Multistate electron transfer dynamics in the condensed phase: Exact calculations from the reduced hierarchy equations of motion approach
Abstract
Multiple displaced oscillators coupled to an Ohmic heat bath are used to describe electron transfer (ET) in a dissipative environment. By performing a canonical transformation, the model is reduced to a multilevel system coupled to a heat bath with the Brownian spectral distribution. A reduced hierarchy equations of motion approach is introduced for numerically rigorous simulation of the dynamics of the three-level system with various oscillator configurations, for different nonadiabatic coupling strengths and damping rates, and at different temperatures. The time evolution of the reduced density matrix elements illustrates the interplay of coherences between the electronic and vibrational states. The ET reaction rates, defined as a flux-flux correlation function, are calculated using the linear response of the system to an external perturbation as a function of activation energy. The results exhibit an asymmetric inverted parabolic profile in a small activation regime due to the presence of the intermediate state between the reactant and product states and a slowly decaying profile in a large activation energy regime, which arises from the quantum coherent transitions.
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The hierarchy equation method provides an exact treatment of decoherence and employs neither the rotating-wave, Born, nor Markovian approximations.
“The set of hierarchy equations derived here provides an exact treatment of decoherence, and employs neither the rotating-wave, Born, nor Markovian approximations.”
From Quantum speed limits in open systems: Non-Markovian dynamics without rotating-wave approximation (2015) · cites this paper as [22]
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Mangaud, É., de la Lande, A., Meier, C., et al. (2015) · Physical Chemistry Chemical Physics · doi:10.1039/c5cp01194a · Cited by 23 · see citing works
Simulation of femtosecond “double-slit” experiments for a chromophore in a dissipative environment
Gelin, M. F., Tanimura, Y., & Domcke, W. (2013) · The Journal of Chemical Physics · doi:10.1063/1.4832876 · Cited by 22 · see citing works
Self-citationGuan, W.-Z., Bao, P., Peng, J., Lan, Z., & Shi, Q. (2024) · The Journal of Chemical Physics · doi:10.1063/5.0226214 · Cited by 22 · see citing works
Quantum effects in ultrafast electron transfers within cryptochromes
Firmino, T. D. R., Mangaud, É., Cailliez, F., et al. (2016) · Physical Chemistry Chemical Physics · doi:10.1039/c6cp02809h · Cited by 21 · see citing works
Zhang, P.-P., & Eisfeld, A. (2016) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.6b02111 · Cited by 20 · see citing works
Sun, X. Y., & Geva, E. (2016) · The Journal of Chemical Physics · doi:10.1063/1.4940308 · Cited by 20 · see citing works
Open Quantum Dynamics Theory of Spin Relaxation: Application to μSR and Low-Field NMR Spectroscopies
Takahashi, H., & Tanimura, Y. (2020) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.89.064710 · Cited by 20 · see citing works
Self-citationXiong, S.-J., Zhang, Y., Sun, Z., Yu, L. Z., Su, Q.-P., Xu, X., Jin, J., Xu, Q., Liu, J., Chen, K., et al. (2017) · Optica · doi:10.1364/optica.4.001065 · Cited by 19 · see citing works
Modeling of Transient Absorption Spectra in Exciton–Charge-Transfer Systems
Kramer, T., Rodrı́guez, M., & Zelinskyy, Y. (2016) · The Journal of Physical Chemistry B · doi:10.1021/acs.jpcb.6b09858 · Cited by 19 · see citing works
“ynamics of the system driven by the external laser eld is described by the Liouville-von Neumann equation for the total (system+bath) density matrix ^ˆ tot(t), @ @t ˆ^ tot(t) = i h [H(t);ˆ^ tot(t)]: (10) The physical observables are computed from the reduced density matrix ^ˆ(t) by taking the partial trace of the full…”
MethodExciton-Coupled Electron Transfer Process Controlled by Non-Markovian Environments
Sakamoto, S., & Tanimura, Y. (2017) · The Journal of Physical Chemistry Letters · doi:10.1021/acs.jpclett.7b01535 · Cited by 19 · see citing works
Self-citationHughes, K. H., Cahier, B., Martinazzo, R., Tamura, H., & Burghardt, I. (2014) · Chemical Physics · doi:10.1016/j.chemphys.2014.06.015 · Cited by 18 · see citing works
Ueno, S., & Tanimura, Y. (2021) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.1c00104 · Cited by 17 · see citing works
Self-citationNakamura, K., & Tanimura, Y. (2021) · The Journal of Chemical Physics · doi:10.1063/5.0060208 · Cited by 17 · see citing works
Self-citationSensing of molecules using quantum dynamics
Migliore, A., Naaman, R., & Beratan, D. N. (2015) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1502000112 · Cited by 17 · see citing works
“The quantum master equation (2) or the reduced hierarchy equations of motion approaches (47) could be used to develop a general treatment of the incoherent evolution of the system after analyte binding.”
BackgroundZhang, D., Zuo, L., Ye, L., et al. (2022) · The Journal of Chemical Physics · doi:10.1063/5.0131739 · Cited by 16 · see citing works
Minimum-exponents ansatz for molecular dynamics and quantum dissipation
Ding, J., Zhang, H.-D., Wang, Y., Xu, R., Zheng, X. H., & Yan, Y. (2016) · The Journal of Chemical Physics · doi:10.1063/1.4967964 · Cited by 14 · see citing works
Mangaud, É., Lasorne, B., Atabek, O., & Desouter-Lecomte, M. (2019) · The Journal of Chemical Physics · doi:10.1063/1.5128852 · Cited by 14 · see citing works
On the practical truncation tier of fermionic hierarchical equations of motion
Ding, X., Zhang, D., Ye, L., et al. (2022) · The Journal of Chemical Physics · doi:10.1063/5.0130355 · Cited by 13 · see citing works
Coupled excitation energy and charge transfer dynamics in reaction centre inspired model systems
Richter, M., & Fingerhut, B. P. (2019) · Faraday Discussions · doi:10.1039/c8fd00189h · Cited by 13 · see citing works
Control in a Dissipative Environment: The Example of a Cope Rearrangement
Chenel, A., Dive, G., Meier, C., et al. (2012) · The Journal of Physical Chemistry A · doi:10.1021/jp305274y · Cited by 13 · see citing works
Quantifying non-Markovianity for a chromophore–qubit pair in a super-Ohmic bath
Liu, J., Sun, K., Wang, X., & Zhao, Y. (2015) · Physical Chemistry Chemical Physics · doi:10.1039/c4cp04922e · Cited by 12 · see citing works
Green, D., Humphries, B. S., Dijkstra, A. G., & Jones, G. A. (2019) · The Journal of Chemical Physics · doi:10.1063/1.5119300 · Cited by 12 · see citing works
Adiabatic terminator for fermionic hierarchical equations of motion
Zhang, D., Ding, X., Zhang, H.-D., et al. (2021) · Chinese Journal of Chemical Physics · doi:10.1063/1674-0068/cjcp2110212 · Cited by 12 · see citing works
Hierarchical equations of motion for multiple baths (HEOM-MB) and their application to Carnot cycle
Koyanagi, S., & Tanimura, Y. (2024) · The Journal of Chemical Physics · doi:10.1063/5.0232073 · Cited by 11 · see citing works
Self-citationNon-Markovianity in experimentally simulated quantum channels: Role of counterrotating-wave terms
Xiong, S.-J., Hu, Q., Sun, Z., Yu, L. Z., Su, Q.-P., Liu, J., & Yang, C. (2019) · Physical Review A · doi:10.1103/physreva.100.032101 · Cited by 11 · see citing works
Exciton dissociation in the presence of phonons: A reduced hierarchy equations of motion approach
Yao, Y., Yang, W., & Zhao, Y. (2014) · The Journal of Chemical Physics · doi:10.1063/1.4867418 · Cited by 10 · see citing works
Breuil, G., Mangaud, É., Lasorne, B., Atabek, O., & Desouter-Lecomte, M. (2021) · The Journal of Chemical Physics · doi:10.1063/5.0056351 · Cited by 10 · see citing works
Creation of quantum steering by interaction with a common bath
Sun, Z., Xu, X., & Лю, Б. (2018) · Physical Review A · doi:10.1103/physreva.97.052309 · Cited by 10 · see citing works
Non-Markovian entanglement dynamics of open quantum systems with continuous measurement feedback
Jiang, W., Wu, F.-Z., & Yang, G. (2018) · Physical Review A · doi:10.1103/physreva.98.052134 · Cited by 10 · see citing works
Non-Markovian dynamics of the mixed-state geometric phase of dissipative qubits
Guo, W., Ma, J., Yin, X., Zhong, W., & Wang, X. (2014) · Physical Review A · doi:10.1103/physreva.90.062133 · Cited by 8 · see citing works
Hierarchical equations of motion in the Libra software package
Temen, S., Jain, A., & Akimov, A. (2020) · International Journal of Quantum Chemistry · doi:10.1002/qua.26373 · Cited by 9 · see citing works
Cao, J., Ye, L., Xu, R., Zheng, X. H., & Yan, Y. (2023) · JUSTC · doi:10.52396/justc-2022-0164 · Cited by 9 · see citing works
Wyke, E., Aiyejina, A., & Andrews, R. (2020) · Physical Review A · doi:10.1103/physreva.101.062101 · Cited by 9 · see citing works
Wu, W. (2018) · Physical Review A · doi:10.1103/physreva.98.032116 · Cited by 8 · see citing works
Humphries, B. S., Green, D., & Jones, G. A. (2022) · The Journal of Chemical Physics · doi:10.1063/5.0077404 · Cited by 7 · see citing works
Room temperature quantum coherence vs. electron transfer in a rhodanine derivative chromophore
Madrid-Úsuga, D., Susa, C. E., & Reina, J. H. (2019) · Physical Chemistry Chemical Physics · doi:10.1039/c9cp01398a · Cited by 7 · see citing works
On the role of non-diagonal system–environment interactions in bridge-mediated electron transfer
Acharyya, N., Ovcharenko, R., & Fingerhut, B. P. (2020) · The Journal of Chemical Physics · doi:10.1063/5.0027976 · Cited by 6 · see citing works
Quantum interferometric power and non-Markovianity in the decoherence channels
Xiong, S.-J., Sun, Z., & Wang, X. (2023) · Chinese Physics B · doi:10.1088/1674-1056/accb4e · Cited by 5 · see citing works
Shit, A., Chattopadhyay, S., & Chaudhuri, J. R. (2014) · Chemical Physics · doi:10.1016/j.chemphys.2014.01.008 · Cited by 5 · see citing works
Janković, V. (2025) · The Journal of Chemical Physics · doi:10.1063/5.0299359 · Cited by 4 · see citing works
Non-Markovian and Non-Perturbative Entanglement Dynamics of Biomolecular Excitons
Liu, Z., & Liang, X.-T. (2011) · Chinese Physics Letters · doi:10.1088/0256-307x/28/8/089201 · Cited by 4 · see citing works
Novoderezhkin, V. I. (2025) · The Journal of Chemical Physics · doi:10.1063/5.0273301 · Cited by 3 · see citing works
Gustín, I., Kim, C. W., & Franco, I. (2026) · The Journal of Chemical Physics · doi:10.1063/5.0305296 · Cited by 3 · see citing works
Park, K., Yoon, J.-Y., & Tanimura, Y. (2025) · The Journal of Chemical Physics · doi:10.1063/5.0300203 · Cited by 3 · see citing works
Self-citationDissipative Dynamics of Two-Level Systems in Low Temperature Glasses
Sun, K., & Zhao, Y. (2014) · The Journal of Physical Chemistry A · doi:10.1021/jp5000717 · Cited by 2 · see citing works
Role of Quantum Information in HEOM Trajectories
Humphries, B. S., Kinslow, J. C., Green, D., et al. (2024) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.4c00144 · Cited by 2 · see citing works
Park, K., Ueno, S., & Tanimura, Y. (2026) · The Journal of Chemical Physics · doi:10.1063/5.0311987 · Cited by 1 · see citing works
Self-citationA numerically exact, non-Markovian, non-Gaussian noise model for open quantum system dynamics
Lin, Z., & Cheng, Y. (2026) · The Journal of Chemical Physics · doi:10.1063/5.0331181 · Cited by 1 · see citing works
Autobiography of Yoshitaka Tanimura
Tanimura, Y. (2021) · The Journal of Physical Chemistry B · doi:10.1021/acs.jpcb.1c08552 · Cited by 1 · see citing works
Self-citationSmoluchowski Equation in Cumulant Approximation
Pereverzev, Y. V., Pereverzev, A., & Prezhdo, E. (2013) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.82.024001 · Cited by 1 · see citing works
High-frequency behavior of periodically modulated stochastic quantum system: activated escape
Shit, A. (2016) · Journal of Statistical Mechanics Theory and Experiment · doi:10.1088/1742-5468/2016/05/053204 · Cited by 1 · see citing works
Steering-induced coherence in decoherence channels
Xiong, S.-J., Zhang, L.-H., Liu, J.-M., & Sun, Z. (2021) · Laser Physics Letters · doi:10.1088/1612-202x/abf0ab · Cited by 1 · see citing works
Experimental simulation of quantum temporal steering beyond rotating-wave approximationPreprint
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Betancourt, J. C. R., Anderson, M. C., Niu, L., Chen, X. L., & Franco, I. (2026) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.6c00525
“26 In the case of linear system-bath coupling, the bath correlation function (BCF) contains all information about the bath needed to construct the quantum master equation.”
- Authors
- Tanaka, M., & Tanimura, Y.
- Affiliations
- Kyoto University (JP)
- ORCID
- Tanimura, Yoshitaka
- Venue
- The Journal of Chemical Physics
- Publisher
- AIP Publishing
- Published
- 2010-06-01
- Volume
- 132
- Issue
- 21
- Pages
- 214502-214502
- ISSN
- 0021-9606 (ISSN-L)
- 1089-7690
- OpenAlex
- W2139514694
- Semantic Scholar
- 69903c6707d786ada5a405dfbe12237c0fdfecc9
- Cited by
- 101
- References
- 83
- Citations by year
- 2026: 5 · 2025: 3 · 2024: 3 · 2023: 3 · 2022: 4 · 2021: 9 · 2020: 7 · 2019: 5 · 2018: 4 · 2017: 8 · 2016: 11 · 2015: 10 · 2014: 9 · 2013: 7 · 2012: 8
- FWCI
- 3.43
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- The Journal of Chemical Physics (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Spectroscopy and Quantum Chemical Studies
- Physical Sciences › Chemistry › Physical and Theoretical Chemistry › Photochemistry and Electron Transfer Studies
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Keywords
- electron transfer dynamics, multi-state systems, multilevel systems, nonadiabatic coupling, reduced density matrix, linear response, activation energy, quantum coherence, condensed phase
- MeSH terms
- Electrons, Kinetics, Motion, Quantum Theory, Temperature, Vibration
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