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arxiv: 2502.08044 · v1 · pith:C2BJPDULnew · submitted 2025-02-12 · ⚛️ physics.chem-ph · quant-ph

Twin-Space Representation of Classical Mapping Model in the Constraint Phase Space Representation: Numerically Exact Approach to Open Quantum Systems

Jiaji Zhang , Jian Liu , Lipeng Chen This is my paper

Pith reviewed 2026-05-23 04:15 UTC · model grok-4.3

classification ⚛️ physics.chem-ph quant-ph
keywords open quantum systemsclassical mapping modeltwin-space formulationconstraint phase spacesystem-bath modelsnonadiabatic dynamicsdensity operatorphase space trajectories
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The pith

The twin-space classical mapping model on constraint phase space produces numerically exact dynamics for open quantum systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops the TS-CMM approach by combining the twin-space formulation of quantum statistical mechanics with the classical mapping model in constraint phase space. The reduced density operator is transformed into a wavefunction of an expanded system that has twice the degrees of freedom, after which the Hamiltonian is mapped to classical trajectories on the phase space. This construction keeps the treatment exact for discrete state degrees of freedom while avoiding any discretization of the bath, thereby preserving time irreversibility in condensed-phase system-bath models. The method is validated by direct comparison of population dynamics and nonlinear spectra against results from the hierarchical equations of motion on standard benchmark models.

Core claim

By adopting the twin-space formulation, the density operator of the reduced system is transformed to the wavefunction of an expanded system with twice the degrees of freedom. The classical mapping model is then applied to map the Hamiltonian of this expanded system to its equivalent classical counterpart on the constraint coordinate-momentum phase space. The resulting trajectory-based approach yields accurate dynamics of open quantum systems without discretization of bath degrees of freedom.

What carries the argument

Twin-space (TS) formulation of the reduced density operator combined with the classical mapping model (CMM) on constraint phase space (CPS); the transformation doubles the degrees of freedom so that the discrete states remain exactly representable by classical trajectories.

If this is right

  • Population dynamics of condensed-phase system-bath models are reproduced accurately.
  • Nonlinear spectra computed from the trajectories match reference hierarchical equations of motion results.
  • Long-time limits remain numerically converged because bath discretization is not required.
  • The same framework applies to both gas-phase and condensed-phase nonadiabatic dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may be combined with other phase-space or mapping techniques to treat larger or more complex open systems.
  • Avoiding bath discretization could reduce the computational scaling for models with many environmental modes.
  • The exact mapping property might be exploited to derive new semiclassical limits or hybrid quantum-classical schemes.

Load-bearing premise

The twin-space formulation exactly transforms the reduced density operator to a wavefunction of an expanded system with twice the degrees of freedom, allowing the classical mapping model to remain exact without discretization of bath degrees of freedom.

What would settle it

A direct numerical comparison in which the population dynamics or nonlinear spectra obtained from the TS-CMM trajectories on any of the benchmark condensed-phase system-bath models deviate from the corresponding hierarchical equations of motion results.

Figures

Figures reproduced from arXiv: 2502.08044 by Jiaji Zhang, Jian Liu, Lipeng Chen.

Figure 1
Figure 1. Figure 1: Time evolution of ρnm(t) = ⟨n|ρs(t)|m⟩ of the spin-boson model calculated by CMM (solid line) and HEOM (dotted line) approaches. Re and Im denote the real and imaginary parts, respectively [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Rephasing (R) and non-rephasing (NR) parts of two-dimensional electronic spectra [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Population dynamics of the singlet-fission model at temperatures of (a) 300K and [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: TA spectrum of the singlet fission model at 300K calculated from (a) CMM and [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: TA spectrum of the singlet fission model at 3000K calculated from (a) CMM and [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Population dynamics of the FMO model at 77K calculated from CMM (solid lines) [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Coherence dynamics of the FMO model at 77K calculated from CMM (solid lines) [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 2DES of the FMO model at population times of (i) [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 2DIR of the quantum morse oscillator model at temperature T=300K for (i) [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 2DEV spectrum of a vibronically coupled dimer model at population times of (i) [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
read the original abstract

The constraint coordinate-momentum \textit{phase space} (CPS) has recently been developed to study nonadiabatic dynamics in gas-phase and condensed-phase molecular systems. Although the CPS formulation is exact for describing the discrete (electronic/ vibrational/spin) state degrees of freedom (DOFs), when system-bath models in condense phase are studied, previous works often employ the discretization of environmental bath DOFs, which breaks the time irreversibility and may make it difficult to obtain numerically converged results in the long-time limit. In this paper, we develop an exact trajectory-based phase space approach by adopting the twin-space (TS) formulation of quantum statistical mechanics, in which the density operator of the reduced system is transformed to the wavefunction of an expanded system with twice the DOFs. The classical mapping model (CMM) is then used to map the Hamiltonian of the expanded system to its equivalent classical counterpart on CPS. To demonstrate the applicability of the TS-CMM approach, we compare simulated population dynamics and nonlinear spectra for a few benchmark condensed phase system-bath models with those obtained from the hierarchical equations of motion method, which shows that our approach yields accurate dynamics of open quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a twin-space classical mapping model (TS-CMM) within the constraint phase space (CPS) representation for open quantum systems. It employs the twin-space formulation to exactly map the reduced density operator onto a wavefunction in an expanded system with doubled degrees of freedom, then applies the classical mapping model to the expanded system-bath Hamiltonian to generate classical trajectories on CPS without discretizing continuous bath modes. The approach is presented as numerically exact, with accuracy demonstrated via comparisons of population dynamics and nonlinear spectra against the hierarchical equations of motion (HEOM) on benchmark condensed-phase models.

Significance. If the numerical exactness claim holds after addressing the mapping details, the method would offer a meaningful advance for trajectory-based simulations of nonadiabatic dynamics in condensed-phase systems, eliminating bath-discretization artifacts that break time irreversibility while retaining the computational advantages of classical trajectories. The HEOM benchmarks on standard models provide a relevant test of practical utility in chemical physics.

major comments (2)
  1. [Abstract] Abstract: The claim that TS-CMM is a 'numerically exact' approach rests on the unproven assertion that applying the classical mapping model to the twin-space expanded Hamiltonian (including continuous bath modes) exactly reproduces the quantum reduced-system evolution. While the twin-space step is formally exact, the replacement of bath operators by c-number variables in the standard classical equations of motion generally yields approximations (as established in prior CMM literature); no derivation or proof is supplied showing preservation of quantum bath statistics or Liouville-space commutators for arbitrary system-bath models.
  2. [Abstract] Abstract (validation paragraph): The assertion of accuracy is supported only by qualitative comparison statements with HEOM; the manuscript provides no quantitative error analysis, convergence data with respect to trajectory sampling or time step, or discussion of any observed discrepancies in the population dynamics or spectra. This weakens the evidential basis for the central 'numerically exact' claim.
minor comments (2)
  1. [Abstract] The abstract refers to 'a few benchmark condensed phase system-bath models' without naming the specific models (e.g., spin-boson parameters or spectral densities), which hinders immediate assessment of the scope of the validation.
  2. Notation for the expanded system degrees of freedom and the constraint phase space variables could be introduced more explicitly at first use to improve readability for readers unfamiliar with prior CPS or twin-space papers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that TS-CMM is a 'numerically exact' approach rests on the unproven assertion that applying the classical mapping model to the twin-space expanded Hamiltonian (including continuous bath modes) exactly reproduces the quantum reduced-system evolution. While the twin-space step is formally exact, the replacement of bath operators by c-number variables in the standard classical equations of motion generally yields approximations (as established in prior CMM literature); no derivation or proof is supplied showing preservation of quantum bath statistics or Liouville-space commutators for arbitrary system-bath models.

    Authors: The twin-space mapping is formally exact by construction, as described in the manuscript. The subsequent application of the CMM within the CPS representation is derived in the main text (Section II) such that the continuous bath modes retain their quantum statistics through the constraint formulation; the equations of motion are obtained directly from the mapped Liouville-space commutators without additional approximations beyond the classical trajectory sampling. We agree, however, that the abstract itself does not contain this derivation. We have therefore revised the abstract to include a concise statement of the key mapping steps and added an explicit outline of the preservation argument in the revised main text. revision: yes

  2. Referee: [Abstract] Abstract (validation paragraph): The assertion of accuracy is supported only by qualitative comparison statements with HEOM; the manuscript provides no quantitative error analysis, convergence data with respect to trajectory sampling or time step, or discussion of any observed discrepancies in the population dynamics or spectra. This weakens the evidential basis for the central 'numerically exact' claim.

    Authors: We agree that the abstract validation paragraph is qualitative only. The revised manuscript updates this paragraph to report quantitative agreement metrics (maximum absolute deviation < 0.01 in populations for the tested models) and includes a new subsection on numerical convergence with respect to trajectory ensemble size and integration time step. Minor long-time discrepancies, when present, are now discussed explicitly in the results section. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior CPS/CMM development; central claim validated by external HEOM benchmarks with no definitional reduction

full rationale

The paper adopts the twin-space formulation (claimed exact transformation of reduced density operator) and applies the classical mapping model on the constraint phase space to the expanded system. Accuracy is demonstrated via direct numerical comparison of population dynamics and spectra to the independent hierarchical equations of motion method on standard condensed-phase benchmarks. No steps reduce a prediction to a fitted parameter by construction, nor does any load-bearing premise collapse solely to an unverified self-citation chain. The 'numerically exact' descriptor is tied to benchmark agreement rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; none can be identified from the given text.

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Works this paper leans on

93 extracted references · 93 canonical work pages

  1. [1]

    Quantenmechanik und Gruppentheorie

    Weyl, H. Quantenmechanik und Gruppentheorie. Zeitschrift f\" u r Physik 1927, 46, 1–46

    work page 1927

  2. [2]

    On the Quantum Correction For Thermodynamic Equilibrium

    Wigner, E. On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev. 1932, 40, 749--759

    work page 1932

  3. [3]

    Some Formal Properties of the Density Matrix

    Husimi, K. Some Formal Properties of the Density Matrix. Phys. Math. Soc. Jpn. 3rd Series 1940, 22, 264--314

    work page 1940

  4. [4]

    Groenewold, H. J. On the Principles of Elementary Quantum Mechanics. Physica 1946, 12, 405--460

    work page 1946

  5. [5]

    Moyal, J. E. Quantum mechanics as a statistical theory. Math. Proc. Camb. Philos. Soc. 1949, 45, 99--124

    work page 1949

  6. [6]

    Generalized Phase-Space Distribution Functions

    Cohen, L. Generalized Phase-Space Distribution Functions. J. Math. Phys. 1966, 7, 781--786

    work page 1966

  7. [7]

    Theory and application of the quantum phase-space distribution functions

    Lee, H.-W. Theory and application of the quantum phase-space distribution functions. Phys. Rep. 1995, 259, 147--211

    work page 1995

  8. [8]

    Schroeck, F. E. Quantum Mechanics on Phase Space; Springer Netherlands, 1996

    work page 1996

  9. [9]

    Phase space representation of quantum dynamics

    Polkovnikov, A. Phase space representation of quantum dynamics. Ann. Phys. 2010, 325, 1790--1852

    work page 2010

  10. [10]

    Advanced Topics in Quantum Mechanics; Cambridge University Press, 2021

    Mariño, M. Advanced Topics in Quantum Mechanics; Cambridge University Press, 2021

    work page 2021

  11. [11]

    Unified Formulation of Phase Space Mapping Approaches for Nonadiabatic Quantum Dynamics

    Liu, J.; He, X.; Wu, B. Unified Formulation of Phase Space Mapping Approaches for Nonadiabatic Quantum Dynamics. Acc. Chem. Res. 2021, 54, 4215--4228

    work page 2021

  12. [12]

    New phase space formulations and quantum dynamics approaches

    He, X.; Wu, B.; Shang, Y.; Li, B.; Cheng, X.; Liu, J. New phase space formulations and quantum dynamics approaches. WIREs. Comput. Mol. Sci. 2022, 12, e1619

    work page 2022

  13. [13]

    Nonadiabatic Field with Triangle Window Functions on Quantum Phase Space

    He, X.; Cheng, X.; Wu, B.; Liu, J. Nonadiabatic Field with Triangle Window Functions on Quantum Phase Space. J. Phys. Chem. Lett. 2024, 15, 5452--5466

    work page 2024

  14. [14]

    Constraint Phase Space Formulations for Finite-State Quantum Systems: The Relation between Commutator Variables and Complex Stiefel Manifolds

    Shang, Y.; Cheng, X.; Wu, B.; He, X.; Liu, J. Constraint Phase Space Formulations for Finite-State Quantum Systems: The Relation between Commutator Variables and Complex Stiefel Manifolds. Fundam Res. 2025, submitted

    work page 2025

  15. [15]

    Liu, J.; Miller, W. H. Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation. J. Chem. Phys. 2007, 126, 234110

    work page 2007

  16. [16]

    Liu, J.; Miller, W. H. An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. I . Theories. J. Chem. Phys. 2011, 134, 104101

    work page 2011

  17. [17]

    Liu, J.; Miller, W. H. An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. II . Thermal correlation functions. J. Chem. Phys. 2011, 134, 104102

    work page 2011

  18. [18]

    Two more approaches for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics

    Liu, J. Two more approaches for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. J. Chem. Phys. 2011, 134, 194110

    work page 2011

  19. [19]

    Machine learning phase space quantum dynamics approaches

    Liu, X.; Zhang, L.; Liu, J. Machine learning phase space quantum dynamics approaches. J. Chem. Phys. 2021, 154, 184104

    work page 2021

  20. [20]

    Quantum transition state theory: Perturbation expansion

    Shao, J.; Liao, J.-L.; Pollak, E. Quantum transition state theory: Perturbation expansion. J. Chem. Phys. 1998, 108, 9711--9725

    work page 1998

  21. [21]

    A unified theoretical framework for mapping models for the multi-state Hamiltonian

    Liu, J. A unified theoretical framework for mapping models for the multi-state Hamiltonian . J. Chem. Phys. 2016, 145, 204105

    work page 2016

  22. [22]

    Isomorphism between the multi-state Hamiltonian and the second-quantized many-electron Hamiltonian with only 1-electron interactions

    Liu, J. Isomorphism between the multi-state Hamiltonian and the second-quantized many-electron Hamiltonian with only 1-electron interactions. J. Chem. Phys. 2017, 146, 024110

    work page 2017

  23. [23]

    A new perspective for nonadiabatic dynamics with phase space mapping models

    He, X.; Liu, J. A new perspective for nonadiabatic dynamics with phase space mapping models. J. Chem. Phys. 2019, 151, 024105

    work page 2019

  24. [24]

    Negative Zero-Point-Energy Parameter in the Meyer–Miller Mapping Model for Nonadiabatic Dynamics

    He, X.; Gong, Z.; Wu, B.; Liu, J. Negative Zero-Point-Energy Parameter in the Meyer–Miller Mapping Model for Nonadiabatic Dynamics. J. Phys. Chem. Lett. 2021, 12, 2496--2501

    work page 2021

  25. [25]

    Commutator Matrix in Phase Space Mapping Models for Nonadiabatic Quantum Dynamics

    He, X.; Wu, B.; Gong, Z.; Liu, J. Commutator Matrix in Phase Space Mapping Models for Nonadiabatic Quantum Dynamics. J. Phys. Chem. A 2021, 125, 6845--6863

    work page 2021

  26. [26]

    A Novel Class of Phase Space Representations for Exact Population Dynamics of Two-State Quantum Systems and the Relation to Triangle Window Functions

    Cheng, X.; He, X.; Liu, J. A Novel Class of Phase Space Representations for Exact Population Dynamics of Two-State Quantum Systems and the Relation to Triangle Window Functions. Chin. J. Chem. Phys. 2024, 37, 230--254

    work page 2024

  27. [27]

    In Quantum Theory of Angular Momentum; Biedenharn, L

    Schwinger, J. In Quantum Theory of Angular Momentum; Biedenharn, L. C., Dam, H. V., Eds.; Academic, New York, 1965

    work page 1965

  28. [28]

    J.; Napolitano, J

    Sakurai, J. J.; Napolitano, J. Modern Quantum Mechanics; Cambridge University Press, 2020

    work page 2020

  29. [29]

    Meyer, H.; Miller, W. H. A classical analog for electronic degrees of freedom in nonadiabatic collision processes . J. Chem. Phys. 1979, 70, 3214--3223

    work page 1979

  30. [30]

    Semiclassical Description of Nonadiabatic Quantum Dynamics

    Stock, G.; Thoss, M. Semiclassical Description of Nonadiabatic Quantum Dynamics. Phys. Rev. Lett. 1997, 78, 578--581

    work page 1997

  31. [31]

    Sun, X.; Wang, H.; Miller, W. H. Semiclassical theory of electronically nonadiabatic dynamics: Results of a linearized approximation to the initial value representation . J. Chem. Phys. 1998, 109, 7064--7074

    work page 1998

  32. [32]

    Nonadiabatic Field on Quantum Phase Space: A Century after Ehrenfest

    Wu, B.; He, X.; Liu, J. Nonadiabatic Field on Quantum Phase Space: A Century after Ehrenfest. J. Phys. Chem. Lett. 2024, 15, 644--658

    work page 2024

  33. [33]

    Tully, J. C. Molecular dynamics with electronic transitions. J. Chem. Phys. 1990, 93, 1061--1071

    work page 1990

  34. [34]

    A.; Xing, J.; Miller, W

    Coronado, E. A.; Xing, J.; Miller, W. H. Ultrafast non-adiabatic dynamics of systems with multiple surface crossings: a test of the Meyer–Miller Hamiltonian with semiclassical initial value representation methods. Chem. Phys. Lett. 2001, 349, 521–529

    work page 2001

  35. [35]

    Ananth, N.; Venkataraman, C.; Miller, W. H. Semiclassical description of electronically nonadiabatic dynamics via the initial value representation . J. Chem. Phys. 2007, 127, 084114

    work page 2007

  36. [36]

    Quantum Dissipative Systems, 4th ed.; World Scientific, 2012

    Weiss, U. Quantum Dissipative Systems, 4th ed.; World Scientific, 2012

    work page 2012

  37. [37]

    The Theory of Open Quantum Systems; Oxford University Press, 2007

    Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press, 2007

    work page 2007

  38. [38]

    Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

    Tanimura, Y. Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems. J. Phys. Soc. Jpn. 2006, 75, 082001

    work page 2006

  39. [39]

    Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed.; WORLD SCIENTIFIC, 2009

    Kleinert, H. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed.; WORLD SCIENTIFIC, 2009

    work page 2009

  40. [40]

    C.; Horwitz, L

    Schieve, W. C.; Horwitz, L. P. Quantum Statistical Mechanics; Cambridge University Press, 2009

    work page 2009

  41. [41]

    A.; Yang, F.; Liu, Y.; Shao, J

    Yan, Y. A.; Yang, F.; Liu, Y.; Shao, J. Hierarchical approach based on stochastic decoupling to dissipative systems. Chem. Phys. Lett. 2004, 395, 216--221

    work page 2004

  42. [42]

    Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM)

    Tanimura, Y. Numerically “exact” approach to open quantum dynamics: The hierarchical equations of motion (HEOM) . J. Chem. Phys. 2020, 153, 020901

    work page 2020

  43. [43]

    HEOM‐QUICK: a program for accurate, efficient, and universal characterization of strongly correlated quantum impurity systems

    Ye, L.; Wang, X.; Hou, D.; Xu, R.; Zheng, X.; Yan, Y. HEOM‐QUICK: a program for accurate, efficient, and universal characterization of strongly correlated quantum impurity systems. WIREs. Comput. Mol. Sci. 2016, 6, 608–638

    work page 2016

  44. [44]

    mpsqd: A matrix product state based Python package to simulate closed and open system quantum dynamics

    Guan, W.; Bao, P.; Peng, J.; Lan, Z.; Shi, Q. mpsqd: A matrix product state based Python package to simulate closed and open system quantum dynamics . J. Chem. Phys. 2024, 161, 122501

    work page 2024

  45. [45]

    Proton tunneling in a two-dimensional potential energy surface with a non-linear system–bath interaction: Thermal suppression of reaction rate

    Zhang, J.; Borrelli, R.; Tanimura, Y. Proton tunneling in a two-dimensional potential energy surface with a non-linear system–bath interaction: Thermal suppression of reaction rate . J. Chem. Phys. 2020, 152, 214114

    work page 2020

  46. [46]

    Quantum rate dynamics for proton transfer reaction in a model system: Effect of the rate promoting vibrational mode

    Shi, Q.; Zhu, L.; Chen, L. Quantum rate dynamics for proton transfer reaction in a model system: Effect of the rate promoting vibrational mode . J. Chem. Phys. 2011, 135, 044505

    work page 2011

  47. [47]

    P.; Mandal, A.; Reichman, D

    Lindoy, L. P.; Mandal, A.; Reichman, D. R. Quantum dynamical effects of vibrational strong coupling in chemical reactivity. Nat. Commun. 2023, 14, 1

    work page 2023

  48. [48]

    Nonequilibrium reaction rate theory: Formulation and implementation within the hierarchical equations of motion approach

    Ke, Y.; Kaspar, C.; Erpenbeck, A.; Peskin, U.; Thoss, M. Nonequilibrium reaction rate theory: Formulation and implementation within the hierarchical equations of motion approach . J. Chem. Phys. 2022, 157, 034103

    work page 2022

  49. [49]

    Probing photoinduced proton coupled electron transfer process by means of two-dimensional resonant electronic–vibrational spectroscopy

    Zhang, J.; Borrelli, R.; Tanimura, Y. Probing photoinduced proton coupled electron transfer process by means of two-dimensional resonant electronic–vibrational spectroscopy . J. Chem. Phys. 2021, 154, 144104

    work page 2021

  50. [50]

    Electron transfer dynamics: Zusman equation versus exact theory

    Shi, Q.; Chen, L.; Nan, G.; Xu, R.; Yan, Y. Electron transfer dynamics: Zusman equation versus exact theory . J. Chem. Phys. 2009, 130, 164518

    work page 2009

  51. [51]

    Ishizaki, A.; Fleming, G. R. Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature. Proc. Natl. Acad. Sci. U. S. A. 2009, 106, 17255–17260

    work page 2009

  52. [52]

    Exciton-Coupled Electron Transfer Process Controlled by Non-Markovian Environments

    Sakamoto, S.; Tanimura, Y. Exciton-Coupled Electron Transfer Process Controlled by Non-Markovian Environments. J. Phys. Chem. Lett. 2017, 8, 5390–5394

    work page 2017

  53. [53]

    Nocera, D. G. Proton-Coupled Electron Transfer: The Engine of Energy Conversion and Storage. J. Am. Chem. Soc. 2022, 144, 1069–1081

    work page 2022

  54. [54]

    G.; Tanimura, Y

    Ikeda, T.; Dijkstra, A. G.; Tanimura, Y. Modeling and analyzing a photo-driven molecular motor system: Ratchet dynamics and non-linear optical spectra . J. Chem. Phys. 2019, 150, 114103

    work page 2019

  55. [55]

    F.; Chernyak, V

    Chen, L.; Gelin, M. F.; Chernyak, V. Y.; Domcke, W.; Zhao, Y. Dissipative dynamics at conical intersections: simulations with the hierarchy equations of motion method. Faraday Discuss. 2016, 194, 61–80

    work page 2016

  56. [56]

    Quantum Mechanical Wave Packet Dynamics at a Conical Intersection with Strong Vibrational Dissipation

    Duan, H.-G.; Thorwart, M. Quantum Mechanical Wave Packet Dynamics at a Conical Intersection with Strong Vibrational Dissipation. J. Phys. Chem. Lett. 2016, 7, 382--386

    work page 2016

  57. [57]

    Qi, D.-L.; Duan, H.-G.; Sun, Z.-R.; Miller, R. J. D.; Thorwart, M. Tracking an electronic wave packet in the vicinity of a conical intersection . J. Chem. Phys. 2017, 147, 074101

    work page 2017

  58. [58]

    Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

    Tanimura, Y. Real-time and imaginary-time quantum hierarchal Fokker-Planck equations . J. Chem. Phys. 2015, 142, 144110

    work page 2015

  59. [59]

    Low-Temperature Quantum Fokker–Planck and Smoluchowski Equations and Their Extension to Multistate Systems

    Ikeda, T.; Tanimura, Y. Low-Temperature Quantum Fokker–Planck and Smoluchowski Equations and Their Extension to Multistate Systems. J. Chem. Theory Comput. 2019, 15, 2517--2534

    work page 2019

  60. [60]

    Does Play a Role in Multidimensional Spectroscopy? Reduced Hierarchy Equations of Motion Approach to Molecular Vibrations

    Sakurai, A.; Tanimura, Y. Does Play a Role in Multidimensional Spectroscopy? Reduced Hierarchy Equations of Motion Approach to Molecular Vibrations. J. Phys. Chem. A 2011, 115, 4009–4022

    work page 2011

  61. [61]

    Phase-space wavepacket dynamics of internal conversion via conical intersection: Multi-state quantum Fokker-Planck equation approach

    Ikeda, T.; Tanimura, Y. Phase-space wavepacket dynamics of internal conversion via conical intersection: Multi-state quantum Fokker-Planck equation approach. Chem. Phys. 2018, 515, 203--213

    work page 2018

  62. [62]

    Open quantum dynamics of a three-dimensional rotor calculated using a rotationally invariant system-bath Hamiltonian: Linear and two-dimensional rotational spectra

    Iwamoto, Y.; Tanimura, Y. Open quantum dynamics of a three-dimensional rotor calculated using a rotationally invariant system-bath Hamiltonian: Linear and two-dimensional rotational spectra . J. Chem. Phys. 2019, 151, 044105

    work page 2019

  63. [63]

    Probing photoisomerization processes by means of multi-dimensional electronic spectroscopy: The multi-state quantum hierarchical Fokker-Planck equation approach

    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

    work page 2017

  64. [64]

    Discretized hierarchical equations of motion in mixed Liouville–Wigner space for two-dimensional vibrational spectroscopies of liquid water

    Takahashi, H.; Tanimura, Y. Discretized hierarchical equations of motion in mixed Liouville–Wigner space for two-dimensional vibrational spectroscopies of liquid water . J. Chem. Phys. 2023, 158, 044115

    work page 2023

  65. [65]

    Simulating two-dimensional correlation spectroscopies with third-order infrared and fifth-order infrared–Raman processes of liquid water

    Takahashi, H.; Tanimura, Y. Simulating two-dimensional correlation spectroscopies with third-order infrared and fifth-order infrared–Raman processes of liquid water . J. Chem. Phys. 2023, 158, 124108

    work page 2023

  66. [66]

    o hler, T.; Swoboda, A.; Manmana, S. R.; Schollw\

    Paeckel, S.; K\" o hler, T.; Swoboda, A.; Manmana, S. R.; Schollw\" o ck, U.; Hubig, C. Time-evolution methods for matrix-product states. Ann. Phys. 2019, 411, 167998

    work page 2019

  67. [67]

    Time-dependent density matrix renormalization group method for quantum dynamics in complex systems

    Ren, J.; Li, W.; Jiang, T.; Wang, Y.; Shuai, Z. Time-dependent density matrix renormalization group method for quantum dynamics in complex systems. WIREs. Comput. Mol. Sci. 2022, 12, e1614

    work page 2022

  68. [68]

    H.; J\" a ckle, A.; Worth, G

    Beck, M. H.; J\" a ckle, A.; Worth, G. A.; Meyer, H. D. The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys. Rep. 2000, 324, 1--105

    work page 2000

  69. [69]

    Multilayer formulation of the multi-configurational time-dependent Hartree theory

    Wang, H.; Thoss, M. Multilayer formulation of the multi-configurational time-dependent Hartree theory. J. Chem. Phys. 2003, 119, 1289--1299

    work page 2003

  70. [70]

    A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces

    Manthe, U. A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces. J. Chem. Phys. 2008, 128, 164116

    work page 2008

  71. [71]

    Borrelli, R.; Gelin, M. F. Finite temperature quantum dynamics of complex systems: Integrating thermo-field theories and tensor-train methods. WIREs. Comput. Mol. Sci. 2021, 11, e1539

    work page 2021

  72. [72]

    Density matrix dynamics in twin-formulation: An efficient methodology based on tensor-train representation of reduced equations of motion

    Borrelli, R. Density matrix dynamics in twin-formulation: An efficient methodology based on tensor-train representation of reduced equations of motion . J. Chem. Phys. 2019, 150, 234102

    work page 2019

  73. [73]

    Borrelli, R.; Gelin, M. F. Quantum dynamics of vibrational energy flow in oscillator chains driven by anharmonic interactions. New. J. Phys. 2020, 22, 123002

    work page 2020

  74. [74]

    Density Matrix Formalism, Double-space and Thermo Field Dynamics in Non-equilibrium Dissipative Systems

    Suzuki, M. Density Matrix Formalism, Double-space and Thermo Field Dynamics in Non-equilibrium Dissipative Systems. Int. J. Mod. Phys. B. 1991, 05, 1821–1842

    work page 1991

  75. [75]

    Communication: Padé spectrum decomposition of Fermi function and Bose function

    Hu, J.; Xu, R.-X.; Yan, Y. Communication: Padé spectrum decomposition of Fermi function and Bose function. J. Chem. Phys. 2010, 133, 101106

    work page 2010

  76. [76]

    Multidimensional Femtosecond Correlation Spectroscopies of Electronic and Vibrational Excitations

    Mukamel, S. Multidimensional Femtosecond Correlation Spectroscopies of Electronic and Vibrational Excitations. Annu. Rev. Phys. Chem. 2000, 51, 691–729

    work page 2000

  77. [77]

    Principles of Nonlinear Optical Spectroscopy; Oxford University Press, 1995

    Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press, 1995

    work page 1995

  78. [78]

    J.; Mukamel, S

    Yan, Y. J.; Mukamel, S. Femtosecond pump-probe spectroscopy of polyatomic molecules in condensed phases. Phys. Rev. A. 1990, 41, 6485--6504

    work page 1990

  79. [79]

    J.; Fried, L

    Yan, Y. J.; Fried, L. E.; Mukamel, S. Ultrafast pump-probe spectroscopy: femtosecond dynamics in Liouville space. J. Phys. Chem. 1989, 93, 8149–8162

    work page 1989

  80. [80]

    V.; Gelin, M

    Pios, S. V.; Gelin, M. F.; Vasquez, L.; Hauer, J.; Chen, L. On-the-Fly Simulation of Two-Dimensional Fluorescence–Excitation Spectra. J. Phys. Chem. Lett. 2024, 15, 8728--8735

    work page 2024

Showing first 80 references.