few (two to three) electronic variables is robust and can describe a variety of situations: from a chemisorbed atom on an insulator to an atom on a metallic surface. Our molecular model also reveals that the friction approach is prone to unpredictable and catastrophic failures.
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Affiliation(s)
- Ilya G Ryabinkin
- Department of Physical and Environmental Sciences, University of Toronto Scarborough , Toronto, Ontario M1C 1A4, Canada
| | - Artur F Izmaylov
- Chemical Physics Theory Group, Department of Chemistry, University of Toronto , Toronto, Ontario M5S 3H6, Canada
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46
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An analysis of nonadiabatic ring-polymer molecular dynamics and its application to vibronic spectra. Chem Phys 2017. [DOI: 10.1016/j.chemphys.2016.09.036] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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47
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Hirshberg B, Gerber RB. Mean-Field Methods for Time-Dependent Quantum Dynamics of Many-Atom Systems. ADVANCES IN QUANTUM CHEMISTRY 2017. [DOI: 10.1016/bs.aiq.2017.01.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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48
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Kananenka AA, Hsieh CY, Cao J, Geva E. Accurate Long-Time Mixed Quantum-Classical Liouville Dynamics via the Transfer Tensor Method. J Phys Chem Lett 2016; 7:4809-4814. [PMID: 27934045 DOI: 10.1021/acs.jpclett.6b02389] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
In this Letter, we combine the recently introduced transfer tensor method with the mixed quantum-classical Liouville method. The resulting protocol provides an accurate, general, flexible and robust new route for simulating the reduced dynamics of the quantum subsystem for arbitrarily long times, starting with computationally feasible short-time mixed quantum-classical Liouville dynamical maps. The accuracy and feasibility of the methodology are demonstrated on a spin-boson benchmark model.
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Affiliation(s)
- Alexei A Kananenka
- Department of Chemistry, University of Michigan , Ann Arbor, Michigan 48109, United States
| | - Chang-Yu Hsieh
- Department of Chemistry, Massachusetts Institute of Technology , Cambridge, Massachusetts 02319, United States
| | - Jianshu Cao
- Department of Chemistry, Massachusetts Institute of Technology , Cambridge, Massachusetts 02319, United States
| | - Eitan Geva
- Department of Chemistry, University of Michigan , Ann Arbor, Michigan 48109, United States
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49
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Walters PL, Allen TC, Makri N. Direct determination of discrete harmonic bath parameters from molecular dynamics simulations. J Comput Chem 2016; 38:110-115. [DOI: 10.1002/jcc.24527] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2016] [Revised: 09/07/2016] [Accepted: 10/02/2016] [Indexed: 12/12/2022]
Affiliation(s)
- Peter L. Walters
- Department of Chemistry600 S. Goodwin Avenue, University of IllinoisUrbana Illinois61801
| | - Thomas C. Allen
- Department of Chemistry600 S. Goodwin Avenue, University of IllinoisUrbana Illinois61801
| | - Nancy Makri
- Department of Chemistry600 S. Goodwin Avenue, University of IllinoisUrbana Illinois61801
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50
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Lee MK, Huo P, Coker DF. Semiclassical Path Integral Dynamics: Photosynthetic Energy Transfer with Realistic Environment Interactions. Annu Rev Phys Chem 2016; 67:639-68. [DOI: 10.1146/annurev-physchem-040215-112252] [Citation(s) in RCA: 96] [Impact Index Per Article: 12.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
Affiliation(s)
- Mi Kyung Lee
- Department of Chemistry, Boston University, Boston, Massachusetts 02215;
| | - Pengfei Huo
- Department of Chemistry, University of Rochester, Rochester, New York 14627;
| | - David F. Coker
- Department of Chemistry, Boston University, Boston, Massachusetts 02215;
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