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arxiv: 2603.08677 · v2 · pith:PL47ISWHnew · submitted 2026-03-09 · ⚛️ physics.comp-ph · physics.chem-ph

NATPS: Nonadiabatic Transition Path Sampling Using Time-Reversible MASH Dynamics

Xiran Yang , Madlen Maria Reiner , Brigitta Bachmair , Leticia Gonz\'alez , Johannes C. B. Dietschreit , Christoph Dellago This is my paper

Pith reviewed 2026-05-21 11:33 UTC · model grok-4.3

classification ⚛️ physics.comp-ph physics.chem-ph
keywords nonadiabatic dynamicstransition path samplingMASHexcited statesphotochemistryrare eventsmolecular simulationsurface hopping
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0 comments X

The pith

A deterministic time-reversible version of MASH dynamics allows transition path sampling to generate ensembles of rare nonadiabatic reactive trajectories efficiently.

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

Rare nonadiabatic events drive photochemistry yet are hard to simulate because excited-state calculations are costly and the events themselves occur infrequently. The paper develops a deterministic implementation of the Mapping Approach to Surface Hopping that is time-reversible and obeys detailed balance. This implementation meets the requirements for applying transition path sampling to nonadiabatic processes and produces the new method NATPS. A reader would care because the approach replaces the need to run many long simulations until a rare crossing happens with direct sampling of the reactive paths themselves, cutting computational effort while still yielding mechanistic details on a model system of coupled potential energy surfaces.

Core claim

The authors establish that a deterministic and time-reversible implementation of MASH dynamics satisfies the conditions required for path ensemble sampling, in particular time reversibility and detailed balance. When this dynamics is combined with the transition path sampling framework, the resulting NATPS method efficiently generates ensembles of reactive trajectories on electronically coupled potential energy surfaces. On the tested model system, NATPS supplies mechanistic insight into nonadiabatic pathways and reduces the computational effort needed compared with both brute-force trajectory simulations and forward-flux sampling.

What carries the argument

Deterministic time-reversible MASH dynamics that ensures reversibility and detailed balance so transition path sampling can be applied to nonadiabatic processes.

If this is right

  • Reactive trajectory ensembles are obtained with substantially lower computational cost than brute-force simulation of rare events.
  • Mechanistic details of nonadiabatic pathways become directly accessible through the sampled paths rather than inferred from infrequent occurrences.
  • NATPS outperforms forward-flux sampling in efficiency for the same model system of coupled surfaces.
  • The method produces properly weighted path ensembles suitable for further statistical analysis of nonadiabatic rates and mechanisms.

Where Pith is reading between the lines

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

  • NATPS could be applied to larger molecular systems to study photochemical reactions whose mechanisms are currently out of reach.
  • The reversibility property may allow equilibrium properties of excited electronic states to be extracted from the same sampled paths.
  • Combining NATPS with existing rare-event techniques might further extend the range of accessible nonadiabatic processes.

Load-bearing premise

The deterministic MASH dynamics must actually be time-reversible and preserve detailed balance so that the sampled paths form the correct ensemble.

What would settle it

Generate path ensembles with NATPS and with sufficiently long brute-force runs on the same model system; if the distributions of reactive paths or the fraction of time-reversed paths that remain valid differ systematically, the central claim is falsified.

Figures

Figures reproduced from arXiv: 2603.08677 by Brigitta Bachmair, Christoph Dellago, Johannes C. B. Dietschreit, Leticia Gonz\'alez, Madlen Maria Reiner, Xiran Yang.

Figure 1
Figure 1. Figure 1: a) Illustration of the analytic potential along the nuclear coordinate [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a) Lineage history of trajectories during ten successive Monte Carlo steps in path space. Earlier trajectories [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Distribution of transition times at a) T = 12000 K and b) T = 1000 K. The average transition time ⟨τ ⟩ is indicated by a red line. The golden line in a) indicates the mean transition time obtained from the brute-force MASH simulation, which matches with the TPS-accelerated result. c) Distribution of hopping positions at 12000 K. Hops are mainly concentrated symmetrically near the crossing seam where q = 1.… view at source ↗
Figure 4
Figure 4. Figure 4: Temperature dependence of TPS ensemble statistics: (a) mean transition time [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of hopping positions for (a) varying temperature and (b) varying electronic coupling and state [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Path transition time with MC steps in the path space overlaying with the moving average of transition time of [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
read the original abstract

Rare nonadiabatic events play a central role in photochemistry but remain difficult to simulate because excited-state dynamics is computationally demanding and often stochastic. Here we introduce a deterministic and time-reversible implementation of nonadiabatic dynamics that enables the application of transition path sampling (TPS) to excited-state processes. Our approach builds on the Mapping Approach to Surface Hopping (MASH) and establishes the conditions required for path ensemble sampling, in particular time reversibility and detailed balance. Combining this dynamics with the TPS framework yields a new method, termed nonadiabatic transition path sampling (NATPS). Using a model system of electronically coupled potential energy surfaces, we demonstrate that NATPS efficiently generates ensembles of reactive trajectories and provides mechanistic insight into nonadiabatic pathways. Compared with brute-force trajectory simulations and forward-flux sampling approaches, NATPS substantially reduces the computational effort required to obtain reactive trajectories.

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

1 major / 2 minor

Summary. The manuscript introduces NATPS, a method combining a deterministic, time-reversible implementation of Mapping Approach to Surface Hopping (MASH) dynamics with the transition path sampling (TPS) framework to efficiently generate ensembles of reactive trajectories for rare nonadiabatic events. It claims to establish time-reversibility and detailed balance conditions required for correct path ensemble sampling and demonstrates the approach on a model system of electronically coupled potential energy surfaces, reporting reduced computational effort relative to brute-force trajectory simulations and forward-flux sampling while providing mechanistic insight into nonadiabatic pathways.

Significance. If the central claims hold, NATPS would provide a valuable advance for simulating photochemistry by enabling unbiased sampling of rare excited-state processes with lower effort than existing methods. The work builds directly on established MASH and TPS ingredients and focuses on the reversibility properties needed for TPS correctness; this is a strength when the verification is rigorous. The model-system demonstration, if accompanied by quantitative efficiency metrics and error bars, would support the efficiency claim.

major comments (1)
  1. [Dynamics implementation and TPS combination sections] The load-bearing premise is that the deterministic MASH implementation strictly satisfies time-reversibility and detailed balance so that TPS generates the correct path ensemble. The abstract asserts that the authors 'establish the conditions required for path ensemble sampling, in particular time reversibility and detailed balance,' but the provided text supplies no explicit equations, integrator details, or verification steps (e.g., forward-backward trajectory tests or balance checks on the electronic mapping variables and hopping rule). Without these, the correctness of the sampled reactive trajectories cannot be confirmed and the reported efficiency gains rest on an unverified foundation.
minor comments (2)
  1. [Abstract and results] The abstract and summary statements would benefit from explicit quantitative comparisons (e.g., number of trajectories or CPU hours saved versus brute-force and FFS) with error bars to substantiate the 'substantially reduces' claim.
  2. [Methods] Notation for the electronic mapping variables and the deterministic hopping rule should be introduced with clear definitions before use in the TPS context to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below and have revised the manuscript to strengthen the presentation of the time-reversibility and detailed balance properties.

read point-by-point responses

  1. Referee: [Dynamics implementation and TPS combination sections] The load-bearing premise is that the deterministic MASH implementation strictly satisfies time-reversibility and detailed balance so that TPS generates the correct path ensemble. The abstract asserts that the authors 'establish the conditions required for path ensemble sampling, in particular time reversibility and detailed balance,' but the provided text supplies no explicit equations, integrator details, or verification steps (e.g., forward-backward trajectory tests or balance checks on the electronic mapping variables and hopping rule). Without these, the correctness of the sampled reactive trajectories cannot be confirmed and the reported efficiency gains rest on an unverified foundation.

    Authors: We agree that explicit verification is necessary to confirm the correctness of the sampled path ensemble. In the revised manuscript we have added a dedicated subsection detailing the deterministic MASH integrator, including the explicit equations for the time evolution of the mapping variables and the time-reversible hopping rule. We also report results of forward-backward trajectory tests that demonstrate exact retracing of paths under time reversal, together with numerical checks confirming that the electronic mapping variables and hopping probabilities satisfy detailed balance to within statistical error. These additions directly address the verification steps requested and place the efficiency claims on a firmer foundation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; NATPS combines established MASH and TPS without self-referential reduction

full rationale

The paper presents NATPS as a combination of deterministic time-reversible MASH dynamics with the TPS framework. It states that the approach establishes time reversibility and detailed balance for path ensemble sampling, then demonstrates efficiency gains on a model system of coupled potential energy surfaces. No equations or parameters are shown to be fitted in a way that makes claimed predictions or efficiencies reduce to inputs by construction. The central claims rest on the dynamics satisfying TPS requirements, which is presented as derived rather than assumed tautologically. Self-citations to MASH are not load-bearing in a circular chain, as the method is externally verifiable through trajectory sampling properties. This yields a minor score consistent with normal use of prior techniques without forcing the result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Crespo-Hernández, Boiko Cohen, Patrick M

    Carlos E. Crespo-Hernández, Boiko Cohen, Patrick M. Hare, and Bern Kohler. Ultrafast excited-state dynamics in nucleic acids.Chemical Reviews, 104(4):1977–2020, April 2004. ISSN 1520-6890. doi:10.1021/cr0206770. URLhttp://dx.doi.org/10.1021/cr0206770

    work page doi:10.1021/cr0206770 1977

  2. [2]

    Introduction: Photochemical catalytic processes.Chemical Reviews, 122(2):1483–1484, January 2022

    Paolo Melchiorre. Introduction: Photochemical catalytic processes.Chemical Reviews, 122(2):1483–1484, January 2022. ISSN 1520-6890. doi:10.1021/acs.chemrev.1c00993. URL http://dx.doi.org/10.1021/acs. chemrev.1c00993

    work page doi:10.1021/acs.chemrev.1c00993 2022

  3. [3]

    Springer International Publishing, Cham, Switzerland, 2014

    Rebeca de Nalda and Luis Bañares, editors.Ultrafast Phenomena in Molecular Sciences: Femtosecond Physics and Chemistry, volume 107 ofSpringer Series in Chemical Physics. Springer International Publishing, Cham, Switzerland, 2014. ISBN 978-3-319-02051-8. Proceedings or edited volume

    work page 2014

  4. [4]

    Yarkony, and Horst Köppel, editors.Conical Intersections: Theory, Computation and Experiment, volume 17 ofAdvanced Series in Physical Chemistry

    Wolfgang Domcke, David R. Yarkony, and Horst Köppel, editors.Conical Intersections: Theory, Computation and Experiment, volume 17 ofAdvanced Series in Physical Chemistry. World Scientific Publishing Co., Singapore,

    work page

  5. [5]

    doi:10.1142/7803

    ISBN 978-981-431345 2. doi:10.1142/7803

    work page doi:10.1142/7803

  6. [6]

    Rudolph A. Marcus. Chemical and electrochemical electron-transfer theory.Annual Review of Physical Chemistry, 15(1):155–196, 1964. doi:10.1146/annurev.pc.15.100164.001103

    work page doi:10.1146/annurev.pc.15.100164.001103 1964

  7. [7]

    Kato and Masaaki

    Hajime. Kato and Masaaki. Baba. Dynamics of excited molecules: Predissociation.Chemical Reviews, 95(7): 2311–2349, November 1995. ISSN 1520-6890. doi:10.1021/cr00039a003. URL http://dx.doi.org/10. 1021/cr00039a003

    work page doi:10.1021/cr00039a003 1995

  8. [8]

    Photodissociation of simple molecules in the gas phase.Chemical Reviews, 101(9):2687–2726, September 2001

    Hiroyasu Sato. Photodissociation of simple molecules in the gas phase.Chemical Reviews, 101(9):2687–2726, September 2001. ISSN 1520-6890. doi:10.1021/cr990403l. URL http://dx.doi.org/10.1021/cr990403l

    work page doi:10.1021/cr990403l 2001

  9. [9]

    Güida, M.A

    J.A. Güida, M.A. Ramos, O.E. Piro, and P.J. Aymonino. Infrared spectra of k2[rucl5no] in two excited metastable states and the evidence for the no linkage photoisomerization of metastable state i (msi) in [rux5 no]2− (x=cl, cn). Journal of Molecular Structure, 609(1–3):39–46, May 2002. ISSN 0022-2860. doi:10.1016/s0022-2860(01)00897-

    work page doi:10.1016/s0022-2860(01)00897- 2002

  10. [10]

    URLhttp://dx.doi.org/10.1016/S0022-2860(01)00897-3

    work page doi:10.1016/s0022-2860(01)00897-3

  11. [11]

    Premadasa, Vyacheslav S

    Ying-Zhong Ma, Uvinduni I. Premadasa, Vyacheslav S. Bryantsev, Audrey R. Miles, Ilia N. Ivanov, Adnan Elgattar, Yi Liao, and Benjamin Doughty. Unravelling photoisomerization dynamics in a metastable-state photoacid. Physical Chemistry Chemical Physics, 26(5):4062–4070, 2024. ISSN 1463-9084. doi:10.1039/d3cp04454h. URL http://dx.doi.org/10.1039/D3CP04454H

    work page doi:10.1039/d3cp04454h 2024

  12. [12]

    Premadasa, May Waters, Nitesh Kumar, Benjamin Doughty, Melyse Laud, Yi Liao, and Vyacheslav S

    Ying-Zhong Ma, Uvinduni I. Premadasa, May Waters, Nitesh Kumar, Benjamin Doughty, Melyse Laud, Yi Liao, and Vyacheslav S. Bryantsev. Accessing transient isomers in the photoreaction of metastable-state photoacid. ChemPhysChem, 26(13), June 2025. ISSN 1439-7641. doi:10.1002/cphc.202500184. URL http://dx.doi. org/10.1002/cphc.202500184

    work page doi:10.1002/cphc.202500184 2025

  13. [13]

    Penfold, Etienne Gindensperger, Chantal Daniel, and Christel M

    Thomas J. Penfold, Etienne Gindensperger, Chantal Daniel, and Christel M. Marian. Spin-vibronic mech- anism for intersystem crossing.Chemical Reviews, 118(15):6975–7025, March 2018. ISSN 1520-6890. doi:10.1021/acs.chemrev.7b00617. URLhttp://dx.doi.org/10.1021/acs.chemrev.7b00617

    work page doi:10.1021/acs.chemrev.7b00617 2018

  14. [14]

    Bolhuis, Félix S

    Christoph Dellago, Peter G. Bolhuis, Félix S. Csajka, and David Chandler. Transition path sampling and the calculation of rate constants.The Journal of Chemical Physics, 108(5):1964–1977, February 1998. ISSN 1089-7690. doi:10.1063/1.475562. URLhttp://dx.doi.org/10.1063/1.475562

    work page doi:10.1063/1.475562 1964

  15. [15]

    Transition path sampling: throwing ropes over mountains in the dark.Journal of Physics: Condensed Matter, 12(8A):A147–A152, February 2000

    Peter G Bolhuis, Christoph Dellago, Phillip L Geissler, and David Chandler. Transition path sampling: throwing ropes over mountains in the dark.Journal of Physics: Condensed Matter, 12(8A):A147–A152, February 2000. ISSN 1361-648X. doi:10.1088/0953-8984/12/8a/316. URL http://dx.doi.org/10.1088/0953-8984/12/ 8A/316

    work page doi:10.1088/0953-8984/12/8a/316 2000

  16. [16]

    Bolhuis, and Phillip L

    Christoph Dellago, Peter G. Bolhuis, and Phillip L. Geissler. Transition path sampling, July 2002. ISSN 1934-4791. URLhttp://dx.doi.org/10.1002/0471231509.ch1. 10

    work page doi:10.1002/0471231509.ch1 2002

  17. [17]

    Bolhuis.Transition Path Sampling and other Advanced Simulation Techniques for Rare Events, pages 167–233

    Christoph Dellago and Peter G. Bolhuis.Transition Path Sampling and other Advanced Simulation Techniques for Rare Events, pages 167–233. Springer Berlin Heidelberg, 2009. ISBN 1424406285

    work page 2009

  18. [18]

    van Erp, Daniele Moroni, and Peter G

    Titus S. van Erp, Daniele Moroni, and Peter G. Bolhuis. A novel path sampling method for the calcula- tion of rate constants.The Journal of Chemical Physics, 118(17):7762–7774, May 2003. ISSN 1089-7690. doi:10.1063/1.1562614. URLhttp://dx.doi.org/10.1063/1.1562614

    work page doi:10.1063/1.1562614 2003

  19. [19]

    van Erp, and Peter G

    Daniele Moroni, Titus S. van Erp, and Peter G. Bolhuis. Investigating rare events by transition interface sampling. Physica A: Statistical Mechanics and its Applications, 340(1–3):395–401, September 2004. ISSN 0378-4371. doi:10.1016/j.physa.2004.04.033. URLhttp://dx.doi.org/10.1016/j.physa.2004.04.033

    work page doi:10.1016/j.physa.2004.04.033 2004

  20. [20]

    van Erp and Peter G

    Titus S. van Erp and Peter G. Bolhuis. Elaborating transition interface sampling methods.Journal of Computational Physics, 205(1):157–181, May 2005. ISSN 0021-9991. doi:10.1016/j.jcp.2004.11.003. URL http://dx.doi. org/10.1016/j.jcp.2004.11.003

    work page doi:10.1016/j.jcp.2004.11.003 2005

  21. [21]

    Allen, Patrick B

    Rosalind J. Allen, Patrick B. Warren, and Pieter Rein Ten Wolde. Sampling rare switching events in biochemical networks.Physical Review Letters, 94(1):018104, 2005. doi:10.1103/PhysRevLett.94.018104

    work page doi:10.1103/physrevlett.94.018104 2005

  22. [22]

    Allen, Daan Frenkel, and Pieter Rein Ten Wolde

    Rosalind J. Allen, Daan Frenkel, and Pieter Rein Ten Wolde. Simulating rare events in equilibrium or nonequilib- rium stochastic systems.The Journal of Chemical Physics, 124(2):024102, 2006. doi:10.1063/1.2140273

    work page doi:10.1063/1.2140273 2006

  23. [23]

    Allen, Daan Frenkel, and Pieter Rein Ten Wolde

    Rosalind J. Allen, Daan Frenkel, and Pieter Rein Ten Wolde. Forward flux sampling–type schemes for simulating rare events: Efficiency analysis.The Journal of Chemical Physics, 124(19):194111, 2006. doi:10.1063/1.2198827

    work page doi:10.1063/1.2198827 2006

  24. [24]

    Allen, Chantal Valeriani, and Pieter Rein Ten Wolde

    Rosalind J. Allen, Chantal Valeriani, and Pieter Rein Ten Wolde. Forward flux sampling for rare event simulations. Journal of Physics Condensed Matter, 21:1–40, 2009. ISSN 09538984. doi:10.1088/0953-8984/21/46/463102

    work page doi:10.1088/0953-8984/21/46/463102 2009

  25. [25]

    Subotnik and Young Min Rhee

    Joseph E. Subotnik and Young Min Rhee. On surface hopping and time-reversal.The Journal of Physical Chemistry A, 119(6):990–995, January 2015. ISSN 1520-5215. doi:10.1021/jp512024w. URL http://dx.doi. org/10.1021/jp512024w

    work page doi:10.1021/jp512024w 2015

  26. [26]

    John C. Tully. Molecular dynamics with electronic transitions.The Journal of Chemical Physics, 93(2):1061–1071, July 1990. ISSN 1089-7690. doi:10.1063/1.459170. URLhttp://dx.doi.org/10.1063/1.459170

    work page doi:10.1063/1.459170 1990

  27. [27]

    Sherman and Corcelli S.A

    M.C. Sherman and Corcelli S.A. Nonadiabatic transition path sampling.The Journal of Chemical Physics, 2016

    work page 2016

  28. [28]

    Nonadiabatic forward flux sampling for excited-state rare events.Journal of Chemical Theory and Computation, 19(6):1657–1671, March 2023

    Madlen Maria Reiner, Brigitta Bachmair, Maximilian Xaver Tiefenbacher, Sebastian Mai, Leticia González, Philipp Marquetand, and Christoph Dellago. Nonadiabatic forward flux sampling for excited-state rare events.Journal of Chemical Theory and Computation, 19(6):1657–1671, March 2023. ISSN 1549-9626. doi:10.1021/acs.jctc.2c01088. URLhttp://dx.doi.org/10.10...

    work page doi:10.1021/acs.jctc.2c01088 2023

  29. [29]

    Mannouch and Jeremy O

    Jonathan R. Mannouch and Jeremy O. Richardson. A mapping approach to surface hopping.The Journal of Chemical Physics, 158(10), March 2023. ISSN 1089-7690. doi:10.1063/5.0139734. URL http://dx.doi.org/ 10.1063/5.0139734

    work page doi:10.1063/5.0139734 2023

  30. [30]

    Mannouch, and Jeremy O

    Graziano Amati, Jonathan R. Mannouch, and Jeremy O. Richardson. Detailed balance in mixed quan- tum–classical mapping approaches.The Journal of Chemical Physics, 159(21), December 2023. ISSN 1089-7690. doi:10.1063/5.0176291. URLhttp://dx.doi.org/10.1063/5.0176291

    work page doi:10.1063/5.0176291 2023

  31. [31]

    Amira Geuther, Kasra Asnaashari, and Jeremy O

    J. Amira Geuther, Kasra Asnaashari, and Jeremy O. Richardson. Time-reversible implementation of mash for efficient nonadiabatic molecular dynamics.Journal of Chemical Theory and Computation, 21(5):2179–2188, February 2025. ISSN 1549-9626. doi:10.1021/acs.jctc.4c01684. URL http://dx.doi.org/10.1021/acs. jctc.4c01684

    work page doi:10.1021/acs.jctc.4c01684 2025

  32. [32]

    Richardson, Joseph E

    Jeremy O. Richardson, Joseph E. Lawrence, and Jonathan R. Mannouch. Nonadiabatic dynamics with the mapping approach to surface hopping (mash).Annual Review of Physical Chemistry, 76(1):663–687, April

    work page

  33. [33]

    doi:10.1146/annurev-physchem-082423-120631

    ISSN 1545-1593. doi:10.1146/annurev-physchem-082423-120631. URL http://dx.doi.org/10. 1146/annurev-physchem-082423-120631

    work page doi:10.1146/annurev-physchem-082423-120631

  34. [34]

    Johannes C. B. Dietschreit, Dennis J. Diestler, and Christian Ochsenfeld. How to obtain reaction free energies from free-energy profiles.The Journal of Chemical Physics, 156(11):114105, March

    work page

  35. [35]

    URL https://pubs.aip.org/jcp/article/156/11/114105/2840874/ How-to-obtain-reaction-free-energies-from-free

    doi:10.1063/5.0083423. URL https://pubs.aip.org/jcp/article/156/11/114105/2840874/ How-to-obtain-reaction-free-energies-from-free

    work page doi:10.1063/5.0083423

  36. [36]

    Bolhuis, David Chandler, Christoph Dellago, and Phillip L

    Peter G. Bolhuis, David Chandler, Christoph Dellago, and Phillip L. Geissler. Transition path sampling: Throwing ropes over rough mountain passes, in the dark.Annual Review of Physical Chemistry, 53:291–318, 2002. ISSN 0066426X. doi:10.1146/annurev.physchem.53.082301.113146

    work page doi:10.1146/annurev.physchem.53.082301.113146 2002

  37. [37]

    Lawrence, Jonathan R

    Joseph E. Lawrence, Jonathan R. Mannouch, and Jeremy O. Richardson. A size-consistent multi-state map- ping approach to surface hopping.The Journal of Chemical Physics, 160(24), June 2024. ISSN 1089-7690. doi:10.1063/5.0208575. URLhttp://dx.doi.org/10.1063/5.0208575. 11

    work page doi:10.1063/5.0208575 2024

  38. [38]

    Escobedo, Ernesto E

    Fernando A. Escobedo, Ernesto E. Borrero, and Juan C. Araque. Transition path sampling and forward flux sampling. applications to biological systems.Journal of Physics Condensed Matter, 21:333101, 2009. ISSN 09538984. doi:10.1088/0953-8984/21/33/333101

    work page doi:10.1088/0953-8984/21/33/333101 2009

  39. [39]

    Bolhuis, Christoph Dellago, and Alessandro Coretti

    Magdalena Häupl, Sebastian Falkner, Peter G. Bolhuis, Christoph Dellago, and Alessandro Coretti. An always- accepting algorithm for transition path sampling, 2026. URLhttps://arxiv.org/abs/2602.13130

    work page arXiv 2026

  40. [40]

    Buijsman and P

    P. Buijsman and P. G. Bolhuis. Transition path sampling for non-equilibrium dynamics without predefined reaction coordinates.Journal of Chemical Physics, 152, 2020. ISSN 00219606. doi:10.1063/1.5130760. URL https://doi.org/10.1063/1.5130760

    work page doi:10.1063/1.5130760 2020

  41. [41]

    G. E. Uhlenbeck and L. S. Ornstein. On the theory of the brownian motion.Phys. Rev., 36:823–841, Sep 1930. doi:10.1103/PhysRev.36.823. URLhttps://link.aps.org/doi/10.1103/PhysRev.36.823

    work page doi:10.1103/physrev.36.823 1930

  42. [42]

    Felix Plasser, Giovanni Granucci, Jiri Pittner, Mario Barbatti, Persico Maurizio, and Lischka Hans. Surface hopping dynamics using a locally diabatic formalism: Charge transfer in the ethylene dimer cation and excited state dynamics in the 2-pyridone dimer.The Journal of Chemical Physics, 2012

    work page 2012

  43. [43]

    Rudolph A. Marcus. Electron transfer reactions in chemistry. theory and experiment.Reviews of Modern Physics, 65(3):599–610, July 1993. ISSN 1539-0756. doi:10.1103/revmodphys.65.599. URL http://dx.doi.org/10. 1103/RevModPhys.65.599

    work page doi:10.1103/revmodphys.65.599 1993

  44. [44]

    Hennefarth, Nicolai Machholdt Høyer, Feven A

    Sebastian Mai, Brigitta Bachmair, Laura Gagliardi, Hans Georg Gallmetzer, Lorenz Grünewald, Matthew R. Hennefarth, Nicolai Machholdt Høyer, Feven A. Korsaye, Sascha Mausenberger, Markus Oppel, Tomislav Piteša, Severin Polonius, Eduarda Sangiogo Gil, Yinan Shu, Nadja K. Singer, Maximilian X. Tiefenbacher, Donald G. Truhlar, Dóra Vörös, Linyao Zhang, and Le...

    work page 2025

  45. [45]

    Nonadiabatic dynamics: The SHARC approach.Wiley Interdisciplinary Reviews: Computational Molecular Science, 8(6):1–23, 2018

    Sebastian Mai, Philipp Marquetand, and Leticia González. Nonadiabatic dynamics: The SHARC approach.Wiley Interdisciplinary Reviews: Computational Molecular Science, 8(6):1–23, 2018. doi:10.1002/wcms.1370

    work page doi:10.1002/wcms.1370 2018

  46. [46]

    away from the equator

    Crispin W. Gardiner. Handbook of stochastic methods for physics, chemistry and the natural sciences.Springer Series in Synergetics, 1985. ISSN 0172-7389. doi:10.1007/978-3-662-02452-2. URL https://cir.nii.ac. jp/crid/1360855570504742656. 12 Appendix A Invariant Distribution of MASH To find the stationary distribution generated by the Mapping Approach to S...

    work page doi:10.1007/978-3-662-02452-2 1985